Bayesian Approach to Inverse Problems
2008-01-01
Many scientific, medical or engineering problems raise the issue of recovering some physical quantities from indirect measurements; for instance, detecting or quantifying flaws or cracks within a material from acoustic or electromagnetic measurements at its surface is an essential problem of non-destructive evaluation. The concept of inverse problems precisely originates from the idea of inverting the laws of physics to recover a quantity of interest from measurable data.Unfortunately, most inverse problems are ill-posed, which means that precise and stable solutions are not easy to devise. Regularization is the key concept to solve inverse problems.The goal of this book is to deal with inverse problems and regularized solutions using the Bayesian statistical tools, with a particular view to signal and image estimation
Minimax approach to inverse problems of geophysics
Balk, P. I.; Dolgal, A. S.; Balk, T. V.; Khristenko, L. A.
2016-03-01
A new approach is suggested for solving the inverse problems that arise in the different fields of applied geophysics (gravity, magnetic, and electrical prospecting, geothermy) and require assessing the spatial region occupied by the anomaly-generating masses in the presence of different types of a priori information. The interpretation which provides the maximum guaranteed proximity of the model field sources to the real perturbing object is treated as the best interpretation. In some fields of science (game theory, economics, operations research), the decision-making principle that lies in minimizing the probable losses which cannot be prevented if the situation develops by the worst-case scenario is referred to as minimax. The minimax criterion of choice is interesting as, instead of being confined to the indirect (and sometimes doubtful) signs of the "optimal" solution, it relies on the actual properties of the information in the results of a particular interpretation. In the hierarchy of the approaches to the solution of the inverse problems of geophysics ordered by the volume and quality of the retrieved information about the sources of the field, the minimax approach should take special place.
Inverse problem approaches for digital hologram reconstruction
Fournier, Corinne; Denis, Loic; Thiebaut, Eric; Fournel, Thierry; Seifi, Mozhdeh
2011-06-01
Digital holography (DH) is being increasingly used for its time-resolved three-dimensional (3-D) imaging capabilities. A 3-D volume can be numerically reconstructed from a single 2-D hologram. Applications of DH range from experimental mechanics, biology, and fluid dynamics. Improvement and characterization of the 3-D reconstruction algorithms is a current issue. Over the past decade, numerous algorithms for the analysis of holograms have been proposed. They are mostly based on a common approach to hologram processing: digital reconstruction based on the simulation of hologram diffraction. They suffer from artifacts intrinsic to holography: twin-image contamination of the reconstructed images, image distortions for objects located close to the hologram borders. The analysis of the reconstructed planes is therefore limited by these defects. In contrast to this approach, the inverse problems perspective does not transform the hologram but performs object detection and location by matching a model of the hologram. Information is thus extracted from the hologram in an optimal way, leading to two essential results: an improvement of the axial accuracy and the capability to extend the reconstructed field beyond the physical limit of the sensor size (out-of-field reconstruction). These improvements come at the cost of an increase of the computational load compared to (typically non iterative) classical approaches.
Structured Sparsity Regularization Approach to the EEG Inverse Problem
DEFF Research Database (Denmark)
Montoya-Martinez, Jair; Artes-Rodriguez, Antonio; Hansen, Lars Kai
2012-01-01
Localization of brain activity involves solving the EEG inverse problem, which is an undetermined ill-posed problem. We propose a novel approach consisting in estimating, using structured sparsity regularization techniques, the Brain Electrical Sources (BES) matrix directly in the spatio-temporal......Localization of brain activity involves solving the EEG inverse problem, which is an undetermined ill-posed problem. We propose a novel approach consisting in estimating, using structured sparsity regularization techniques, the Brain Electrical Sources (BES) matrix directly in the spatio...
Combined approach to the inverse protein folding problem. Final report
Energy Technology Data Exchange (ETDEWEB)
Ruben A. Abagyan
2000-06-01
The main scientific contribution of the project ''Combined approach to the inverse protein folding problem'' submitted in 1996 and funded by the Department of Energy in 1997 is the formulation and development of the idea of the multilink recognition method for identification of functional and structural homologues of newly discovered genes. This idea became very popular after they first announced it and used it in prediction of the threading targets for the CASP2 competition (Critical Assessment of Structure Prediction).
An inverse problem approach to modelling coastal effluent plumes
Lam, D. C. L.; Murthy, C. R.; Miners, K. C.
Formulated as an inverse problem, the diffusion parameters associated with length-scale dependent eddy diffusivities can be viewed as the unknowns in the mass conservation equation for coastal zone transport problems. The values of the diffusion parameters can be optimized according to an error function incorporated with observed concentration data. Examples are given for the Fickian, shear diffusion and inertial subrange diffusion models. Based on a new set of dyeplume data collected in the coastal zone off Bronte, Lake Ontario, it is shown that the predictions of turbulence closure models can be evaluated for different flow conditions. The choice of computational schemes for this diagnostic approach is based on tests with analytic solutions and observed data. It is found that the optimized shear diffusion model produced a better agreement with observations for both high and low advective flows than, e.g., the unoptimized semi-empirical model, Ky=0.075 σy1.2, described by Murthy and Kenney.
Riemann–Hilbert problem approach for two-dimensional flow inverse scattering
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Agaltsov, A. D., E-mail: agalets@gmail.com [Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 119991 Moscow (Russian Federation); Novikov, R. G., E-mail: novikov@cmap.polytechnique.fr [CNRS (UMR 7641), Centre de Mathématiques Appliquées, Ecole Polytechnique, 91128 Palaiseau (France); IEPT RAS, 117997 Moscow (Russian Federation); Moscow Institute of Physics and Technology, Dolgoprudny (Russian Federation)
2014-10-15
We consider inverse scattering for the time-harmonic wave equation with first-order perturbation in two dimensions. This problem arises in particular in the acoustic tomography of moving fluid. We consider linearized and nonlinearized reconstruction algorithms for this problem of inverse scattering. Our nonlinearized reconstruction algorithm is based on the non-local Riemann–Hilbert problem approach. Comparisons with preceding results are given.
A unified approach to the helioseismic forward and inverse problems of differential rotation
Energy Technology Data Exchange (ETDEWEB)
Ritzwoller, M.H.; Lavely, E.M. (Colorado Univ., Boulder (USA) MIT, Cambridge, MA (USA))
1991-03-01
A general, degenerate perturbation theoretic treatment of the helioseismic forward and inverse problem for solar differential rotation is presented. For the forward problem, differential rotation is represented as the axisymmetric component of a general toroidal flow field using velocity spherical harmonics. This approach allows each degree of differential rotation to be estimated independently from all other degrees. In the inverse problem, the splitting caused by differential rotation is expressed as an expansion in a set of orthonormal polynomials that are intimately related to the solution of the forward problem. The combined use of vector spherical harmonics as basis functions for differential ratio and the Clebsch-Gordon coefficients to represent splitting provides a unified approach to the forward and inverse problems of differential rotation which greatly simplify inversion. 43 refs.
Cheng, Jin; Hon, Yiu-Chung; Seo, Jin Keun; Yamamoto, Masahiro
2005-01-01
The Second International Conference on Inverse Problems: Recent Theoretical Developments and Numerical Approaches was held at Fudan University, Shanghai from 16-21 June 2004. The first conference in this series was held at the City University of Hong Kong in January 2002 and it was agreed to hold the conference once every two years in a Pan-Pacific Asian country. The next conference is scheduled to be held at Hokkaido University, Sapporo, Japan in July 2006. The purpose of this series of biennial conferences is to establish and develop constant international collaboration, especially among the Pan-Pacific Asian countries. In recent decades, interest in inverse problems has been flourishing all over the globe because of both the theoretical interest and practical requirements. In particular, in Asian countries, one is witnessing remarkable new trends of research in inverse problems as well as the participation of many young talents. Considering these trends, the second conference was organized with the chairperson Professor Li Tat-tsien (Fudan University), in order to provide forums for developing research cooperation and to promote activities in the field of inverse problems. Because solutions to inverse problems are needed in various applied fields, we entertained a total of 92 participants at the second conference and arranged various talks which ranged from mathematical analyses to solutions of concrete inverse problems in the real world. This volume contains 18 selected papers, all of which have undergone peer review. The 18 papers are classified as follows: Surveys: four papers give reviews of specific inverse problems. Theoretical aspects: six papers investigate the uniqueness, stability, and reconstruction schemes. Numerical methods: four papers devise new numerical methods and their applications to inverse problems. Solutions to applied inverse problems: four papers discuss concrete inverse problems such as scattering problems and inverse problems in
An Inverse Robust Optimisation Approach for a Class of Vehicle Routing Problems under Uncertainty
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Liang Sun
2016-01-01
Full Text Available There is a trade-off between the total penalty paid to customers (TPC and the total transportation cost (TTC in depot for vehicle routing problems under uncertainty (VRPU. The trade-off refers to the fact that the TTC in depot inevitably increases when the TPC decreases and vice versa. With respect to this issue, the vehicle routing problem (VRP with uncertain customer demand and travel time was studied to optimise the TPC and the TTC in depot. In addition, an inverse robust optimisation approach was proposed to solve this kind of VRPU by combining the ideas of inverse optimisation and robust optimisation so as to improve both the TPC and the TTC in depot. The method aimed to improve the corresponding TTC of the robust optimisation solution under the minimum TPC through minimising the adjustment of benchmark road transportation cost. According to the characteristics of the inverse robust optimisation model, a genetic algorithm (GA and column generation algorithm are combined to solve the problem. Moreover, 39 test problems are solved by using an inverse robust optimisation approach: the results show that both the TPC and TTC obtained by using the inverse robust optimisation approach are less than those calculated using a robust optimisation approach.
A Riemann-Hilbert approach to the inverse problem for the Stark operator on the line
Its, A.; Sukhanov, V.
2016-05-01
The paper is concerned with the inverse scattering problem for the Stark operator on the line with a potential from the Schwartz class. In our study of the inverse problem, we use the Riemann-Hilbert formalism. This allows us to overcome the principal technical difficulties which arise in the more traditional approaches based on the Gel’fand-Levitan-Marchenko equations, and indeed solve the problem. We also produce a complete description of the relevant scattering data (which have not been obtained in the previous works on the Stark operator) and establish the bijection between the Schwartz class potentials and the scattering data.
Martinez-Camara, Marta; Dokmanic, Ivan; Ranieri, Juri; Scheibler, Robin; Vetterli, Martin; STOHL Andreas
2013-01-01
Knowing what amount of radioactive material was released from Fukushima in March 2011 and at what time instants is crucial to assess the risk, the pollution, and to understand the scope of the consequences. Moreover, it could be used in forward simulations to obtain accurate maps of deposition. But these data are often not publicly available. We propose to estimate the emission waveforms by solving an inverse problem. Previous approaches have relied on a detailed expert guess of how the relea...
On the new approach to solving the inverse problem of gravimetry
Arsanukaev, Z. Z.
2017-01-01
The results of the studies within the new approach to solving the inverse problem of gravimetry are considered. This approach consists in direct (analytical) continuation of the anomalous gravitational field specified on the Earth's surface into the lower half-space with the use of the method of discrete approximations. The solution of the problem of analytical continuation is demonstrated by the model example. In the solution of the problem of analytical continuation, the developed algorithms and computer programs were implemented in two program packages which are used both in the model computations and in practice.
Variational approach to direct and inverse problems of atmospheric pollution studies
Penenko, Vladimir; Tsvetova, Elena; Penenko, Alexey
2016-04-01
We present the development of a variational approach for solving interrelated problems of atmospheric hydrodynamics and chemistry concerning air pollution transport and transformations. The proposed approach allows us to carry out complex studies of different-scale physical and chemical processes using the methods of direct and inverse modeling [1-3]. We formulate the problems of risk/vulnerability and uncertainty assessment, sensitivity studies, variational data assimilation procedures [4], etc. A computational technology of constructing consistent mathematical models and methods of their numerical implementation is based on the variational principle in the weak constraint formulation specifically designed to account for uncertainties in models and observations. Algorithms for direct and inverse modeling are designed with the use of global and local adjoint problems. Implementing the idea of adjoint integrating factors provides unconditionally monotone and stable discrete-analytic approximations for convection-diffusion-reaction problems [5,6]. The general framework is applied to the direct and inverse problems for the models of transport and transformation of pollutants in Siberian and Arctic regions. The work has been partially supported by the RFBR grant 14-01-00125 and RAS Presidium Program I.33P. References: 1. V. Penenko, A.Baklanov, E. Tsvetova and A. Mahura . Direct and inverse problems in a variational concept of environmental modeling //Pure and Applied Geoph.(2012) v.169: 447-465. 2. V. V. Penenko, E. A. Tsvetova, and A. V. Penenko Development of variational approach for direct and inverse problems of atmospheric hydrodynamics and chemistry, Izvestiya, Atmospheric and Oceanic Physics, 2015, Vol. 51, No. 3, p. 311-319, DOI: 10.1134/S0001433815030093. 3. V.V. Penenko, E.A. Tsvetova, A.V. Penenko. Methods based on the joint use of models and observational data in the framework of variational approach to forecasting weather and atmospheric composition
A hierarchical Bayesian-MAP approach to inverse problems in imaging
Raj, Raghu G.
2016-07-01
We present a novel approach to inverse problems in imaging based on a hierarchical Bayesian-MAP (HB-MAP) formulation. In this paper we specifically focus on the difficult and basic inverse problem of multi-sensor (tomographic) imaging wherein the source object of interest is viewed from multiple directions by independent sensors. Given the measurements recorded by these sensors, the problem is to reconstruct the image (of the object) with a high degree of fidelity. We employ a probabilistic graphical modeling extension of the compound Gaussian distribution as a global image prior into a hierarchical Bayesian inference procedure. Since the prior employed by our HB-MAP algorithm is general enough to subsume a wide class of priors including those typically employed in compressive sensing (CS) algorithms, HB-MAP algorithm offers a vehicle to extend the capabilities of current CS algorithms to include truly global priors. After rigorously deriving the regression algorithm for solving our inverse problem from first principles, we demonstrate the performance of the HB-MAP algorithm on Monte Carlo trials and on real empirical data (natural scenes). In all cases we find that our algorithm outperforms previous approaches in the literature including filtered back-projection and a variety of state-of-the-art CS algorithms. We conclude with directions of future research emanating from this work.
Model-based elastography: a survey of approaches to the inverse elasticity problem
Doyley, M M
2012-01-01
Elastography is emerging as an imaging modality that can distinguish normal versus diseased tissues via their biomechanical properties. This article reviews current approaches to elastography in three areas — quasi-static, harmonic, and transient — and describes inversion schemes for each elastographic imaging approach. Approaches include: first-order approximation methods; direct and iterative inversion schemes for linear elastic; isotropic materials; and advanced reconstruction methods for recovering parameters that characterize complex mechanical behavior. The paper’s objective is to document efforts to develop elastography within the framework of solving an inverse problem, so that elastography may provide reliable estimates of shear modulus and other mechanical parameters. We discuss issues that must be addressed if model-based elastography is to become the prevailing approach to quasi-static, harmonic, and transient elastography: (1) developing practical techniques to transform the ill-posed problem with a well-posed one; (2) devising better forward models to capture the transient behavior of soft tissue; and (3) developing better test procedures to evaluate the performance of modulus elastograms. PMID:22222839
Gladwell, Graham ML
2011-01-01
The papers in this volume present an overview of the general aspects and practical applications of dynamic inverse methods, through the interaction of several topics, ranging from classical and advanced inverse problems in vibration, isospectral systems, dynamic methods for structural identification, active vibration control and damage detection, imaging shear stiffness in biological tissues, wave propagation, to computational and experimental aspects relevant for engineering problems.
Ferrari, André; Ferrari, Chiara; Mary, David; Schutz, Antony; Smirnov, Oleg
2015-01-01
We describe a "spatio-spectral" deconvolution algorithm for wide-band imaging in radio interferometry. In contrast with the existing multi-frequency reconstruction algorithms, the proposed method does not rely on a model of the sky-brightness spectral distribution. This non-parametric approach can be of particular interest for the new generation of low frequency radiotelescopes. The proposed solution formalizes the reconstruction problem as a convex optimization problem with spatial and spectral regularizations. The efficiency of this approach has been already proven for narrow-band image reconstruction and the present contribution can be considered as its extension to the multi-frequency case. Because the number of frequency bands multiplies the size of the inverse problem, particular attention is devoted to the derivation of an iterative large scale optimization algorithm. It is shown that the main computational bottleneck of the approach, which lies in the resolution of a linear system, can be efficiently ...
Multiscale Modelling and Inverse Problems
Nolen, J; Stuart, A M
2010-01-01
The need to blend observational data and mathematical models arises in many applications and leads naturally to inverse problems. Parameters appearing in the model, such as constitutive tensors, initial conditions, boundary conditions, and forcing can be estimated on the basis of observed data. The resulting inverse problems are often ill-posed and some form of regularization is required. These notes discuss parameter estimation in situations where the unknown parameters vary across multiple scales. We illustrate the main ideas using a simple model for groundwater flow. We will highlight various approaches to regularization for inverse problems, including Tikhonov and Bayesian methods. We illustrate three ideas that arise when considering inverse problems in the multiscale context. The first idea is that the choice of space or set in which to seek the solution to the inverse problem is intimately related to whether a homogenized or full multiscale solution is required. This is a choice of regularization. The ...
An Inverse Source Problem for a One-dimensional Wave Equation: An Observer-Based Approach
Asiri, Sharefa M.
2013-05-25
Observers are well known in the theory of dynamical systems. They are used to estimate the states of a system from some measurements. However, recently observers have also been developed to estimate some unknowns for systems governed by Partial differential equations. Our aim is to design an observer to solve inverse source problem for a one dimensional wave equation. Firstly, the problem is discretized in both space and time and then an adaptive observer based on partial field measurements (i.e measurements taken form the solution of the wave equation) is applied to estimate both the states and the source. We see the effectiveness of this observer in both noise-free and noisy cases. In each case, numerical simulations are provided to illustrate the effectiveness of this approach. Finally, we compare the performance of the observer approach with Tikhonov regularization approach.
Generalized emissivity inverse problem.
Ming, DengMing; Wen, Tao; Dai, XianXi; Dai, JiXin; Evenson, William E
2002-04-01
Inverse problems have recently drawn considerable attention from the physics community due to of potential widespread applications [K. Chadan and P. C. Sabatier, Inverse Problems in Quantum Scattering Theory, 2nd ed. (Springer Verlag, Berlin, 1989)]. An inverse emissivity problem that determines the emissivity g(nu) from measurements of only the total radiated power J(T) has recently been studied [Tao Wen, DengMing Ming, Xianxi Dai, Jixin Dai, and William E. Evenson, Phys. Rev. E 63, 045601(R) (2001)]. In this paper, a new type of generalized emissivity and transmissivity inverse (GETI) problem is proposed. The present problem differs from our previous work on inverse problems by allowing the unknown (emissivity) function g(nu) to be temperature dependent as well as frequency dependent. Based on published experimental information, we have developed an exact solution formula for this GETI problem. A universal function set suggested for numerical calculation is shown to be robust, making this inversion method practical and convenient for realistic calculations.
Soulez, Ferréol; Denis, Loïc; Fournier, Corinne; Thiébaut, Eric; Goepfert, Charles
2007-04-01
We propose a microparticle localization scheme in digital holography. Most conventional digital holography methods are based on Fresnel transform and present several problems such as twin-image noise, border effects, and other effects. To avoid these difficulties, we propose an inverse-problem approach, which yields the optimal particle set that best models the observed hologram image. We resolve this global optimization problem by conventional particle detection followed by a local refinement for each particle. Results for both simulated and real digital holograms show strong improvement in the localization of the particles, particularly along the depth dimension. In our simulations, the position precision is > or =1 microm rms. Our results also show that the localization precision does not deteriorate for particles near the edge of the field of view.
Statistical perspectives on inverse problems
DEFF Research Database (Denmark)
Andersen, Kim Emil
of the interior of an object from electrical boundary measurements. One part of this thesis concerns statistical approaches for solving, possibly non-linear, inverse problems. Thus inverse problems are recasted in a form suitable for statistical inference. In particular, a Bayesian approach for regularisation...... is obtained by assuming that the a priori beliefs about the solution before having observed any data can be described by a prior distribution. The solution to the statistical inverse problem is then given by the posterior distribution obtained by Bayes' formula. Hence the solution of an ill-posed inverse...... problem is given in terms of probability distributions. Posterior inference is obtained by Markov chain Monte Carlo methods and new, powerful simulation techniques based on e.g. coupled Markov chains and simulated tempering is developed to improve the computational efficiency of the overall simulation...
2010-02-28
illuminations. Inverse medium problems are encountered in acoustic, elastic, and electromagnetic wave propagation. We use a Lippmann- Schwinger formulation...Na. (5) This is a Born-approximation Lippmann- Schwinger scattering equation, where G(-, •;u) is the Green’s function (in the reference medium
A Direct Solution Approach to the Inverse Shallow-Water Problem
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Alelign Gessese
2012-01-01
Full Text Available The study of open channel flow modelling often requires an accurate representation of the channel bed topography to accurately predict the flow hydrodynamics. Experimental techniques are the most widely used approaches to measure the bed topographic elevation of open channels. However, they are usually cost and time consuming. Free surface measurement is, on the other hand, relatively easy to obtain using airborne photographic techniques. We present in this work an easy to implement and fast to solve numerical technique to identify the underlying bedrock topography from given free surface elevation data in shallow open channel flows. The main underlying idea is to derive explicit partial differential equations which govern this inverse reconstruction problem. The technique described here is a “one-shot technique” in the sense that the solution of the partial differential equation provides the solution to the inverse problem directly. The idea is tested on a set of artificial data obtained by first solving the forward problem governed by the shallow-water equations. Numerical results show that the channel bed topographic elevation can be reconstructed with a level of accuracy less than 3%. The method is also shown to be robust when noise is present in the input data.
Posterior Consistency of the Bayesian Approach to Linear Ill-Posed Inverse Problems
Agapiou, Sergios; Stuart, Andrew M
2012-01-01
We consider a Bayesian nonparametric approach to a family of linear inverse problems in a separable Hilbert space setting, with Gaussian prior and noise distribution. A method of identifying the posterior distribution using its precision operator is presented. Working with the unbounded precision operator enables us to use partial differential equations (PDE) methodology to study posterior consistency in a frequentist sense, and in particular to obtain rates of contraction of the posterior distribution to a Dirac measure centered on the true solution. We show how these rates may be optimized by a choice of the scale parameter in the prior covariance operator. Our methods assume a relatively weak relation between the prior covariance operator, the forward operator and the noise covariance operator; more precisely, we assume that appropriate powers of these operators induce equivalent norms. We compare our results to known minimax rates of convergence in the case where the forward operator and the prior and noi...
Control of plasma profile in microwave discharges via inverse-problem approach
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Yasuyoshi Yasaka
2013-12-01
Full Text Available In the manufacturing process of semiconductors, plasma processing is an essential technology, and the plasma used in the process is required to be of high density, low temperature, large diameter, and high uniformity. This research focuses on the microwave-excited plasma that meets these needs, and the research target is a spatial profile control. Two novel techniques are introduced to control the uniformity; one is a segmented slot antenna that can change radial distribution of the radiated field during operation, and the other is a hyper simulator that can predict microwave power distribution necessary for a desired radial density profile. The control system including these techniques provides a method of controlling radial profiles of the microwave plasma via inverse-problem approach, and is investigated numerically and experimentally.
Optimization and geophysical inverse problems
Energy Technology Data Exchange (ETDEWEB)
Barhen, J.; Berryman, J.G.; Borcea, L.; Dennis, J.; de Groot-Hedlin, C.; Gilbert, F.; Gill, P.; Heinkenschloss, M.; Johnson, L.; McEvilly, T.; More, J.; Newman, G.; Oldenburg, D.; Parker, P.; Porto, B.; Sen, M.; Torczon, V.; Vasco, D.; Woodward, N.B.
2000-10-01
A fundamental part of geophysics is to make inferences about the interior of the earth on the basis of data collected at or near the surface of the earth. In almost all cases these measured data are only indirectly related to the properties of the earth that are of interest, so an inverse problem must be solved in order to obtain estimates of the physical properties within the earth. In February of 1999 the U.S. Department of Energy sponsored a workshop that was intended to examine the methods currently being used to solve geophysical inverse problems and to consider what new approaches should be explored in the future. The interdisciplinary area between inverse problems in geophysics and optimization methods in mathematics was specifically targeted as one where an interchange of ideas was likely to be fruitful. Thus about half of the participants were actively involved in solving geophysical inverse problems and about half were actively involved in research on general optimization methods. This report presents some of the topics that were explored at the workshop and the conclusions that were reached. In general, the objective of a geophysical inverse problem is to find an earth model, described by a set of physical parameters, that is consistent with the observational data. It is usually assumed that the forward problem, that of calculating simulated data for an earth model, is well enough understood so that reasonably accurate synthetic data can be generated for an arbitrary model. The inverse problem is then posed as an optimization problem, where the function to be optimized is variously called the objective function, misfit function, or fitness function. The objective function is typically some measure of the difference between observational data and synthetic data calculated for a trial model. However, because of incomplete and inaccurate data, the objective function often incorporates some additional form of regularization, such as a measure of smoothness
A Bayesian approach to multiscale inverse problems with on-the-fly scale determination
Ellam, Louis; Zabaras, Nicholas; Girolami, Mark
2016-12-01
A Bayesian computational approach is presented to provide a multi-resolution estimate of an unknown spatially varying parameter from indirect measurement data. In particular, we are interested in spatially varying parameters with multiscale characteristics. In our work, we consider the challenge of not knowing the characteristic length scale(s) of the unknown a priori, and present an algorithm for on-the-fly scale determination. Our approach is based on representing the spatial field with a wavelet expansion. Wavelet basis functions are hierarchically structured, localized in both spatial and frequency domains and tend to provide sparse representations in that a large number of wavelet coefficients are approximately zero. For these reasons, wavelet bases are suitable for representing permeability fields with non-trivial correlation structures. Moreover, the intra-scale correlations between wavelet coefficients form a quadtree, and this structure is exploited to identify additional basis functions to refine the model. Bayesian inference is performed using a sequential Monte Carlo (SMC) sampler with a Markov Chain Monte Carlo (MCMC) transition kernel. The SMC sampler is used to move between posterior densities defined on different scales, thereby providing a computationally efficient method for adaptive refinement of the wavelet representation. We gain insight from the marginal likelihoods, by computing Bayes factors, for model comparison and model selection. The marginal likelihoods provide a termination criterion for our scale determination algorithm. The Bayesian computational approach is rather general and applicable to several inverse problems concerning the estimation of a spatially varying parameter. The approach is demonstrated with permeability estimation for groundwater flow using pressure sensor measurements.
A statistical physics approach to learning curves for the inverse Ising problem
Bachschmid-Romano, Ludovica; Opper, Manfred
2017-06-01
Using methods of statistical physics, we analyse the error of learning couplings in large Ising models from independent data (the inverse Ising problem). We concentrate on learning based on local cost functions, such as the pseudo-likelihood method for which the couplings are inferred independently for each spin. Assuming that the data are generated from a true Ising model, we compute the reconstruction error of the couplings using a combination of the replica method with the cavity approach for densely connected systems. We show that an explicit estimator based on a quadratic cost function achieves minimal reconstruction error, but requires the length of the true coupling vector as prior knowledge. A simple mean field estimator of the couplings which does not need such knowledge is asymptotically optimal, i.e. when the number of observations is much larger than the number of spins. Comparison of the theory with numerical simulations shows excellent agreement for data generated from two models with random couplings in the high temperature region: a model with independent couplings (Sherrington-Kirkpatrick model), and a model where the matrix of couplings has a Wishart distribution.
Tao Min; Xing Chen; Yao Sun; Qiang Huang
2014-01-01
This paper is intended to provide a numerical algorithm involving the combined use of the Levenberg-Marquardt algorithm and the Galerkin finite element method for estimating the diffusion coefficient in an inverse heat conduction problem (IHCP). In the present study, the functional form of the diffusion coefficient is unknown a priori. The unknown diffusion coefficient is approximated by the polynomial form and the present numerical algorithm is employed to find the solution. Numerical experi...
A Structured Approach to Solve the Inverse Eigenvalue Problem for a Beam with Added Mass
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Farhad Mir Hosseini
2014-01-01
Full Text Available The problem of determining the eigenvalues of a vibrational system having multiple lumped attachments has been investigated extensively. However, most of the research conducted in this field focuses on determining the natural frequencies of the combined system assuming that the characteristics of the combined vibrational system are known (forward problem. A problem of great interest from the point of view of engineering design is the ability to impose certain frequencies on the vibrational system or to avoid certain frequencies by modifying the characteristics of the vibrational system (inverse problem. In this paper, a method to impose two natural frequencies on a dynamical system consisting of an Euler-Bernoulli beam and carrying a single mass attachment is evaluated.
Directory of Open Access Journals (Sweden)
Tao Min
2014-01-01
Full Text Available This paper is intended to provide a numerical algorithm involving the combined use of the Levenberg-Marquardt algorithm and the Galerkin finite element method for estimating the diffusion coefficient in an inverse heat conduction problem (IHCP. In the present study, the functional form of the diffusion coefficient is unknown a priori. The unknown diffusion coefficient is approximated by the polynomial form and the present numerical algorithm is employed to find the solution. Numerical experiments are presented to show the efficiency of the proposed method.
A New Concept for Atmospheric Reentry Optimal Guidance: An Inverse Problem Inspired Approach
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Davood Abbasi
2013-01-01
Full Text Available This paper presents a new concept for atmospheric reentry online optimal guidance and control using a method called MARE G&C that exploits the different time scale featured by reentry dynamics. The new technique reaches a quasi-analytical solution and simplified computations, even considering both lift-to-drag ratio and aerodynamic roll as control variables; in addition, the paper offers a solution for the challenging path constraints issue, getting inspiration from the inverse problem methodology. The final resulting algorithm seems suitable for onboard predictive guidance, a new need for future space missions.
Inverse-problem approach to designing photonic crystals for cavity QED experiments.
Geremia, J M; Williams, Jon; Mabuchi, Hideo
2002-12-01
Photonic band gap (PBG) materials are attractive for cavity QED experiments because they provide extremely small mode volumes and are monolithic, integratable structures. As such, PBG cavities are a promising alternative to Fabry-Perot resonators. However, the cavity requirements imposed by QED experiments, such as the need for high Q (low cavity damping) and small mode volumes, present significant design challenges for photonic band gap materials. Here, we pose the PBG design problem as a mathematical inversion and provide an analytical solution for a two-dimensional (2D) crystal. We then address a planar (2D crystal with finite thickness) structure using numerical techniques.
A linear model approach for ultrasonic inverse problems with attenuation and dispersion.
Carcreff, Ewen; Bourguignon, Sébastien; Idier, Jérôme; Simon, Laurent
2014-07-01
Ultrasonic inverse problems such as spike train deconvolution, synthetic aperture focusing, or tomography attempt to reconstruct spatial properties of an object (discontinuities, delaminations, flaws, etc.) from noisy and incomplete measurements. They require an accurate description of the data acquisition process. Dealing with frequency-dependent attenuation and dispersion is therefore crucial because both phenomena modify the wave shape as the travel distance increases. In an inversion context, this paper proposes to exploit a linear model of ultrasonic data taking into account attenuation and dispersion. The propagation distance is discretized to build a finite set of radiation impulse responses. Attenuation is modeled with a frequency power law and then dispersion is computed to yield physically consistent responses. Using experimental data acquired from attenuative materials, this model outperforms the standard attenuation-free model and other models of the literature. Because of model linearity, robust estimation methods can be implemented. When matched filtering is employed for single echo detection, the model that we propose yields precise estimation of the attenuation coefficient and of the sound velocity. A thickness estimation problem is also addressed through spike deconvolution, for which the proposed model also achieves accurate results.
Directory of Open Access Journals (Sweden)
Wong William WL
2009-04-01
Full Text Available Abstract Background The inverse-QSAR problem seeks to find a new molecular descriptor from which one can recover the structure of a molecule that possess a desired activity or property. Surprisingly, there are very few papers providing solutions to this problem. It is a difficult problem because the molecular descriptors involved with the inverse-QSAR algorithm must adequately address the forward QSAR problem for a given biological activity if the subsequent recovery phase is to be meaningful. In addition, one should be able to construct a feasible molecule from such a descriptor. The difficulty of recovering the molecule from its descriptor is the major limitation of most inverse-QSAR methods. Results In this paper, we describe the reversibility of our previously reported descriptor, the vector space model molecular descriptor (VSMMD based on a vector space model that is suitable for kernel studies in QSAR modeling. Our inverse-QSAR approach can be described using five steps: (1 generate the VSMMD for the compounds in the training set; (2 map the VSMMD in the input space to the kernel feature space using an appropriate kernel function; (3 design or generate a new point in the kernel feature space using a kernel feature space algorithm; (4 map the feature space point back to the input space of descriptors using a pre-image approximation algorithm; (5 build the molecular structure template using our VSMMD molecule recovery algorithm. Conclusion The empirical results reported in this paper show that our strategy of using kernel methodology for an inverse-Quantitative Structure-Activity Relationship is sufficiently powerful to find a meaningful solution for practical problems.
Directory of Open Access Journals (Sweden)
Calvez V.
2010-12-01
Full Text Available We consider the radiative transfer equation (RTE with reflection in a three-dimensional domain, infinite in two dimensions, and prove an existence result. Then, we study the inverse problem of retrieving the optical parameters from boundary measurements, with help of existing results by Choulli and Stefanov. This theoretical analysis is the framework of an attempt to model the color of the skin. For this purpose, a code has been developed to solve the RTE and to study the sensitivity of the measurements made by biophysicists with respect to the physiological parameters responsible for the optical properties of this complex, multi-layered material. On étudie l’équation du transfert radiatif (ETR dans un domaine tridimensionnel infini dans deux directions, et on prouve un résultat d’existence. On s’intéresse ensuite à la reconstruction des paramètres optiques à partir de mesures faites au bord, en s’appuyant sur des résultats de Choulli et Stefanov. Cette analyse sert de cadre théorique à un travail de modélisation de la couleur de la peau. Dans cette perspective, un code à été développé pour résoudre l’ETR et étudier la sensibilité des mesures effectuées par les biophysiciens par rapport aux paramètres physiologiques tenus pour responsables des propriétés optiques de ce complexe matériau multicouche.
Inverse problem in hydrogeology
Carrera, Jesús; Alcolea, Andrés; Medina, Agustín; Hidalgo, Juan; Slooten, Luit J.
2005-03-01
The state of the groundwater inverse problem is synthesized. Emphasis is placed on aquifer characterization, where modelers have to deal with conceptual model uncertainty (notably spatial and temporal variability), scale dependence, many types of unknown parameters (transmissivity, recharge, boundary conditions, etc.), nonlinearity, and often low sensitivity of state variables (typically heads and concentrations) to aquifer properties. Because of these difficulties, calibration cannot be separated from the modeling process, as it is sometimes done in other fields. Instead, it should be viewed as one step in the process of understanding aquifer behavior. In fact, it is shown that actual parameter estimation methods do not differ from each other in the essence, though they may differ in the computational details. It is argued that there is ample room for improvement in groundwater inversion: development of user-friendly codes, accommodation of variability through geostatistics, incorporation of geological information and different types of data (temperature, occurrence and concentration of isotopes, age, etc.), proper accounting of uncertainty, etc. Despite this, even with existing codes, automatic calibration facilitates enormously the task of modeling. Therefore, it is contended that its use should become standard practice. L'état du problème inverse des eaux souterraines est synthétisé. L'accent est placé sur la caractérisation de l'aquifère, où les modélisateurs doivent jouer avec l'incertitude des modèles conceptuels (notamment la variabilité spatiale et temporelle), les facteurs d'échelle, plusieurs inconnues sur différents paramètres (transmissivité, recharge, conditions aux limites, etc.), la non linéarité, et souvent la sensibilité de plusieurs variables d'état (charges hydrauliques, concentrations) des propriétés de l'aquifère. A cause de ces difficultés, le calibrage ne peut êtreséparé du processus de modélisation, comme c'est le
Inverse problems in stochastic computational dynamics
Capiez-Lernout, Evangéline; Soize, Christian
2008-01-01
International audience; This paper deals with robust updating of dynamical systems using stochastic computational models for which model and parameter uncertainties are taken into account by the nonparametric probabilistic approach. Such a problem is formulated as an inverse problem consisting in identifying the parameters of the mean computational model and the parameters of the probabilistic model of uncertainties. This inverse problem leads us to solve an optimization problem for which the...
Inverse problems for Maxwell's equations
Romanov, V G
1994-01-01
The Inverse and Ill-Posed Problems Series is a series of monographs publishing postgraduate level information on inverse and ill-posed problems for an international readership of professional scientists and researchers. The series aims to publish works which involve both theory and applications in, e.g., physics, medicine, geophysics, acoustics, electrodynamics, tomography, and ecology.
Llibre, Jaume; Ramírez, Rafael; Ramírez, Valentín
2017-09-01
We consider polynomial vector fields X with a linear type and with homogenous nonlinearities. It is well-known that X has a center at the origin if and only if X has an analytic first integral of the form H =1/2 (x2 +y2) + ∑ j = 3 ∞Hj, where Hj =Hj (x , y) is a homogenous polynomial of degree j. The classical center-focus problem already studied by H. Poincaré consists in distinguishing when the origin of X is either a center or a focus. In this paper we study the inverse center-focus problem. In particular for a given analytic function H defined in a neighborhood of the origin we want to determine the homogenous polynomials in such a way that H is a first integral of X and consequently the origin of X will be a center. We study the particular case of centers which have a local analytic first integral of the form H =1/2 (x2 +y2) (1 + ∑ j = 1 ∞ϒj) , in a neighborhood of the origin, where ϒj is a convenient homogenous polynomial of degree j, for j ≥ 1. These centers are called weak centers, they contain the class of center studied by Alwash and Lloyd, the uniform isochronous centers and the isochronous holomorphic centers, but they do not coincide with the class of isochronous centers. We give a classification of the weak centers for quadratic and cubic vector fields with homogenous nonlinearities.
Parameter estimation and inverse problems
Aster, Richard C; Thurber, Clifford H
2005-01-01
Parameter Estimation and Inverse Problems primarily serves as a textbook for advanced undergraduate and introductory graduate courses. Class notes have been developed and reside on the World Wide Web for faciliting use and feedback by teaching colleagues. The authors'' treatment promotes an understanding of fundamental and practical issus associated with parameter fitting and inverse problems including basic theory of inverse problems, statistical issues, computational issues, and an understanding of how to analyze the success and limitations of solutions to these probles. The text is also a practical resource for general students and professional researchers, where techniques and concepts can be readily picked up on a chapter-by-chapter basis.Parameter Estimation and Inverse Problems is structured around a course at New Mexico Tech and is designed to be accessible to typical graduate students in the physical sciences who may not have an extensive mathematical background. It is accompanied by a Web site that...
Velimsky, J.
2011-12-01
Inversion of observatory and low-orbit satellite geomagnetic data in terms of the three-dimensional distribution of electrical conductivity in the Earth's mantle can provide an independent constraint on the physical, chemical, and mineralogical composition of the Earth's mantle. This problem has been recently approached by different numerical methods. There are several key challenges from the numerical and algorithmic point of view, in particular the accuracy and speed of the forward solver, the effective evaluation of sensitivities of data to changes of model parameters, and the dependence of results on the a-priori knowledge of the spatio-temporal structure of the primary ionospheric and magnetospheric electric currents. Here I present recent advancements of the time-domain, spherical harmonic-finite element approach. The forward solver has been adapted to distributed-memory parallel architecture using band-matrix routines from the ScaLapack library. The evaluation of gradient of data misfit in the model space using adjoint approach has been also paralellized. Finally, the inverse problem has been reformulated in a way which allows for simultaneous reconstruction of conductivity model and external field model directly from the data.
Mijovic, S
2003-01-01
Computer-supported techniques are introduced in the evaluation of experimental data and obtaining the real profile of spectral lines. The direct and inverse approaches were used. The MINUIT program from the packets of CERN's library was used to solve direct problems. Tikhonov's regularization method was also applied to solve the same problems in an inverse manner. Model functions were introduced to check the applicability limitation of these methods and make a comparison between them as well. The advantages and disadvantages of these approaches were shown. The procedures were applied to the measured profiles of He II's spectral lines in a pulsed low-pressure arc. The chosen lines are He II Paschen-alpha (468.6 nm) in the visible region and Balmer-beta (121.5 nm) in the VUV spectral region. The range of experimental errors was determined where both approaches have given reliable results. It was found that we can obtain the real profile of He II 468.6 nm and He II 121.5 nm spectral lines, using the regularizati...
A statistical approach for isolating fossil fuel emissions in atmospheric inverse problems
Yadav, Vineet; Michalak, Anna M.; Ray, Jaideep; Shiga, Yoichi P.
2016-10-01
Independent verification and quantification of fossil fuel (FF) emissions constitutes a considerable scientific challenge. By coupling atmospheric observations of CO2 with models of atmospheric transport, inverse models offer the possibility of overcoming this challenge. However, disaggregating the biospheric and FF flux components of terrestrial fluxes from CO2 concentration measurements has proven to be difficult, due to observational and modeling limitations. In this study, we propose a statistical inverse modeling scheme for disaggregating winter time fluxes on the basis of their unique error covariances and covariates, where these covariances and covariates are representative of the underlying processes affecting FF and biospheric fluxes. The application of the method is demonstrated with one synthetic and two real data prototypical inversions by using in situ CO2 measurements over North America. Inversions are performed only for the month of January, as predominance of biospheric CO2 signal relative to FF CO2 signal and observational limitations preclude disaggregation of the fluxes in other months. The quality of disaggregation is assessed primarily through examination of a posteriori covariance between disaggregated FF and biospheric fluxes at regional scales. Findings indicate that the proposed method is able to robustly disaggregate fluxes regionally at monthly temporal resolution with a posteriori cross covariance lower than 0.15 µmol m-2 s-1 between FF and biospheric fluxes. Error covariance models and covariates based on temporally varying FF inventory data provide a more robust disaggregation over static proxies (e.g., nightlight intensity and population density). However, the synthetic data case study shows that disaggregation is possible even in absence of detailed temporally varying FF inventory data.
BOOK REVIEW: Inverse Problems. Activities for Undergraduates
Yamamoto, Masahiro
2003-06-01
This book is a valuable introduction to inverse problems. In particular, from the educational point of view, the author addresses the questions of what constitutes an inverse problem and how and why we should study them. Such an approach has been eagerly awaited for a long time. Professor Groetsch, of the University of Cincinnati, is a world-renowned specialist in inverse problems, in particular the theory of regularization. Moreover, he has made a remarkable contribution to educational activities in the field of inverse problems, which was the subject of his previous book (Groetsch C W 1993 Inverse Problems in the Mathematical Sciences (Braunschweig: Vieweg)). For this reason, he is one of the most qualified to write an introductory book on inverse problems. Without question, inverse problems are important, necessary and appear in various aspects. So it is crucial to introduce students to exercises in inverse problems. However, there are not many introductory books which are directly accessible by students in the first two undergraduate years. As a consequence, students often encounter diverse concrete inverse problems before becoming aware of their general principles. The main purpose of this book is to present activities to allow first-year undergraduates to learn inverse theory. To my knowledge, this book is a rare attempt to do this and, in my opinion, a great success. The author emphasizes that it is very important to teach inverse theory in the early years. He writes; `If students consider only the direct problem, they are not looking at the problem from all sides .... The habit of always looking at problems from the direct point of view is intellectually limiting ...' (page 21). The book is very carefully organized so that teachers will be able to use it as a textbook. After an introduction in chapter 1, sucessive chapters deal with inverse problems in precalculus, calculus, differential equations and linear algebra. In order to let one gain some insight
Optical tomography: forward and inverse problems
Arridge, Simon
2009-01-01
This paper is a review of recent mathematical and computational advances in optical tomography. We discuss the physical foundations of forward models for light propagation on microscopic, mesoscopic and macroscopic scales. We also consider direct and numerical approaches to the inverse problems that arise at each of these scales. Finally, we outline future directions and open problems in the field.
DEM analysis for AIA/SDO EUV channels using a probabilistic approach to the spectral inverse problem
Goryaev, Farid; Parenti, Susanna; Hochedez, Jean-François; Urnov, Alexander
The Atmospheric Imaging Assembly (AIA) for the Solar Dynamics Observatory (SDO) mis-sion is designed to observe the Sun from the photosphere to the flaring corona. These data have to improve our understanding of processes in the solar atmosphere. The differential emis-sion measure (DEM) analysis is one of the main methods to derive information about coronal optically thin plasma characteristics from EUV and SXR emission. In this work we analyze AIA/SDO EUV channels to estimate their ability to reconstruct DEM(T) distributions. We use an iterative method (called Bayesian iterative method, BIM) within the framework of a probabilistic approach to the spectral inverse problem for determining the thermal structures of the emitting plasma sources (Goryaev et al., submitted to AA). The BIM is an iterative procedure based on Bayes' theorem and used for the reconstruction of DEM profiles. Using the BIM algorithm we performed various numerical tests and model simulations demonstrating abilities of our inversion approach for DEM analysis with AIA/SDO EUV channels.
Size Estimates in Inverse Problems
Di Cristo, Michele
2014-01-06
Detection of inclusions or obstacles inside a body by boundary measurements is an inverse problems very useful in practical applications. When only finite numbers of measurements are available, we try to detect some information on the embedded object such as its size. In this talk we review some recent results on several inverse problems. The idea is to provide constructive upper and lower estimates of the area/volume of the unknown defect in terms of a quantity related to the work that can be expressed with the available boundary data.
Bradshaw, L A; Wijesinghe, R S; Wikswo, J P
2001-03-01
We present an analysis of the relative information content of cortical current source reconstructions from electroencephalogram (EEG) and magnetoencephalogram (MEG) forward calculations by examining the spatial filters that relate the internal sources with the externally measured electric potentials and magnetic fields. The forward spatial filters are seen to be low-pass functions of spatial frequency and spatial resolution degrades in external measurements. Inverse spatial filters may be used to reconstruct cortical sources from external data, but since they are high-pass functions of spatial frequency, they must be regularized to avoid instabilities caused by noise at higher spatial frequencies. The regularization process limits the spatial resolution of source reconstructions. EEG forward spatial filters fall off at lower spatial frequencies than MEG filters; hence, there is less information available in higher spatial frequencies resulting in lower spatial resolution in inverse reconstructions. The tangential component of the magnetic field provides even higher spatial resolution than can be obtained using the radial component. An accompanying article examines the surface Laplacian for both the EEG and the MEG.
Inverse problems and uncertainty quantification
Litvinenko, Alexander
2013-12-18
In a Bayesian setting, inverse problems and uncertainty quantification (UQ)— the propagation of uncertainty through a computational (forward) model—are strongly connected. In the form of conditional expectation the Bayesian update becomes computationally attractive. This is especially the case as together with a functional or spectral approach for the forward UQ there is no need for time- consuming and slowly convergent Monte Carlo sampling. The developed sampling- free non-linear Bayesian update is derived from the variational problem associated with conditional expectation. This formulation in general calls for further discretisa- tion to make the computation possible, and we choose a polynomial approximation. After giving details on the actual computation in the framework of functional or spectral approximations, we demonstrate the workings of the algorithm on a number of examples of increasing complexity. At last, we compare the linear and quadratic Bayesian update on the small but taxing example of the chaotic Lorenz 84 model, where we experiment with the influence of different observation or measurement operators on the update.
Inverse Problems and Uncertainty Quantification
Litvinenko, Alexander
2014-01-06
In a Bayesian setting, inverse problems and uncertainty quantification (UQ) - the propagation of uncertainty through a computational (forward) modelare strongly connected. In the form of conditional expectation the Bayesian update becomes computationally attractive. This is especially the case as together with a functional or spectral approach for the forward UQ there is no need for time- consuming and slowly convergent Monte Carlo sampling. The developed sampling- free non-linear Bayesian update is derived from the variational problem associated with conditional expectation. This formulation in general calls for further discretisa- tion to make the computation possible, and we choose a polynomial approximation. After giving details on the actual computation in the framework of functional or spectral approximations, we demonstrate the workings of the algorithm on a number of examples of increasing complexity. At last, we compare the linear and quadratic Bayesian update on the small but taxing example of the chaotic Lorenz 84 model, where we experiment with the influence of different observation or measurement operators on the update.
Stoklasová, Pavla; Sedlák, Petr; Seiner, Hanuš; Landa, Michal
2015-02-01
We show that the Ritz-Rayleigh method can be used for calculation of velocity of surface acoustic waves (SAWs) propagating in a general direction of an anisotropic medium of arbitrary symmetry class. The main advantage of this method is that expanding the displacement field of SAW into a fixed functional basis transforms the calculation of SAW velocities into a simple linear eigenvalue problem. The correctness and reliability of the proposed approach are verified on experimental SAW data obtained for generally oriented planes of an indium phosphide single crystal. The same experimental datasets are then used to discuss the invertibility of the method, i.e. the possibility of determination of elastic coefficients from SAW measurements in general directions. It is shown that the SAW data obtained on a single generally oriented plane are sufficient for such an inverse calculation for a cubic material only if they are complemented by measurements of velocities of bulk quasi-longitudinal (qL) waves propagating along the same free surface. Moreover, when the SAW and qL data are available from three almost perpendicular faces of a single specimen, the complete elastic tensor (21 independent constants) can be inversely determined, without considering a priori any symmetry constraints to the material.
The role of nonlinearity in inverse problems
Snieder, Roel
1998-06-01
In many practical inverse problems, one aims to retrieve a model that has infinitely many degrees of freedom from a finite amount of data. It follows from a simple variable count that this cannot be done in a unique way. Therefore, inversion entails more than estimating a model: any inversion is not complete without a description of the class of models that is consistent with the data; this is called the appraisal problem. Nonlinearity makes the appraisal problem particularly difficult. The first reason for this is that nonlinear error propagation is a difficult problem. The second reason is that for some nonlinear problems the model parameters affect the way in which the model is being interrogated by the data. Two examples are given of this, and it is shown how the nonlinearity may make the problem more ill-posed. Finally, three attempts are shown to carry out the model appraisal for nonlinear inverse problems that are based on an analytical approach, a numerical approach and a common sense approach.
Integrated Approaches to Parallelism in Optimization and the Solution of Inverse Problems
1992-09-30
Aplicadas y Primera Escuela , Chile - CEE de Optimizaci6n; Santiago, Chile; August 28, 1992 - September 6, 1992. * "On Alternative Problem...the Iteration Sequence in Primal-Dual Interior-Point Methods for Linear Programming," Fourth SIAM Conference on Optimization; Chicago , Illinois; May
A unified approach for inversion problems in intensity-modulated radiation therapy
Censor, Yair; Bortfeld, Thomas; Martin, Benjamin; Trofimov, Alexei
2006-05-01
We propose and study a unified model for handling dose constraints (physical dose, equivalent uniform dose (EUD), etc) and radiation source constraints in a single mathematical framework based on the split feasibility problem. The model does not impose on the constraints an exogenous objective (merit) function. The optimization algorithm minimizes a weighted proximity function that measures the sum of the squares of the distances to the constraint sets. This guarantees convergence to a feasible solution point if the split feasibility problem is consistent (i.e., has a solution), or, otherwise, convergence to a solution that minimally violates the physical dose constraints and EUD constraints. We present computational results that demonstrate the validity of the model and the power of the proposed algorithmic scheme.
An error function minimization approach for the inverse problem of adaptive mirrors tuning
Vannoni, Maurizio; Yang, Fan; Siewert, Frank; Sinn, Harald
2014-09-01
Adaptive x-ray optics are more and more used in synchrotron beamlines, and it is probable that they will be considered for the future high-power free-electron laser sources, as the European XFEL now under construction in Hamburg, or similar projects now in discussion. These facilities will deliver a high power x-ray beam, with an expected high heat load delivered on the optics. For this reason, bendable mirrors are required to actively compensate the resulting wavefront distortion. On top of that, the mirror could have also intrinsic surface defects, as polishing errors or mounting stresses. In order to be able to correct the mirror surface with a high precision to maintain its challenging requirements, the mirror surface is usually characterized with a high accuracy metrology to calculate the actuators pulse functions and to assess its initial shape. After that, singular value decomposition (SVD) is used to find the signals to be applied into the actuators, to reach the desired surface deformation or correction. But in some cases this approach could be not robust enough for the needed performance. We present here a comparison between the classical SVD method and an error function minimization based on root-mean-square calculation. Some examples are provided, using a simulation of the European XFEL mirrors design as a case of study, and performances of the algorithms are evaluated in order to reach the ultimate quality in different scenarios. The approach could be easily generalized to other situations as well.
Riemann Zeros and the Inverse Phase Problem
Tourigny, David S.
2013-10-01
Finding a universal method of crystal structure solution and proving the Riemann hypothesis are two outstanding challenges in apparently unrelated fields. For centro-symmetric crystals however, a connection arises as the result of a statistical approach to the inverse phase problem. It is shown that parameters of the phase distribution are related to the non-trivial Riemann zeros by a Mellin transform.
RIEMANN ZEROS AND THE INVERSE PHASE PROBLEM
TOURIGNY, DAVID S.
2013-01-01
Finding a universal method of crystal structure solution and proving the Riemann hypothesis are two outstanding challenges in apparently unrelated fields. For centrosymmetric crystals however, a connection arises as the result of a statistical approach to the inverse phase problem. It is shown that parameters of the phase distribution are related to the non-trivial Riemann zeros by a Mellin transform. PMID:24293780
Iterative optimization in inverse problems
Byrne, Charles L
2014-01-01
Iterative Optimization in Inverse Problems brings together a number of important iterative algorithms for medical imaging, optimization, and statistical estimation. It incorporates recent work that has not appeared in other books and draws on the author's considerable research in the field, including his recently developed class of SUMMA algorithms. Related to sequential unconstrained minimization methods, the SUMMA class includes a wide range of iterative algorithms well known to researchers in various areas, such as statistics and image processing. Organizing the topics from general to more
Inverse Problems in Classical and Quantum Physics
Almasy, Andrea A
2009-01-01
The subject of this thesis is in the area of Applied Mathematics known as Inverse Problems. Inverse problems are those where a set of measured data is analysed in order to get as much information as possible on a model which is assumed to represent a system in the real world. We study two inverse problems in the fields of classical and quantum physics: QCD condensates from tau-decay data and the inverse conductivity problem. We use a functional method which allows us to extract within rather general assumptions phenomenological parameters of QCD (the condensates) from a comparison of the time-like experimental data with asymptotic space-like results from theory. The price to be paid for the generality of assumptions is relatively large errors in the values of the extracted parameters. Although we do not claim that our method is superior to other approaches, we hope that our results lend additional confidence to the numerical results obtained with the help of methods based on QCD sum rules. In this thesis, als...
Crestel, Benjamin; Alexanderian, Alen; Stadler, Georg; Ghattas, Omar
2017-07-01
The computational cost of solving an inverse problem governed by PDEs, using multiple experiments, increases linearly with the number of experiments. A recently proposed method to decrease this cost uses only a small number of random linear combinations of all experiments for solving the inverse problem. This approach applies to inverse problems where the PDE solution depends linearly on the right-hand side function that models the experiment. As this method is stochastic in essence, the quality of the obtained reconstructions can vary, in particular when only a small number of combinations are used. We develop a Bayesian formulation for the definition and computation of encoding weights that lead to a parameter reconstruction with the least uncertainty. We call these weights A-optimal encoding weights. Our framework applies to inverse problems where the governing PDE is nonlinear with respect to the inversion parameter field. We formulate the problem in infinite dimensions and follow the optimize-then-discretize approach, devoting special attention to the discretization and the choice of numerical methods in order to achieve a computational cost that is independent of the parameter discretization. We elaborate our method for a Helmholtz inverse problem, and derive the adjoint-based expressions for the gradient of the objective function of the optimization problem for finding the A-optimal encoding weights. The proposed method is potentially attractive for real-time monitoring applications, where one can invest the effort to compute optimal weights offline, to later solve an inverse problem repeatedly, over time, at a fraction of the initial cost.
Selg, M
2005-01-01
Elegant and mathematically rigorous methods of the quantum inverse theory are difficult to put into practice because there is always some lack of needful input information. In this situation, one may try to construct a reference potential, whose spectral characteristics would be in a reasonable agreement with the available data of the system's properties. Since the reference potential is fixed, it is always possible to calculate all its spectral characteristics, including phase shift for scattering states and Jost function, the main key to solve the inverse problem. Thereafter, one can calculate a Bargmann potential whose Jost function differs from the initial one only by a rational factor. This way it is possible, at least in principle, to construct a more reliable potential for the system. The model system investigated in this paper is diatomic xenon molecule in ground electronic state. Its reference potential is built up of several smoothly joined Morse type components, which enables to solve the related e...
Computationally efficient Bayesian inference for inverse problems.
Energy Technology Data Exchange (ETDEWEB)
Marzouk, Youssef M.; Najm, Habib N.; Rahn, Larry A.
2007-10-01
Bayesian statistics provides a foundation for inference from noisy and incomplete data, a natural mechanism for regularization in the form of prior information, and a quantitative assessment of uncertainty in the inferred results. Inverse problems - representing indirect estimation of model parameters, inputs, or structural components - can be fruitfully cast in this framework. Complex and computationally intensive forward models arising in physical applications, however, can render a Bayesian approach prohibitive. This difficulty is compounded by high-dimensional model spaces, as when the unknown is a spatiotemporal field. We present new algorithmic developments for Bayesian inference in this context, showing strong connections with the forward propagation of uncertainty. In particular, we introduce a stochastic spectral formulation that dramatically accelerates the Bayesian solution of inverse problems via rapid evaluation of a surrogate posterior. We also explore dimensionality reduction for the inference of spatiotemporal fields, using truncated spectral representations of Gaussian process priors. These new approaches are demonstrated on scalar transport problems arising in contaminant source inversion and in the inference of inhomogeneous material or transport properties. We also present a Bayesian framework for parameter estimation in stochastic models, where intrinsic stochasticity may be intermingled with observational noise. Evaluation of a likelihood function may not be analytically tractable in these cases, and thus several alternative Markov chain Monte Carlo (MCMC) schemes, operating on the product space of the observations and the parameters, are introduced.
Analog fault diagnosis by inverse problem technique
Ahmed, Rania F.
2011-12-01
A novel algorithm for detecting soft faults in linear analog circuits based on the inverse problem concept is proposed. The proposed approach utilizes optimization techniques with the aid of sensitivity analysis. The main contribution of this work is to apply the inverse problem technique to estimate the actual parameter values of the tested circuit and so, to detect and diagnose single fault in analog circuits. The validation of the algorithm is illustrated through applying it to Sallen-Key second order band pass filter and the results show that the detecting percentage efficiency was 100% and also, the maximum error percentage of estimating the parameter values is 0.7%. This technique can be applied to any other linear circuit and it also can be extended to be applied to non-linear circuits. © 2011 IEEE.
Inverse problems in classical and quantum physics
Energy Technology Data Exchange (ETDEWEB)
Almasy, A.A.
2007-06-29
The subject of this thesis is in the area of Applied Mathematics known as Inverse Problems. Inverse problems are those where a set of measured data is analysed in order to get as much information as possible on a model which is assumed to represent a system in the real world. We study two inverse problems in the fields of classical and quantum physics: QCD condensates from tau-decay data and the inverse conductivity problem. Despite a concentrated effort by physicists extending over many years, an understanding of QCD from first principles continues to be elusive. Fortunately, data continues to appear which provide a rather direct probe of the inner workings of the strong interactions. We use a functional method which allows us to extract within rather general assumptions phenomenological parameters of QCD (the condensates) from a comparison of the time-like experimental data with asymptotic space-like results from theory. The price to be paid for the generality of assumptions is relatively large errors in the values of the extracted parameters. Although we do not claim that our method is superior to other approaches, we hope that our results lend additional confidence to the numerical results obtained with the help of methods based on QCD sum rules. EIT is a technology developed to image the electrical conductivity distribution of a conductive medium. The technique works by performing simultaneous measurements of direct or alternating electric currents and voltages on the boundary of an object. These are the data used by an image reconstruction algorithm to determine the electrical conductivity distribution within the object. In this thesis, two approaches of EIT image reconstruction are proposed. The first is based on reformulating the inverse problem in terms of integral equations. This method uses only a single set of measurements for the reconstruction. The second approach is an algorithm based on linearisation which uses more then one set of measurements. A
DEFF Research Database (Denmark)
Montoya-Martinez, Jair; Artes-Rodriguez, Antonio; Pontil, Massimiliano;
2014-01-01
We consider the estimation of the Brain Electrical Sources (BES) matrix from noisy electroencephalographic (EEG) measurements, commonly named as the EEG inverse problem. We propose a new method to induce neurophysiological meaningful solutions, which takes into account the smoothness, structured...... sparsity, and low rank of the BES matrix. The method is based on the factorization of the BES matrix as a product of a sparse coding matrix and a dense latent source matrix. The structured sparse-low-rank structure is enforced by minimizing a regularized functional that includes the ℓ21-norm of the coding...... matrix and the squared Frobenius norm of the latent source matrix. We develop an alternating optimization algorithm to solve the resulting nonsmooth-nonconvex minimization problem. We analyze the convergence of the optimization procedure, and we compare, under different synthetic scenarios...
Estimating uncertainties in complex joint inverse problems
Afonso, Juan Carlos
2016-04-01
Sources of uncertainty affecting geophysical inversions can be classified either as reflective (i.e. the practitioner is aware of her/his ignorance) or non-reflective (i.e. the practitioner does not know that she/he does not know!). Although we should be always conscious of the latter, the former are the ones that, in principle, can be estimated either empirically (by making measurements or collecting data) or subjectively (based on the experience of the researchers). For complex parameter estimation problems in geophysics, subjective estimation of uncertainty is the most common type. In this context, probabilistic (aka Bayesian) methods are commonly claimed to offer a natural and realistic platform from which to estimate model uncertainties. This is because in the Bayesian approach, errors (whatever their nature) can be naturally included as part of the global statistical model, the solution of which represents the actual solution to the inverse problem. However, although we agree that probabilistic inversion methods are the most powerful tool for uncertainty estimation, the common claim that they produce "realistic" or "representative" uncertainties is not always justified. Typically, ALL UNCERTAINTY ESTIMATES ARE MODEL DEPENDENT, and therefore, besides a thorough characterization of experimental uncertainties, particular care must be paid to the uncertainty arising from model errors and input uncertainties. We recall here two quotes by G. Box and M. Gunzburger, respectively, of special significance for inversion practitioners and for this session: "…all models are wrong, but some are useful" and "computational results are believed by no one, except the person who wrote the code". In this presentation I will discuss and present examples of some problems associated with the estimation and quantification of uncertainties in complex multi-observable probabilistic inversions, and how to address them. Although the emphasis will be on sources of uncertainty related
A Forward Glimpse into Inverse Problems through a Geology Example
Winkel, Brian J.
2012-01-01
This paper describes a forward approach to an inverse problem related to detecting the nature of geological substrata which makes use of optimization techniques in a multivariable calculus setting. The true nature of the related inverse problem is highlighted. (Contains 2 figures.)
CUDA Parallel Algorithms for Forward and Inverse Structural Gravity Problems
2015-01-01
This paper describes usage of CUDA parallelization scheme for forward and inverse gravity problems for structural boundaries. Forward problem is calculated using the finite elements approach. This means that the whole calculation volume is split into parallelepipeds and then the gravity effect of each is calculated using known formula. Inverse problem solution is found using iteration local corrections method. This method requires only forward problem calculation on each iteration and does no...
Goryaev, F F; Urnov, A M; Oparin, S N; Hochedez, J -F; Reale, F; 10.1051/0004-6361/201014280
2010-01-01
Inverse problems are of great importance in astrophysics for deriving information about the physical characteristics of hot optically thin plasma sources from their EUV and X-ray spectra. We describe and test an iterative method developed within the framework of a probabilistic approach to the spectral inverse problem for determining the thermal structures of the emitting plasma. We also demonstrate applications of this method to both high resolution line spectra and broadband imaging data. Our so-called Bayesian iterative method (BIM) is an iterative procedure based on Bayes' theorem and is used to reconstruct differential emission measure (DEM) distributions. To demonstrate the abilities of the BIM, we performed various numerical tests and model simulations establishing its robustness and usefulness. We then applied the BIM to observable data for several active regions (AR) previously analyzed with other DEM diagnostic techniques: both SUMER/SOHO (Landi and Feldman, 2008) and SPIRIT/CORONAS-F (Shestov et al...
Nonlinear Least Squares for Inverse Problems
Chavent, Guy
2009-01-01
Presents an introduction into the least squares resolution of nonlinear inverse problems. This title intends to develop a geometrical theory to analyze nonlinear least square (NLS) problems with respect to their quadratic wellposedness, that is, both wellposedness and optimizability
Metaheuristic optimization of acoustic inverse problems.
van Leijen, A.V.; Rothkrantz, L.; Groen, F.
2011-01-01
Swift solving of geoacoustic inverse problems strongly depends on the application of a global optimization scheme. Given a particular inverse problem, this work aims to answer the questions how to select an appropriate metaheuristic search strategy, and how to configure it for optimal performance.
Metaheuristic optimization of acoustic inverse problems.
van Leijen, A.V.; Rothkrantz, L.; Groen, F.
2011-01-01
Swift solving of geoacoustic inverse problems strongly depends on the application of a global optimization scheme. Given a particular inverse problem, this work aims to answer the questions how to select an appropriate metaheuristic search strategy, and how to configure it for optimal performance. F
Optimization and inverse problems in electromagnetism
Wiak, Sławomir
2003-01-01
From 12 to 14 September 2002, the Academy of Humanities and Economics (AHE) hosted the workshop "Optimization and Inverse Problems in Electromagnetism". After this bi-annual event, a large number of papers were assembled and combined in this book. During the workshop recent developments and applications in optimization and inverse methodologies for electromagnetic fields were discussed. The contributions selected for the present volume cover a wide spectrum of inverse and optimal electromagnetic methodologies, ranging from theoretical to practical applications. A number of new optimal and inverse methodologies were proposed. There are contributions related to dedicated software. Optimization and Inverse Problems in Electromagnetism consists of three thematic chapters, covering: -General papers (survey of specific aspects of optimization and inverse problems in electromagnetism), -Methodologies, -Industrial Applications. The book can be useful to students of electrical and electronics engineering, computer sci...
INVERSE COEFFICIENT PROBLEMS FOR PARABOLIC HEMIVARIATIONAL INEQUALITIES
Institute of Scientific and Technical Information of China (English)
Liu Zhenhai; I.Szántó
2011-01-01
This paper is devoted to the class of inverse problems for a nonlinear parabolic hemivariational inequality.The unknown coefficient of the operator depends on the gradient of the solution and belongs to a set of admissible coefficients.It is proved that the convergence of solutions for the corresponding direct problems continuously depends on the coefficient convergence.Based on this result the existence of a quasisolution of the inverse problem is obtained.
3rd Annual Workshop on Inverse Problem
2015-01-01
This proceeding volume is based on papers presented on the Third Annual Workshop on Inverse Problems which was organized by the Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, and took place in May 2013 in Stockholm. The purpose of this workshop was to present new analytical developments and numerical techniques for solution of inverse problems for a wide range of applications in acoustics, electromagnetics, optical fibers, medical imaging, geophysics, etc. The contributions in this volume reflect these themes and will be beneficial to researchers who are working in the area of applied inverse problems.
The inverse gravimetric problem in gravity modelling
Sanso, F.; Tscherning, C. C.
1989-01-01
One of the main purposes of geodesy is to determine the gravity field of the Earth in the space outside its physical surface. This purpose can be pursued without any particular knowledge of the internal density even if the exact shape of the physical surface of the Earth is not known, though this seems to entangle the two domains, as it was in the old Stoke's theory before the appearance of Molodensky's approach. Nevertheless, even when large, dense and homogeneous data sets are available, it was always recognized that subtracting from the gravity field the effect of the outer layer of the masses (topographic effect) yields a much smoother field. This is obviously more important when a sparse data set is bad so that any smoothing of the gravity field helps in interpolating between the data without raising the modeling error, this approach is generally followed because it has become very cheap in terms of computing time since the appearance of spectral techniques. The mathematical description of the Inverse Gravimetric Problem (IGP) is dominated mainly by two principles, which in loose terms can be formulated as follows: the knowledge of the external gravity field determines mainly the lateral variations of the density; and the deeper the density anomaly giving rise to a gravity anomaly, the more improperly posed is the problem of recovering the former from the latter. The statistical relation between rho and n (and its inverse) is also investigated in its general form, proving that degree cross-covariances have to be introduced to describe the behavior of rho. The problem of the simultaneous estimate of a spherical anomalous potential and of the external, topographic masses is addressed criticizing the choice of the mixed collection approach.
Analysis of nonlinear channel friction inverse problem
Institute of Scientific and Technical Information of China (English)
CHENG Weiping; LIU Guohua
2007-01-01
Based on the Backus-Gilbert inverse theory, the singular value decomposition (SVD) for general inverse matrices and the optimization algorithm are used to solve the channel friction inverse problem. The resolution and covari- ance friction inverse model in matrix form is developed to examine the reliability of solutions. Theoretical analyses demonstrate that the convergence rate of the general Newton optimization algorithm is in the second-order. The Wiggins method is also incorporated into the algorithm. Using the method, noise can be suppressed effectively, and the results are close to accurate solutions with proper control parameters. Also, the numerical stability can be improved.
An Inverse Problem Statistical Methodology Summary
2008-01-12
R. Vogel, Computational Methods for Inverse Problems, SIAM, Philadelphia, 2002. [36] D. D. Wackerly, W. Mendenhall III, and R. L. Scheaffer , Mathematical Statistics with Applications, Duxbury Thompson Learning, USA, 2002. 56
Inverse feasibility problems of the inverse maximum ﬂow problems
Indian Academy of Sciences (India)
Adrian Deaconu; Eleonor Ciurea
2013-04-01
A linear time method to decide if any inverse maximum ﬂow (denoted General Inverse Maximum Flow problems (IMFG)) problem has solution is deduced. If IMFG does not have solution, methods to transform IMFG into a feasible problem are presented. The methods consist of modifying as little as possible the restrictions to the variation of the bounds of the ﬂow. New inverse combinatorial optimization problems are introduced and solved.
An inverse problem in analytical dynamics
Institute of Scientific and Technical Information of China (English)
Li Guang-Cheng; Mei-Feng-Xiang
2006-01-01
This paper presents an inverse problem in analytical dynamics.The inverse problem is to construct the Lagrangian when the integrals of a system are given.Firstly,the differential equations are obtained by using the time derivative of the integrals.Secondly,the differential equations can be written in the Lagrange equations under certain conditions and the Lagrangian can be obtained.Finally,two examples are given to illustrate the application of the result.
The inverse maximum dynamic flow problem
Institute of Scientific and Technical Information of China (English)
BAGHERIAN; Mehri
2010-01-01
We consider the inverse maximum dynamic flow (IMDF) problem.IMDF problem can be described as: how to change the capacity vector of a dynamic network as little as possible so that a given feasible dynamic flow becomes a maximum dynamic flow.After discussing some characteristics of this problem,it is converted to a constrained minimum dynamic cut problem.Then an efficient algorithm which uses two maximum dynamic flow algorithms is proposed to solve the problem.
Aneesur Rahman Prize: The Inverse Ising Problem
Swendsen, Robert
2014-03-01
Many methods are available for carrying out computer simulations of a model Hamiltonian to obtain thermodynamic information by generating a set of configurations. The inverse problem consists of recreating the parameters of the Hamiltonian, given a set of configurations. The problem arises in a variety of contexts, and there has been much interest recently in the inverse Ising problem, in which the configurations consist of Ising spins. I will discuss an efficient method for solving the problem and what it can tell us about the Sherrington-Kirkpatrick model.
Inverse problem in Parker's dynamo
Reshetnyak, M Yu
2015-01-01
The inverse solution of the 1D Parker dynamo equations is considered. The method is based on minimization of the cost-function, which characterize deviation of the model solution properties from the desired ones. The output is the latitude distribution of the magnetic field generation sources: the $\\alpha$- and $\\omega$-effects. Minimization is made using the Monte-Carlo method. The details of the method, as well as some applications, which can be interesting for the broad dynamo community, are considered: conditions when the invisible for the observer at the surface of the planet toroidal part of the magnetic field is much larger than the poloidal counterpart. It is shown that at some particular distributions of $\\alpha$ and $\\omega$ the well-known thesis that sign of the dynamo-number defines equatorial symmetry of the magnetic field to the equator plane, is violated. It is also demonstrated in what circumstances magnetic field in the both hemispheres have different properties, and simple physical explanati...
Linear inverse problem of the reactor dynamics
Volkov, N. P.
2017-01-01
The aim of this work is the study transient processes in nuclear reactors. The mathematical model of the reactor dynamics excluding reverse thermal coupling is investigated. This model is described by a system of integral-differential equations, consisting of a non-stationary anisotropic multispeed kinetic transport equation and a delayed neutron balance equation. An inverse problem was formulated to determine the stationary part of the function source along with the solution of the direct problem. The author obtained sufficient conditions for the existence and uniqueness of a generalized solution of this inverse problem.
Solving Direct and Inverse Heat Conduction Problems
Taler, Jan
2006-01-01
Presents a solution for direct and inverse heat conduction problems. This work discusses the theoretical basis for the heat transfer process in the first part. It presents selected theoretical and numerical problems in the form of exercises with their subsequent solutions in the second part
Inverse Problems in a Bayesian Setting
Matthies, Hermann G.
2016-02-13
In a Bayesian setting, inverse problems and uncertainty quantification (UQ)—the propagation of uncertainty through a computational (forward) model—are strongly connected. In the form of conditional expectation the Bayesian update becomes computationally attractive. We give a detailed account of this approach via conditional approximation, various approximations, and the construction of filters. Together with a functional or spectral approach for the forward UQ there is no need for time-consuming and slowly convergent Monte Carlo sampling. The developed sampling-free non-linear Bayesian update in form of a filter is derived from the variational problem associated with conditional expectation. This formulation in general calls for further discretisation to make the computation possible, and we choose a polynomial approximation. After giving details on the actual computation in the framework of functional or spectral approximations, we demonstrate the workings of the algorithm on a number of examples of increasing complexity. At last, we compare the linear and nonlinear Bayesian update in form of a filter on some examples.
Applications of elliptic Carleman inequalities to Cauchy and inverse problems
Choulli, Mourad
2016-01-01
This book presents a unified approach to studying the stability of both elliptic Cauchy problems and selected inverse problems. Based on elementary Carleman inequalities, it establishes three-ball inequalities, which are the key to deriving logarithmic stability estimates for elliptic Cauchy problems and are also useful in proving stability estimates for certain elliptic inverse problems. The book presents three inverse problems, the first of which consists in determining the surface impedance of an obstacle from the far field pattern. The second problem investigates the detection of corrosion by electric measurement, while the third concerns the determination of an attenuation coefficient from internal data, which is motivated by a problem encountered in biomedical imaging.
Inverse Eigenvalue Problem in Structural Dynamics Design
Institute of Scientific and Technical Information of China (English)
Huiqing Xie; Hua Dai
2006-01-01
A kind of inverse eigenvalue problem in structural dynamics design is considered. The problem is formulated as an optimization problem. The properties of this problem are analyzed, and the existence of the optimum solution is proved. The directional derivative of the objective function is obtained and a necessary condition for a point to be a local minimum point is given. Then a numerical algorithm for solving the problem is presented and a plane-truss problem is discussed to show the applications of the theories and the algorithm.
Kozunov, Vladimir V; Ossadtchi, Alexei
2015-01-01
Although MEG/EEG signals are highly variable between subjects, they allow characterizing systematic changes of cortical activity in both space and time. Traditionally a two-step procedure is used. The first step is a transition from sensor to source space by the means of solving an ill-posed inverse problem for each subject individually. The second is mapping of cortical regions consistently active across subjects. In practice the first step often leads to a set of active cortical regions whose location and timecourses display a great amount of interindividual variability hindering the subsequent group analysis. We propose Group Analysis Leads to Accuracy (GALA)-a solution that combines the two steps into one. GALA takes advantage of individual variations of cortical geometry and sensor locations. It exploits the ensuing variability in electromagnetic forward model as a source of additional information. We assume that for different subjects functionally identical cortical regions are located in close proximity and partially overlap and their timecourses are correlated. This relaxed similarity constraint on the inverse solution can be expressed within a probabilistic framework, allowing for an iterative algorithm solving the inverse problem jointly for all subjects. A systematic simulation study showed that GALA, as compared with the standard min-norm approach, improves accuracy of true activity recovery, when accuracy is assessed both in terms of spatial proximity of the estimated and true activations and correct specification of spatial extent of the activated regions. This improvement obtained without using any noise normalization techniques for both solutions, preserved for a wide range of between-subject variations in both spatial and temporal features of regional activation. The corresponding activation timecourses exhibit significantly higher similarity across subjects. Similar results were obtained for a real MEG dataset of face-specific evoked responses.
Directory of Open Access Journals (Sweden)
Vladimir eKozunov
2015-04-01
Full Text Available Although MEG/EEG signals are highly variable between subjects, they allow characterizing systematic changes of cortical activity in both space and time. Traditionally a two-step procedure is used. The first step is a transition from sensor to source space by the means of solving an ill-posed inverse problem for each subject individually. The second is mapping of cortical regions consistently active across subjects. In practice the first step often leads to a set of active cortical regions whose location and timecourses display a great amount of interindividual variability hindering the subsequent group analysis.We propose Group Analysis Leads to Accuracy (GALA - a solution that combines the two steps into one. GALA takes advantage of individual variations of cortical geometry and sensor locations. It exploits the ensuing variability in electromagnetic forward model as a source of additional information. We assume that for different subjects functionally identical cortical regions are located in close proximity and partially overlap and their timecourses are correlated. This relaxed similarity constraint on the inverse solution can be expressed within a probabilistic framework, allowing for an iterative algorithm solving the inverse problem jointly for all subjects.A systematic simulation study showed that GALA, as compared with the standard min-norm approach, improves accuracy of true activity recovery, when accuracy is assessed both in terms of spatial proximity of the estimated and true activations and correct specification of spatial extent of the activated regions. This improvement obtained without using any noise normalization techniques for both solutions, preserved for a wide range of between-subject variations in both spatial and temporal features of regional activation. The corresponding activation timecourses exhibit significantly higher similarity across subjects. Similar results were obtained for a real MEG dataset of face
Iterative Reconstruction Methods for Hybrid Inverse Problems in Impedance Tomography
DEFF Research Database (Denmark)
Hoffmann, Kristoffer; Knudsen, Kim
2014-01-01
For a general formulation of hybrid inverse problems in impedance tomography the Picard and Newton iterative schemes are adapted and four iterative reconstruction algorithms are developed. The general problem formulation includes several existing hybrid imaging modalities such as current density...... impedance imaging, magnetic resonance electrical impedance tomography, and ultrasound modulated electrical impedance tomography, and the unified approach to the reconstruction problem encompasses several algorithms suggested in the literature. The four proposed algorithms are implemented numerically in two...... be based on a theoretical analysis of the underlying inverse problem....
INVERSE CENTER LOCATION PROBLEM ON A TREE
Institute of Scientific and Technical Information of China (English)
无
2008-01-01
This paper discusses the inverse center location problem restricted on a tree with different costs and bound constraints.The authors first show that the problem can be formulated as a series of combinatorial linear programs,then an O(|V|2 log |V|)time algorithm to solve the problem is presented.For the equal cost case,the authors further give an O(|V|)time algorithm.
TOPICAL REVIEW: Optical tomography: forward and inverse problems
Arridge, Simon R.; Schotland, John C.
2009-12-01
This is a review of recent mathematical and computational advances in optical tomography. We discuss the physical foundations of forward models for light propagation on microscopic, mesoscopic and macroscopic scales. We also consider direct and numerical approaches to the inverse problems that arise at each of these scales. Finally, we outline future directions and open problems in the field.
Direct Problems and Inverse Problems in Biometric Systems
Directory of Open Access Journals (Sweden)
Mihailescu Marius Iulian
2013-10-01
Full Text Available The article purpose is to describe the two sides of biometrics technologies, direct problems and inverse problems. The advance that we face today in field of Information Technology makes Information Security an inseparable part. The authentication has a huge role when we deal about security. The problems that can appear in implementing and developing biometrics systems is raising many problems, and one of the goal of this article is to focus on direct and inverse problems which is a new and challenging branch in biometrics technologies.
Energy Technology Data Exchange (ETDEWEB)
Sorrentino, A [Dipartimento di Fisica, Universita di Genova (Italy); Pascarella, A; Campi, C [Dipartimento di Matematica, Universita di Genova (Italy); Piana, M [Dipartimento di Informatica, Universita di Verona (Italy)], E-mail: sorrentino@fisica.unige.it
2008-07-15
We consider the problem of dynamically estimating the parameters of point-like neural sources from magnetoencephalography data. Since the problem is non-linear, we apply the sequential Monte Carlo algorithms known as particle filters for solving the Bayesian filtering problem. We suggest that the linear dependence of the data on a subset of the parameters allows the analytic computation of the posterior density for these parameters, i.e. Rao-Blackwellization; this considerably improves the accuracy of the method and its statistical efficiency.
The relativistic inverse stellar structure problem
Energy Technology Data Exchange (ETDEWEB)
Lindblom, Lee [Theoretical Astrophysics, California Institute of Technology, Pasadena, CA 91125 (United States)
2014-01-14
The observable macroscopic properties of relativistic stars (whose equations of state are known) can be predicted by solving the stellar structure equations that follow from Einstein’s equation. For neutron stars, however, our knowledge of the equation of state is poor, so the direct stellar structure problem can not be solved without modeling the highest density part of the equation of state in some way. This talk will describe recent work on developing a model independent approach to determining the high-density neutron-star equation of state by solving an inverse stellar structure problem. This method uses the fact that Einstein’s equation provides a deterministic relationship between the equation of state and the macroscopic observables of the stars which are composed of that material. This talk illustrates how this method will be able to determine the high-density part of the neutron-star equation of state with few percent accuracy when high quality measurements of the masses and radii of just two or three neutron stars become available. This talk will also show that this method can be used with measurements of other macroscopic observables, like the masses and tidal deformabilities, which can (in principle) be measured by gravitational wave observations of binary neutron-star mergers.
The Relativistic Inverse Stellar Structure Problem
Lindblom, Lee
2014-01-01
The observable macroscopic properties of relativistic stars (whose equations of state are known) can be predicted by solving the stellar structure equations that follow from Einstein's equation. For neutron stars, however, our knowledge of the equation of state is poor, so the direct stellar structure problem can not be solved without modeling the highest density part of the equation of state in some way. This talk will describe recent work on developing a model independent approach to determining the high-density neutron-star equation of state by solving an inverse stellar structure problem. This method uses the fact that Einstein's equation provides a deterministic relationship between the equation of state and the macroscopic observables of the stars which are composed of that material. This talk illustrates how this method will be able to determine the high-density part of the neutron-star equation of state with few percent accuracy when high quality measurements of the masses and radii of just two or thr...
Introduction to inverse problems for differential equations
Hasanov Hasanoğlu, Alemdar
2017-01-01
This book presents a systematic exposition of the main ideas and methods in treating inverse problems for PDEs arising in basic mathematical models, though it makes no claim to being exhaustive. Mathematical models of most physical phenomena are governed by initial and boundary value problems for PDEs, and inverse problems governed by these equations arise naturally in nearly all branches of science and engineering. The book’s content, especially in the Introduction and Part I, is self-contained and is intended to also be accessible for beginning graduate students, whose mathematical background includes only basic courses in advanced calculus, PDEs and functional analysis. Further, the book can be used as the backbone for a lecture course on inverse and ill-posed problems for partial differential equations. In turn, the second part of the book consists of six nearly-independent chapters. The choice of these chapters was motivated by the fact that the inverse coefficient and source problems considered here a...
Dark Energy as an Inverse Problem
Espana-Bonet, C; Espana-Bonet, Cristina; Ruiz-Lapuente, Pilar
2005-01-01
A model--independent approach to dark energy is here developed by considering the determination of its equation of state as an inverse problem. The reconstruction of w(z) as a non--parametric function using the current SNe Ia data is explored. It is investigated as well how results would improve when considering other samples of cosmic distance indicators at higher redshift. This approach reveals the lack of information in the present samples to conclude on the behavior of w(z) at z > 0.6. At low level of significance a preference is found for w_{0} 0 at z ~ 0.2--0.3. The solution of w(z) along redshift never departs more than 1.95\\sigma from the cosmological constant w(z)=-1, and this only occurs when using various cosmic distance indicators. The determination of w(z) as a function is readdressed considering samples of large number of SNe Ia as those to be provided by SNAP. It is found an improvement in the resolution of w(z) when using those synthetic samples, which is favored by adding data at very high z...
Inverse acoustic problem of N homogeneous scatterers
DEFF Research Database (Denmark)
Berntsen, Svend
2002-01-01
The three-dimensional inverse acoustic medium problem of N homogeneous objects with known geometry and location is considered. It is proven that one scattering experiment is sufficient for the unique determination of the complex wavenumbers of the objects. The mapping from the scattered fields...
Inverse scattering problem in turbulent magnetic fluctuations
Treumann, R A; Narita, Y
2016-01-01
We apply a particular form of the inverse scattering theory to turbulent magnetic fluctuations in a plasma. In the present note we develop the theory, formulate the magnetic fluctuation problem in terms of its electrodynamic turbulent response function, and reduce it to the solution of a special form of the famous Gel$'$fand-Levitan-Marchenko equation of quantum mechanical scattering theory.
The inverse problem of bioelectricity: an evaluation
Oosterom, A. van
2012-01-01
This invited paper presents a personal view on the current status of the solution to the inverse problem of bioelectricity. Its focus lies on applications in the field of electrocardiography. The topic discussed is also relevant in other medical domains, such as electroencephalography, electroneurog
Direct and inverse problems of infrared tomography
DEFF Research Database (Denmark)
Sizikov, Valery S.; Evseev, Vadim; Fateev, Alexander
2016-01-01
The problems of infrared tomography-direct (the modeling of measured functions) and inverse (the reconstruction of gaseous medium parameters)-are considered with a laboratory burner flame as an example of an application. The two measurement modes are used: active (ON) with an external IR source...
General inverse problems for regular variation
DEFF Research Database (Denmark)
Damek, Ewa; Mikosch, Thomas Valentin; Rosinski, Jan
2014-01-01
Regular variation of distributional tails is known to be preserved by various linear transformations of some random structures. An inverse problem for regular variation aims at understanding whether the regular variation of a transformed random object is caused by regular variation of components ...
On the Stewart-Lyth Inverse Problem
Ayón-Beato, E; Mansilla, R; Terrero-Escalante, C A; Ay\\'on-Beato, Eloy; Garc\\'{\\i}a, Alberto; Mansilla, Ricardo
2000-01-01
In this paper the Stewart-Lyth inverse problem is rewritten using the comoving scales as the basic parameter. It is shown that some information on the inflaton potential can be obtained from observations taking into account only the scalar power spectrum.
Inverse acoustic problem of N homogeneous scatterers
DEFF Research Database (Denmark)
Berntsen, Svend
2002-01-01
The three-dimensional inverse acoustic medium problem of N homogeneous objects with known geometry and location is considered. It is proven that one scattering experiment is sufficient for the unique determination of the complex wavenumbers of the objects. The mapping from the scattered fields...
Numerical linear algebra for reconstruction inverse problems
Nachaoui, Abdeljalil
2004-01-01
Our goal in this paper is to discuss various issues we have encountered in trying to find and implement efficient solvers for a boundary integral equation (BIE) formulation of an iterative method for solving a reconstruction problem. We survey some methods from numerical linear algebra, which are relevant for the solution of this class of inverse problems. We motivate the use of our constructing algorithm, discuss its implementation and mention the use of preconditioned Krylov methods.
PUBLISHER'S ANNOUNCEMENT: New developments for Inverse Problems
2006-12-01
2006 has proved to be a very successful year for Inverse Problems. After an increase for the fourth successive year, we achieved our highest impact factor to date, 1.541 (Source: 2005 ISI® Journal Citation Report), and the Editorial Board is keen to build on this success by continuing to improve the service we offer to our readers and authors. The Board has observed that Inverse Problems receives very few Letters to the Editor submissions, and that moreover those that we do receive rarely conform to the requirements for Letters to the Editor set out in the journal's editorial policy. The Board has therefore decided to merge the current Letters to the Editor section into our regular Papers section, which will now accommodate all research articles that meet the journal's high quality standards. Any submissions that would previously have been Letters to the Editor are still very welcome as Papers, and can be submitted by e-mail to ip@iop.org or online using our online submissions form at authors.iop.org/submit. Inverse Problems' processing times are already among the fastest in the field—on average, authors receive our decision on their paper in less than three months. Thanks to our easy-to-use online refereeing system, publishing a Paper is now just as fast as publishing a Letter to the Editor, and we are striving to ensure that the journal's high standards are applied consistently to all our Papers, maintaining Inverse Problems' position as the leading journal in the field. Our highly acclaimed Topical Review section will also continue and grow; providing timely insights into the development of all topical fields within Inverse Problems. We have many exciting Topical Reviews currently in preparation for 2007 and will continue to commission articles at the cutting edge of research. We look forward to receiving your contributions and to continuing to provide the best publication service available.
TOPICAL REVIEW: Inverse problems in elasticity
Bonnet, Marc; Constantinescu, Andrei
2005-04-01
This review is devoted to some inverse problems arising in the context of linear elasticity, namely the identification of distributions of elastic moduli, model parameters or buried objects such as cracks. These inverse problems are considered mainly for three-dimensional elastic media under equilibrium or dynamical conditions, and also for thin elastic plates. The main goal is to overview some recent results, in an effort to bridge the gap between studies of a mathematical nature and problems defined from engineering practice. Accordingly, emphasis is given to formulations and solution techniques which are well suited to general-purpose numerical methods for solving elasticity problems on complex configurations, in particular the finite element method and the boundary element method. An underlying thread of the discussion is the fact that useful tools for the formulation, analysis and solution of inverse problems arising in linear elasticity, namely the reciprocity gap and the error in constitutive equation, stem from variational and virtual work principles, i.e., fundamental principles governing the mechanics of deformable solid continua. In addition, the virtual work principle is shown to be instrumental for establishing computationally efficient formulae for parameter or geometrical sensitivity, based on the adjoint solution method. Sensitivity formulae are presented for various situations, especially in connection with contact mechanics, cavity and crack shape perturbations, thus enriching the already extensive known repertoire of such results. Finally, the concept of topological derivative and its implementation for the identification of cavities or inclusions are expounded.
Compressed word problems for inverse monoids
Lohrey, Markus
2011-01-01
The compressed word problem for a finitely generated monoid M asks whether two given compressed words over the generators of M represent the same element of M. For string compression, straight-line programs, i.e., context-free grammars that generate a single string, are used in this paper. It is shown that the compressed word problem for a free inverse monoid of finite rank at least two is complete for Pi^p_2 (second universal level of the polynomial time hierarchy). Moreover, it is shown that there exists a fixed finite idempotent presentation (i.e., a finite set of relations involving idempotents of a free inverse monoid), for which the corresponding quotient monoid has a PSPACE-complete compressed word problem. It was shown previously that the ordinary uncompressed word problem for such a quotient can be solved in logspace. Finally, a PSPACE-algorithm that checks whether a given element of a free inverse monoid belongs to a given rational subset is presented. This problem is also shown to be PSPACE-complet...
Inverse problems in ordinary differential equations and applications
Llibre, Jaume
2016-01-01
This book is dedicated to study the inverse problem of ordinary differential equations, that is it focuses in finding all ordinary differential equations that satisfy a given set of properties. The Nambu bracket is the central tool in developing this approach. The authors start characterizing the ordinary differential equations in R^N which have a given set of partial integrals or first integrals. The results obtained are applied first to planar polynomial differential systems with a given set of such integrals, second to solve the 16th Hilbert problem restricted to generic algebraic limit cycles, third for solving the inverse problem for constrained Lagrangian and Hamiltonian mechanical systems, fourth for studying the integrability of a constrained rigid body. Finally the authors conclude with an analysis on nonholonomic mechanics, a generalization of the Hamiltonian principle, and the statement an solution of the inverse problem in vakonomic mechanics.
The inverse variational problem in classical mechanics
Lopuszánski, Jan T
1999-01-01
This book provides a concise description of the current status of a fascinating scientific problem - the inverse variational problem in classical mechanics. The essence of this problem is as follows: one is given a set of equations of motion describing a certain classical mechanical system, and the question to be answered is: Do these equations of motion correspond to some Lagrange function as its Euler-Lagrange equations? In general, not for every system of equations of motion does a Lagrange function exist; it can, however, happen that one may modify the given equations of motion in such a w
Inverse problems biomechanical imaging (Conference Presentation)
Oberai, Assad A.
2016-03-01
It is now well recognized that a host of imaging modalities (a list that includes Ultrasound, MRI, Optical Coherence Tomography, and optical microscopy) can be used to "watch" tissue as it deforms in response to an internal or external excitation. The result is a detailed map of the deformation field in the interior of the tissue. This deformation field can be used in conjunction with a material mechanical response to determine the spatial distribution of material properties of the tissue by solving an inverse problem. Images of material properties thus obtained can be used to quantify the health of the tissue. Recently, they have been used to detect, diagnose and monitor cancerous lesions, detect vulnerable plaque in arteries, diagnose liver cirrhosis, and possibly detect the onset of Alzheimer's disease. In this talk I will describe the mathematical and computational aspects of solving this class of inverse problems, and their applications in biology and medicine. In particular, I will discuss the well-posedness of these problems and quantify the amount of displacement data necessary to obtain a unique property distribution. I will describe an efficient algorithm for solving the resulting inverse problem. I will also describe some recent developments based on Bayesian inference in estimating the variance in the estimates of material properties. I will conclude with the applications of these techniques in diagnosing breast cancer and in characterizing the mechanical properties of cells with sub-cellular resolution.
Solving probabilistic inverse problems rapidly with prior samples
Käufl, Paul; Valentine, Andrew P.; de Wit, Ralph W.; Trampert, Jeannot
2016-01-01
Owing to the increasing availability of computational resources, in recent years the probabilistic solution of non-linear, geophysical inverse problems by means of sampling methods has become increasingly feasible. Nevertheless, we still face situations in which a Monte Carlo approach is not
Solving probabilistic inverse problems rapidly with prior samples
Käufl, Paul; Valentine, Andrew P.|info:eu-repo/dai/nl/364418680; de Wit, Ralph W.|info:eu-repo/dai/nl/344668908; Trampert, Jeannot|info:eu-repo/dai/nl/304829250
2016-01-01
Owing to the increasing availability of computational resources, in recent years the probabilistic solution of non-linear, geophysical inverse problems by means of sampling methods has become increasingly feasible. Nevertheless, we still face situations in which a Monte Carlo approach is not practic
Variational Bayesian Approximation methods for inverse problems
Mohammad-Djafari, Ali
2012-09-01
Variational Bayesian Approximation (VBA) methods are recent tools for effective Bayesian computations. In this paper, these tools are used for inverse problems where the prior models include hidden variables and where where the estimation of the hyper parameters has also to be addressed. In particular two specific prior models (Student-t and mixture of Gaussian models) are considered and details of the algorithms are given.
The Stewart-Lyth Inverse Problem
Ayón-Beato, E; Mansilla, R; Terrero-Escalante, C A; Ay\\'on-Beato, Eloy; Garc\\'{\\i}a, Alberto; Mansilla, Ricardo
2000-01-01
In this paper the Stewart-Lyth inverse problem is introduced. It consists of solving two non-linear differential equations for the first slow-roll parameter and finding the inflaton potential. The equations are derived from the Stewart-Lyth equations for the scalar and tensorial perturbations produced during the inflationary period. The geometry of the phase planes transverse to the trajectories is analyzed, and conclusions about the possible behaviour for general solutions are drawn.
INVERSE SCATTERING PROBLEMS BY SINGULAR SOURCE METHODS
Institute of Scientific and Technical Information of China (English)
无
2005-01-01
The inverse scattering problems are to detect the property of obstacles from the measurements outside the obstacles. One of important research areas in this topic is the recovery of boundary property for impenetrable obstacles. In this paper, we would like to give a brief review about the recently developed singular source methods. There are three different methods in this category, namely, linear sampling method, pointsource method and probe method. We also present some recent new results about the probe method.
Inverse scattering problem for quantum graph vertices
Cheon, Taksu; Turek, Ondrej
2011-01-01
We demonstrate how the inverse scattering problem of a quantum star graph can be solved by means of diagonalization of Hermitian unitary matrix when the vertex coupling is of the scale invariant (or F\\"ul\\H{o}p-Tsutsui) form. This enables the construction of quantum graphs with desired properties in a tailor-made fashion. The procedure is illustrated on the example of quantum vertices with equal transmission probabilities.
Differential equations inverse and direct problems
Favini, Angelo
2006-01-01
DEGENERATE FIRST ORDER IDENTIFICATION PROBLEMS IN BANACH SPACES A NONISOTHERMAL DYNAMICAL GINZBURG-LANDAU MODEL OF SUPERCONDUCTIVITY. EXISTENCE AND UNIQUENESS THEOREMSSOME GLOBAL IN TIME RESULTS FOR INTEGRODIFFERENTIAL PARABOLIC INVERSE PROBLEMSFOURTH ORDER ORDINARY DIFFERENTIAL OPERATORS WITH GENERAL WENTZELL BOUNDARY CONDITIONSTUDY OF ELLIPTIC DIFFERENTIAL EQUATIONS IN UMD SPACESDEGENERATE INTEGRODIFFERENTIAL EQUATIONS OF PARABOLIC TYPE EXPONENTIAL ATTRACTORS FOR SEMICONDUCTOR EQUATIONSCONVERGENCE TO STATIONARY STATES OF SOLUTIONS TO THE SEMILINEAR EQUATION OF VISCOELASTICITY ASYMPTOTIC BEHA
Voltammetry: mathematical modelling and Inverse Problem
Koshev, N A; Kuzina, V V
2016-01-01
We propose the fast semi-analytical method of modelling the polarization curves in the voltammetric experiment. The method is based on usage of the special func- tions and shows a big calculation speed and a high accuracy and stability. Low computational needs of the proposed algorithm allow us to state the set of Inverse Problems of voltammetry for the reconstruction of metal ions concentrations or the other parameters of the electrolyte under investigation.
About some inverse problems of nuclear physics
Belashev, B Z
2002-01-01
Some inverse problems of high energy physics and NMR spectroscopy are observed. The methods of the Fourier transformation and the maximum entropy technique have been applied for their solutions. The integral images of the experimental distributions are informative for determination of the space-time characteristics of the particles generation domain and for the analysis of blurring spectra. These methods have been tested in comparison with the results which have been obtained independently
Homometric Point Sets and Inverse Problems
Grimm, Uwe
2008-01-01
The inverse problem of diffraction theory in essence amounts to the reconstruction of the atomic positions of a solid from its diffraction image. From a mathematical perspective, this is a notoriously difficult problem, even in the idealised situation of perfect diffraction from an infinite structure. Here, the problem is analysed via the autocorrelation measure of the underlying point set, where two point sets are called homometric when they share the same autocorrelation. For the class of mathematical quasicrystals within a given cut and project scheme, the homometry problem becomes equivalent to Matheron's covariogram problem, in the sense of determining the window from its covariogram. Although certain uniqueness results are known for convex windows, interesting examples of distinct homometric model sets already emerge in the plane. The uncertainty level increases in the presence of diffuse scattering. Already in one dimension, a mixed spectrum can be compatible with structures of different entropy. We ex...
Bayesian approach to inverse statistical mechanics.
Habeck, Michael
2014-05-01
Inverse statistical mechanics aims to determine particle interactions from ensemble properties. This article looks at this inverse problem from a Bayesian perspective and discusses several statistical estimators to solve it. In addition, a sequential Monte Carlo algorithm is proposed that draws the interaction parameters from their posterior probability distribution. The posterior probability involves an intractable partition function that is estimated along with the interactions. The method is illustrated for inverse problems of varying complexity, including the estimation of a temperature, the inverse Ising problem, maximum entropy fitting, and the reconstruction of molecular interaction potentials.
Aydin, Umit; Dogrusoz, Yesim Serinagaoglu
2011-09-01
In this article, we aimed to reduce the effects of geometric errors and measurement noise on the inverse problem of Electrocardiography (ECG) solutions. We used the Kalman filter to solve the inverse problem in terms of epicardial potential distributions. The geometric errors were introduced into the problem via wrong determination of the size and location of the heart in simulations. An error model, which is called the enhanced error model (EEM), was modified to be used in inverse problem of ECG to compensate for the geometric errors. In this model, the geometric errors are modeled as additive Gaussian noise and their noise variance is added to the measurement noise variance. The Kalman filter method includes a process noise component, whose variance should also be estimated along with the measurement noise. To estimate these two noise variances, two different algorithms were used: (1) an algorithm based on residuals, (2) expectation maximization algorithm. The results showed that it is important to use the correct noise variances to obtain accurate results. The geometric errors, if ignored in the inverse solution procedure, yielded incorrect epicardial potential distributions. However, even with a noise model as simple as the EEM, the solutions could be significantly improved.
An Entropic Estimator for Linear Inverse Problems
Directory of Open Access Journals (Sweden)
Amos Golan
2012-05-01
Full Text Available In this paper we examine an Information-Theoretic method for solving noisy linear inverse estimation problems which encompasses under a single framework a whole class of estimation methods. Under this framework, the prior information about the unknown parameters (when such information exists, and constraints on the parameters can be incorporated in the statement of the problem. The method builds on the basics of the maximum entropy principle and consists of transforming the original problem into an estimation of a probability density on an appropriate space naturally associated with the statement of the problem. This estimation method is generic in the sense that it provides a framework for analyzing non-normal models, it is easy to implement and is suitable for all types of inverse problems such as small and or ill-conditioned, noisy data. First order approximation, large sample properties and convergence in distribution are developed as well. Analytical examples, statistics for model comparisons and evaluations, that are inherent to this method, are discussed and complemented with explicit examples.
PREFACE: International Conference on Inverse Problems 2010
Hon, Yiu-Chung; Ling, Leevan
2011-03-01
Following the first International Conference on Inverse Problems - Recent Theoretical Development and Numerical Approaches held at the City University of Hong Kong in 2002, the fifth International Conference was held again at the City University during December 13-17, 2010. This fifth conference was jointly organized by Professor Yiu-Chung Hon (Co-Chair, City University of Hong Kong, HKSAR), Dr Leevan Ling (Co-Chair, Hong Kong Baptist University, HKSAR), Professor Jin Cheng (Fudan University, China), Professor June-Yub Lee (Ewha Womans University, South Korea), Professor Gui-Rong Liu (University of Cincinnati, USA), Professor Jenn-Nan Wang (National Taiwan University, Taiwan), and Professor Masahiro Yamamoto (The University of Tokyo, Japan). It was agreed to alternate holding the conference among the above places (China, Japan, Korea, Taiwan, and Hong Kong) once every two years. The next conference has been scheduled to be held at the Southeast University (Nanjing, China) in 2012. The purpose of this series of conferences is to establish a strong collaborative link among the universities of the Asian-Pacific regions and worldwide leading researchers in inverse problems. The conference addressed both theoretical (mathematics), applied (engineering) and developmental aspects of inverse problems. The conference was intended to nurture Asian-American-European collaborations in the evolving interdisciplinary areas and it was envisioned that the conference would lead to long-term commitments and collaborations among the participating countries and researchers. There was a total of more than 100 participants. A call for the submission of papers was sent out after the conference, and a total of 19 papers were finally accepted for publication in this proceedings. The papers included in the proceedings cover a wide scope, which reflects the current flourishing theoretical and numerical research into inverse problems. Finally, as the co-chairs of the Inverse Problems
Bayesian Inference Applied to the Electromagnetic Inverse Problem
Schmidt, D M; Wood, C C; Schmidt, David M.; George, John S.
1998-01-01
We present a new approach to the electromagnetic inverse problem that explicitly addresses the ambiguity associated with its ill-posed character. Rather than calculating a single ``best'' solution according to some criterion, our approach produces a large number of likely solutions that both fit the data and any prior information that is used. While the range of the different likely results is representative of the ambiguity in the inverse problem even with prior information present, features that are common across a large number of the different solutions can be identified and are associated with a high degree of probability. This approach is implemented and quantified within the formalism of Bayesian inference which combines prior information with that from measurement in a common framework using a single measure. To demonstrate this approach, a general neural activation model is constructed that includes a variable number of extended regions of activation and can incorporate a great deal of prior informati...
Deep Convolutional Neural Network for Inverse Problems in Imaging
Jin, Kyong Hwan; McCann, Michael T.; Froustey, Emmanuel; Unser, Michael
2017-09-01
In this paper, we propose a novel deep convolutional neural network (CNN)-based algorithm for solving ill-posed inverse problems. Regularized iterative algorithms have emerged as the standard approach to ill-posed inverse problems in the past few decades. These methods produce excellent results, but can be challenging to deploy in practice due to factors including the high computational cost of the forward and adjoint operators and the difficulty of hyper parameter selection. The starting point of our work is the observation that unrolled iterative methods have the form of a CNN (filtering followed by point-wise non-linearity) when the normal operator (H*H, the adjoint of H times H) of the forward model is a convolution. Based on this observation, we propose using direct inversion followed by a CNN to solve normal-convolutional inverse problems. The direct inversion encapsulates the physical model of the system, but leads to artifacts when the problem is ill-posed; the CNN combines multiresolution decomposition and residual learning in order to learn to remove these artifacts while preserving image structure. We demonstrate the performance of the proposed network in sparse-view reconstruction (down to 50 views) on parallel beam X-ray computed tomography in synthetic phantoms as well as in real experimental sinograms. The proposed network outperforms total variation-regularized iterative reconstruction for the more realistic phantoms and requires less than a second to reconstruct a 512 x 512 image on GPU.
THE INVERSE PROBLEM FOR BOOLEAN EQUATIONS
Directory of Open Access Journals (Sweden)
Hussain Mobarak Albarakati
2012-01-01
Full Text Available The Forward Problem (FB of Boolean equations consists of finding solutions of a system of Boolean equations, or equivalently, a single Boolean equation of the form f(X = 0 where f(X: Bn â B and B is an arbitrary Boolean algebra. By contrast, the Inverse Problem (IB of Boolean equations aims to reconstruct the equation f (X = 0 given the set of solutions and hence to verify the correctness of this set. This study derives methods that handle this inverse problem for the main types of solutions of Boolean equations. These include: (a Subsumptive general solutions, in which each of the variables is expressed as an interval by deriving successive conjunctive or disjunctive eliminants of the original function, (b Parametric general solutions, in which each of the variables is expressed via arbitrary parameters which are freely chosen elements of the underlying Boolean algebra and (c Particular solutions, each of which is an assignment from the underlying Boolean algebra to every pertinent variable that makes the Boolean equation an identity. The reconstructed function f(X in every case is set in a canonical form, such as the complete-sum form, to facilitate proving its equivalence to the original function. The methods presented herein are demonstrated with carefully-chosen illustrative examples over big Boolean algebras of various sizes. Among the methods utilized in handling the inverse problem for Boolean equations, the ones utilizing the variable-entered Karnaugh map offered pictorial insight and exhibited an efficient divide-and-conquer strategy.
Inverse scattering problem in turbulent magnetic fluctuations
Treumann, Rudolf A.; Baumjohann, Wolfgang; Narita, Yasuhito
2016-08-01
We apply a particular form of the inverse scattering theory to turbulent magnetic fluctuations in a plasma. In the present note we develop the theory, formulate the magnetic fluctuation problem in terms of its electrodynamic turbulent response function, and reduce it to the solution of a special form of the famous Gelfand-Levitan-Marchenko equation of quantum mechanical scattering theory. The last of these applies to transmission and reflection in an active medium. The theory of turbulent magnetic fluctuations does not refer to such quantities. It requires a somewhat different formulation. We reduce the theory to the measurement of the low-frequency electromagnetic fluctuation spectrum, which is not the turbulent spectral energy density. The inverse theory in this form enables obtaining information about the turbulent response function of the medium. The dynamic causes of the electromagnetic fluctuations are implicit to it. Thus, it is of vital interest in low-frequency magnetic turbulence. The theory is developed until presentation of the equations in applicable form to observations of turbulent electromagnetic fluctuations as input from measurements. Solution of the final integral equation should be done by standard numerical methods based on iteration. We point to the possibility of treating power law fluctuation spectra as an example. Formulation of the problem to include observations of spectral power densities in turbulence is not attempted. This leads to severe mathematical problems and requires a reformulation of inverse scattering theory. One particular aspect of the present inverse theory of turbulent fluctuations is that its structure naturally leads to spatial information which is obtained from the temporal information that is inherent to the observation of time series. The Taylor assumption is not needed here. This is a consequence of Maxwell's equations, which couple space and time evolution. The inversion procedure takes advantage of a particular
Prior Information in Inverse Boundary Problems
DEFF Research Database (Denmark)
Garde, Henrik
This thesis gives a threefold perspective on the inverse problem of inclusion detection in electrical impedance tomography: depth dependence, monotonicitybased reconstruction, and sparsity-based reconstruction. The depth dependence is given in terms of explicit bounds on the datum norm, which shows...... into how much noise that can be allowed in the datum before an inclusion cannot be detected. The monotonicity method is a direct reconstruction method that utilizes a monotonicity property of the forward problem in order to characterize the inclusions. Here we rigorously prove that the method can...... of the method. Sparsity-based reconstruction is an iterative method, that through an optimization problem with a sparsity prior, approximates the inhomogeneities. Here we make use of prior information, that can cheaply be obtained from the monotonicity method, to improve both the contrast and resolution...
The Inverse Problem for the Dipole Field
Epp, V
2015-01-01
The Inverse problem for an electromagnetic field produced by a dipole is solved. It is assumed that the field of an arbitrary changing dipole is known. Obtained formulae allow calculation of the position and dynamics of the dipole which produces the measured field. The derived results can be used in investigations on radiative process in solids caused by changing of the charge distribution. For example, generation of the electromagnetic field caused by oscillations of atoms or electron gas at the trace of a particle channeling in a crystal, or fields arising at solids cracking or dislocation formation -- in any case when one is interested in the details of the dipole field source.
Diffuse interface methods for inverse problems: case study for an elliptic Cauchy problem
Burger, Martin; Løseth Elvetun, Ole; Schlottbom, Matthias
2015-12-01
Many inverse problems have to deal with complex, evolving and often not exactly known geometries, e.g. as domains of forward problems modeled by partial differential equations. This makes it desirable to use methods which are robust with respect to perturbed or not well resolved domains, and which allow for efficient discretizations not resolving any fine detail of those geometries. For forward problems in partial differential equations methods based on diffuse interface representations have gained strong attention in the last years, but so far they have not been considered systematically for inverse problems. In this work we introduce a diffuse domain method as a tool for the solution of variational inverse problems. As a particular example we study ECG inversion in further detail. ECG inversion is a linear inverse source problem with boundary measurements governed by an anisotropic diffusion equation, which naturally cries for solutions under changing geometries, namely the beating heart. We formulate a regularization strategy using Tikhonov regularization and, using standard source conditions, we prove convergence rates. A special property of our approach is that not only operator perturbations are introduced by the diffuse domain method, but more important we have to deal with topologies which depend on a parameter \\varepsilon in the diffuse domain method, i.e. we have to deal with \\varepsilon -dependent forward operators and \\varepsilon -dependent norms. In particular the appropriate function spaces for the unknown and the data depend on \\varepsilon . This prevents the application of some standard convergence techniques for inverse problems, in particular interpreting the perturbations as data errors in the original problem does not yield suitable results. We consequently develop a novel approach based on saddle-point problems. The numerical solution of the problem is discussed as well and results for several computational experiments are reported. In
Microlocal analysis of a seismic linearized inverse problem
Stolk, C.C.
2001-01-01
The seismic inverse problem is to determine the wavespeed c x in the interior of a medium from measurements at the boundary In this paper we analyze the linearized inverse problem in general acoustic media The problem is to nd a left inverse of the linearized forward map F or equivalently to nd the
Generalized Inverse Eigenvalue Problem for Centrohermitian Matrices
Institute of Scientific and Technical Information of China (English)
刘仲云; 谭艳祥; 田兆录
2004-01-01
In this paper we first consider the existence and the general form of solution to the following generalized inverse eigenvalue problem(GIEP) : given a set of n-dimension complex vectors { xj }jm = 1 and a set of complex numbers { λj} jm = 1, find two n × n centrohermitian matrices A, B such that { xj }jm = 1 and { λj }jm= 1 are the generalized eigenvectors and generalized eigenvalues of Ax = λBx, respectively. We then discuss the optimal approximation problem for the GIEP. More concretely, given two arbitrary matrices, A-, B- ∈Cn×n , we find two matrices A* and B* such that the matrix (A* ,B* ) is closest to (A- ,B-) in the Frobenius norm, where the matrix (A*, B* ) is the solution to the GIEP. We show that the expression of the solution of the optimal approximation is unique and derive the expression for it.
An inverse problem by boundary element method
Energy Technology Data Exchange (ETDEWEB)
Tran-Cong, T.; Nguyen-Thien, T. [University of Southern Queensland, Toowoomba, QLD (Australia); Graham, A.L. [Los Alamos National Lab., NM (United States)
1996-02-01
Boundary Element Methods (BEM) have been established as useful and powerful tools in a wide range of engineering applications, e.g. Brebbia et al. In this paper, we report a particular three dimensional implementation of a direct boundary integral equation (BIE) formulation and its application to numerical simulations of practical polymer processing operations. In particular, we will focus on the application of the present boundary element technology to simulate an inverse problem in plastics processing.by extrusion. The task is to design profile extrusion dies for plastics. The problem is highly non-linear due to material viscoelastic behaviours as well as unknown free surface conditions. As an example, the technique is shown to be effective in obtaining the die profiles corresponding to a square viscoelastic extrudate under different processing conditions. To further illustrate the capability of the method, examples of other non-trivial extrudate profiles and processing conditions are also given.
Inverse problems for partial differential equations
Isakov, Victor
2017-01-01
This third edition expands upon the earlier edition by adding nearly 40 pages of new material reflecting the analytical and numerical progress in inverse problems in last 10 years. As in the second edition, the emphasis is on new ideas and methods rather than technical improvements. These new ideas include use of the stationary phase method in the two-dimensional elliptic problems and of multi frequencies\\temporal data to improve stability and numerical resolution. There are also numerous corrections and improvements of the exposition throughout. This book is intended for mathematicians working with partial differential equations and their applications, physicists, geophysicists, and financial, electrical, and mechanical engineers involved with nondestructive evaluation, seismic exploration, remote sensing, and various kinds of tomography. Review of the second edition: "The first edition of this excellent book appeared in 1998 and became a standard reference for everyone interested in analysis and numerics of...
Data quality for the inverse lsing problem
Decelle, Aurélien; Ricci-Tersenghi, Federico; Zhang, Pan
2016-09-01
There are many methods proposed for inferring parameters of the Ising model from given data, that is a set of configurations generated according to the model itself. However little attention has been paid until now to the data, e.g. how the data is generated, whether the inference error using one set of data could be smaller than using another set of data, etc. In this paper we discuss the data quality problem in the inverse Ising problem, using as a benchmark the kinetic Ising model. We quantify the quality of data using effective rank of the correlation matrix, and show that data gathered in a out-of-equilibrium regime has a better quality than data gathered in equilibrium for coupling reconstruction. We also propose a matrix-perturbation based method for tuning the quality of given data and for removing bad-quality (i.e. redundant) configurations from data.
Inverse Variational Problem for Nonstandard Lagrangians
Saha, A.; Talukdar, B.
2014-06-01
In the mathematical physics literature the nonstandard Lagrangians (NSLs) were introduced in an ad hoc fashion rather than being derived from the solution of the inverse problem of variational calculus. We begin with the first integral of the equation of motion and solve the associated inverse problem to obtain some of the existing results for NSLs. In addition, we provide a number of alternative Lagrangian representations. The case studies envisaged by us include (i) the usual modified Emden-type equation, (ii) Emden-type equation with dissipative term quadratic in velocity, (iii) Lotka-Volterra model and (vi) a number of the generic equations for dissipative-like dynamical systems. Our method works for nonstandard Lagrangians corresponding to the usual action integral of mechanical systems but requires modification for those associated with the modified actions like S =∫abe L(x ,x˙ , t) dt and S =∫abL 1 - γ(x ,x˙ , t) dt because in the latter case one cannot construct expressions for the Jacobi integrals.
Energy Technology Data Exchange (ETDEWEB)
Kılıç, Emre, E-mail: emre.kilic@tum.de; Eibert, Thomas F.
2015-05-01
An approach combining boundary integral and finite element methods is introduced for the solution of three-dimensional inverse electromagnetic medium scattering problems. Based on the equivalence principle, unknown equivalent electric and magnetic surface current densities on a closed surface are utilized to decompose the inverse medium problem into two parts: a linear radiation problem and a nonlinear cavity problem. The first problem is formulated by a boundary integral equation, the computational burden of which is reduced by employing the multilevel fast multipole method (MLFMM). Reconstructed Cauchy data on the surface allows the utilization of the Lorentz reciprocity and the Poynting's theorems. Exploiting these theorems, the noise level and an initial guess are estimated for the cavity problem. Moreover, it is possible to determine whether the material is lossy or not. In the second problem, the estimated surface currents form inhomogeneous boundary conditions of the cavity problem. The cavity problem is formulated by the finite element technique and solved iteratively by the Gauss–Newton method to reconstruct the properties of the object. Regularization for both the first and the second problems is achieved by a Krylov subspace method. The proposed method is tested against both synthetic and experimental data and promising reconstruction results are obtained.
Fuentes, D; Elliott, A; Weinberg, J S; Shetty, A; Hazle, J D; Stafford, R J
2013-01-01
Quantification of local variations in the optical properties of tumor tissue introduced by the presence of gold-silica nanoparticles (NP) presents significant opportunities in monitoring and control of NP-mediated laser induced thermal therapy (LITT) procedures. Finite element methods of inverse parameter recovery constrained by a Pennes bioheat transfer model were applied to estimate the optical parameters. Magnetic resonance temperature imaging (MRTI) acquired during a NP-mediated LITT of a canine transmissible venereal tumor in brain was used in the presented statistical inverse problem formulation. The maximum likelihood (ML) value of the optical parameters illustrated a marked change in the periphery of the tumor corresponding with the expected location of NP and area of selective heating observed on MRTI. Parameter recovery information became increasingly difficult to infer in distal regions of tissue where photon fluence had been significantly attenuated. Finite element temperature predictions using the ML parameter values obtained from the solution of the inverse problem are able to reproduce the NP selective heating within 5 °C of measured MRTI estimations along selected temperature profiles. Results indicate the ML solution found is able to sufficiently reproduce the selectivity of the NP mediated laser induced heating and therefore the ML solution is likely to return useful optical parameters within the region of significant laser fluence.
Relevance vector machine technique for the inverse scattering problem
Institute of Scientific and Technical Information of China (English)
Wang Fang-Fang; Zhang Ye-Rong
2012-01-01
A novel method based on the relevance vector machine(RVM)for the inverse scattering problem is presented in this paper.The nonlinearity and the ill-posedness inherent in this problem are simultaneously considered.The nonlinearity is embodied in the relation between the scattered field and the target property,which can be obtained through the RVM training process.Besides,rather than utilizing regularization,the ill-posed nature of the inversion is naturally accounted for because the RVM can produce a probabilistic output.Simulation results reveal that the proposed RVM-based approach can provide comparative performances in terms of accuracy,convergence,robustness,generalization,and improved performance in terms of sparse property in comparison with the support vector machine(SVM)based approach.
Forward and inverse problems of electrocardiography : clinical investigations
2008-01-01
The non-invasive reconstruction of cardiac activity can significantly improve the quality of cardiac diagnostics. Two major approaches are considered. The model-based method consists in the optimization of an electrophysiological cardiac model until the measured and simulated ECGs are similar. The inverse problem of electrocardiography is solved to compute the cardiac sources distributions from body surface potential maps. The results and their interpretation are shown for several patients.
Burton, Brett M; Tate, Jess D; Erem, Burak; Swenson, Darrell J; Wang, Dafang F; Steffen, Michael; Brooks, Dana H; van Dam, Peter M; Macleod, Rob S
2011-01-01
Computational modeling in electrocardiography often requires the examination of cardiac forward and inverse problems in order to non-invasively analyze physiological events that are otherwise inaccessible or unethical to explore. The study of these models can be performed in the open-source SCIRun problem solving environment developed at the Center for Integrative Biomedical Computing (CIBC). A new toolkit within SCIRun provides researchers with essential frameworks for constructing and manipulating electrocardiographic forward and inverse models in a highly efficient and interactive way. The toolkit contains sample networks, tutorials and documentation which direct users through SCIRun-specific approaches in the assembly and execution of these specific problems.
A hybrid algorithm for solving inverse problems in elasticity
Directory of Open Access Journals (Sweden)
Barabasz Barbara
2014-12-01
Full Text Available The paper offers a new approach to handling difficult parametric inverse problems in elasticity and thermo-elasticity, formulated as global optimization ones. The proposed strategy is composed of two phases. In the first, global phase, the stochastic hp-HGS algorithm recognizes the basins of attraction of various objective minima. In the second phase, the local objective minimizers are closer approached by steepest descent processes executed singly in each basin of attraction. The proposed complex strategy is especially dedicated to ill-posed problems with multimodal objective functionals. The strategy offers comparatively low computational and memory costs resulting from a double-adaptive technique in both forward and inverse problem domains. We provide a result on the Lipschitz continuity of the objective functional composed of the elastic energy and the boundary displacement misfits with respect to the unknown constitutive parameters. It allows common scaling of the accuracy of solving forward and inverse problems, which is the core of the introduced double-adaptive technique. The capability of the proposed method of finding multiple solutions is illustrated by a computational example which consists in restoring all feasible Young modulus distributions minimizing an objective functional in a 3D domain of a photo polymer template obtained during step and flash imprint lithography.
Prior Information in Inverse Boundary Problems
DEFF Research Database (Denmark)
Garde, Henrik
This thesis gives a threefold perspective on the inverse problem of inclusion detection in electrical impedance tomography: depth dependence, monotonicitybased reconstruction, and sparsity-based reconstruction. The depth dependence is given in terms of explicit bounds on the datum norm, which shows...... be regularized against noise with a uniform regularization parameter, and that the method can be generalized to discrete electrode models. We give examples in 2D and 3D with noisy simulated data as well as real measurements, and give a comparison of reconstructions based on a non-linear and a linear formulation...... of the reconstruction. Numerical examples are given in both 2D and 3D for partial data using noisy simulated data as well as real measurements....
Large Deviation Strategy for Inverse Problem
Ojima, Izumi
2011-01-01
Taken traditionally as a no-go theorem against the theorization of inductive processes, Duheme-Quine thesis may interfere with the essence of statistical inference. This difficulty can be resolved by \\textquotedblleft Micro-Macro duality\\textquotedblright\\ \\cite{Oj03, Oj05} which clarifies the importance of specifying the pertinent aspects and accuracy relevant to concrete contexts of scientific discussions and which ensures the matching between what to be described and what to describe in the form of the validity of duality relations. This consolidates the foundations of the inverse problem, induction method, and statistical inference crucial for the sound relations between theory and experiments. To achieve the purpose, we propose here Large Deviation Strategy (LDS for short) on the basis of Micro-Macro duality, quadrality scheme, and large deviation principle. According to the quadrality scheme emphasizing the basic roles played by the dynamics, algebra of observables together with its representations and ...
The inverse problem for Schwinger pair production
Energy Technology Data Exchange (ETDEWEB)
Hebenstreit, F., E-mail: hebenstreit@itp.unibe.ch
2016-02-10
The production of electron–positron pairs in time-dependent electric fields (Schwinger mechanism) depends non-linearly on the applied field profile. Accordingly, the resulting momentum spectrum is extremely sensitive to small variations of the field parameters. Owing to this non-linear dependence it is so far unpredictable how to choose a field configuration such that a predetermined momentum distribution is generated. We show that quantum kinetic theory along with optimal control theory can be used to approximately solve this inverse problem for Schwinger pair production. We exemplify this by studying the superposition of a small number of harmonic components resulting in predetermined signatures in the asymptotic momentum spectrum. In the long run, our results could facilitate the observation of this yet unobserved pair production mechanism in quantum electrodynamics by providing suggestions for tailored field configurations.
Numerical Methods for Bayesian Inverse Problems
Ernst, Oliver
2014-01-06
We present recent results on Bayesian inversion for a groundwater flow problem with an uncertain conductivity field. In particular, we show how direct and indirect measurements can be used to obtain a stochastic model for the unknown. The main tool here is Bayes’ theorem which merges the indirect data with the stochastic prior model for the conductivity field obtained by the direct measurements. Further, we demonstrate how the resulting posterior distribution of the quantity of interest, in this case travel times of radionuclide contaminants, can be obtained by Markov Chain Monte Carlo (MCMC) simulations. Moreover, we investigate new, promising MCMC methods which exploit geometrical features of the posterior and which are suited to infinite dimensions.
An inverse problem for Schwinger pair production
Hebenstreit, Florian
2016-01-01
The production of electron-positron pairs in time-dependent electric fields (Schwinger mechanism) depends non-linearly on the applied field profile. Accordingly, the resulting momentum spectrum is extremely sensitive to small variations of the field parameters. Owing to this non-linear dependence it is so far unpredictable how to choose a field configuration such that a predetermined momentum distribution is generated. We show that quantum kinetic theory along with optimal control theory can be used to approximately solve this inverse problem for Schwinger pair production. We exemplify this by studying the superposition of a small number of harmonic components resulting in predetermined signatures in the asymptotic momentum spectrum. In the long run, our results could facilitate the observation of this yet unobserved pair production mechanism in quantum electrodynamics by providing suggestions for tailored field configurations.
The inverse problem for Schwinger pair production
Directory of Open Access Journals (Sweden)
F. Hebenstreit
2016-02-01
Full Text Available The production of electron–positron pairs in time-dependent electric fields (Schwinger mechanism depends non-linearly on the applied field profile. Accordingly, the resulting momentum spectrum is extremely sensitive to small variations of the field parameters. Owing to this non-linear dependence it is so far unpredictable how to choose a field configuration such that a predetermined momentum distribution is generated. We show that quantum kinetic theory along with optimal control theory can be used to approximately solve this inverse problem for Schwinger pair production. We exemplify this by studying the superposition of a small number of harmonic components resulting in predetermined signatures in the asymptotic momentum spectrum. In the long run, our results could facilitate the observation of this yet unobserved pair production mechanism in quantum electrodynamics by providing suggestions for tailored field configurations.
Stochastic inverse problems: Models and metrics
Energy Technology Data Exchange (ETDEWEB)
Sabbagh, Elias H.; Sabbagh, Harold A.; Murphy, R. Kim [Victor Technologies, LLC, Bloomington, IN 47407-7706 (United States); Aldrin, John C. [Computational Tools, Gurnee, IL 60031 (United States); Annis, Charles [Statistical Engineering, Palm Beach Gardens, FL 33418 (United States); Knopp, Jeremy S. [Air Force Research Laboratory (AFRL/RXCA), Wright Patterson AFB, OH 45433-7817 (United States)
2015-03-31
In past work, we introduced model-based inverse methods, and applied them to problems in which the anomaly could be reasonably modeled by simple canonical shapes, such as rectangular solids. In these cases the parameters to be inverted would be length, width and height, as well as the occasional probe lift-off or rotation. We are now developing a formulation that allows more flexibility in modeling complex flaws. The idea consists of expanding the flaw in a sequence of basis functions, and then solving for the expansion coefficients of this sequence, which are modeled as independent random variables, uniformly distributed over their range of values. There are a number of applications of such modeling: 1. Connected cracks and multiple half-moons, which we have noted in a POD set. Ideally we would like to distinguish connected cracks from one long shallow crack. 2. Cracks of irregular profile and shape which have appeared in cold work holes during bolt-hole eddy-current inspection. One side of such cracks is much deeper than other. 3. L or C shaped crack profiles at the surface, examples of which have been seen in bolt-hole cracks. By formulating problems in a stochastic sense, we are able to leverage the stochastic global optimization algorithms in NLSE, which is resident in VIC-3D®, to answer questions of global minimization and to compute confidence bounds using the sensitivity coefficient that we get from NLSE. We will also address the issue of surrogate functions which are used during the inversion process, and how they contribute to the quality of the estimation of the bounds.
Uhlmann, Gunther
2008-07-01
This volume represents the proceedings of the fourth Applied Inverse Problems (AIP) international conference and the first congress of the Inverse Problems International Association (IPIA) which was held in Vancouver, Canada, June 25 29, 2007. The organizing committee was formed by Uri Ascher, University of British Columbia, Richard Froese, University of British Columbia, Gary Margrave, University of Calgary, and Gunther Uhlmann, University of Washington, chair. The conference was part of the activities of the Pacific Institute of Mathematical Sciences (PIMS) Collaborative Research Group on inverse problems (http://www.pims.math.ca/scientific/collaborative-research-groups/past-crgs). This event was also supported by grants from NSF and MITACS. Inverse Problems (IP) are problems where causes for a desired or an observed effect are to be determined. They lie at the heart of scientific inquiry and technological development. The enormous increase in computing power and the development of powerful algorithms have made it possible to apply the techniques of IP to real-world problems of growing complexity. Applications include a number of medical as well as other imaging techniques, location of oil and mineral deposits in the earth's substructure, creation of astrophysical images from telescope data, finding cracks and interfaces within materials, shape optimization, model identification in growth processes and, more recently, modelling in the life sciences. The series of Applied Inverse Problems (AIP) Conferences aims to provide a primary international forum for academic and industrial researchers working on all aspects of inverse problems, such as mathematical modelling, functional analytic methods, computational approaches, numerical algorithms etc. The steering committee of the AIP conferences consists of Heinz Engl (Johannes Kepler Universität, Austria), Joyce McLaughlin (RPI, USA), William Rundell (Texas A&M, USA), Erkki Somersalo (Helsinki University of Technology
A variational Bayesian method to inverse problems with impulsive noise
Jin, Bangti
2012-01-01
We propose a novel numerical method for solving inverse problems subject to impulsive noises which possibly contain a large number of outliers. The approach is of Bayesian type, and it exploits a heavy-tailed t distribution for data noise to achieve robustness with respect to outliers. A hierarchical model with all hyper-parameters automatically determined from the given data is described. An algorithm of variational type by minimizing the Kullback-Leibler divergence between the true posteriori distribution and a separable approximation is developed. The numerical method is illustrated on several one- and two-dimensional linear and nonlinear inverse problems arising from heat conduction, including estimating boundary temperature, heat flux and heat transfer coefficient. The results show its robustness to outliers and the fast and steady convergence of the algorithm. © 2011 Elsevier Inc.
MAP estimators and their consistency in Bayesian nonparametric inverse problems
Dashti, M.; Law, K. J. H.; Stuart, A. M.; Voss, J.
2013-09-01
We consider the inverse problem of estimating an unknown function u from noisy measurements y of a known, possibly nonlinear, map {G} applied to u. We adopt a Bayesian approach to the problem and work in a setting where the prior measure is specified as a Gaussian random field μ0. We work under a natural set of conditions on the likelihood which implies the existence of a well-posed posterior measure, μy. Under these conditions, we show that the maximum a posteriori (MAP) estimator is well defined as the minimizer of an Onsager-Machlup functional defined on the Cameron-Martin space of the prior; thus, we link a problem in probability with a problem in the calculus of variations. We then consider the case where the observational noise vanishes and establish a form of Bayesian posterior consistency for the MAP estimator. We also prove a similar result for the case where the observation of {G}(u) can be repeated as many times as desired with independent identically distributed noise. The theory is illustrated with examples from an inverse problem for the Navier-Stokes equation, motivated by problems arising in weather forecasting, and from the theory of conditioned diffusions, motivated by problems arising in molecular dynamics.
MAP estimators and their consistency in Bayesian nonparametric inverse problems
Dashti, M.
2013-09-01
We consider the inverse problem of estimating an unknown function u from noisy measurements y of a known, possibly nonlinear, map applied to u. We adopt a Bayesian approach to the problem and work in a setting where the prior measure is specified as a Gaussian random field μ0. We work under a natural set of conditions on the likelihood which implies the existence of a well-posed posterior measure, μy. Under these conditions, we show that the maximum a posteriori (MAP) estimator is well defined as the minimizer of an Onsager-Machlup functional defined on the Cameron-Martin space of the prior; thus, we link a problem in probability with a problem in the calculus of variations. We then consider the case where the observational noise vanishes and establish a form of Bayesian posterior consistency for the MAP estimator. We also prove a similar result for the case where the observation of can be repeated as many times as desired with independent identically distributed noise. The theory is illustrated with examples from an inverse problem for the Navier-Stokes equation, motivated by problems arising in weather forecasting, and from the theory of conditioned diffusions, motivated by problems arising in molecular dynamics. © 2013 IOP Publishing Ltd.
Solving inverse problems of mathematical physics by means of the PHOENICS software package
Energy Technology Data Exchange (ETDEWEB)
Matsevity, Y.; Lushpenko, S. [Institute for Problems in Machinery, National Academy of Sciences of Ukraine Pozharskogo, Kharkov (Ukraine)
1997-12-31
Several approaches on organizing solution of inverse problems by means of PHOENICS on the basis of the technique of automated fitting are proposing. A version of a `nondestructive` method of using PHOENICS in the inverse problem solution regime and the ways of altering the program in the case of introducing optimization facilities in it are under consideration. (author) 12 refs.
Large-Scale Inverse Problems and Quantification of Uncertainty
Biegler, Lorenz; Ghattas, Omar
2010-01-01
Large-scale inverse problems and associated uncertainty quantification has become an important area of research, central to a wide range of science and engineering applications. Written by leading experts in the field, Large-scale Inverse Problems and Quantification of Uncertainty focuses on the computational methods used to analyze and simulate inverse problems. The text provides PhD students, researchers, advanced undergraduate students, and engineering practitioners with the perspectives of researchers in areas of inverse problems and data assimilation, ranging from statistics and large-sca
Inverse problems and inverse scattering of plane waves
Ghosh Roy, Dilip N
2001-01-01
The purpose of this text is to present the theory and mathematics of inverse scattering, in a simple way, to the many researchers and professionals who use it in their everyday research. While applications range across a broad spectrum of disciplines, examples in this text will focus primarly, but not exclusively, on acoustics. The text will be especially valuable for those applied workers who would like to delve more deeply into the fundamentally mathematical character of the subject matter.Practitioners in this field comprise applied physicists, engineers, and technologists, whereas the theory is almost entirely in the domain of abstract mathematics. This gulf between the two, if bridged, can only lead to improvement in the level of scholarship in this highly important discipline. This is the book''s primary focus.
A GPU-Computing Approach to Solar Stokes Profile Inversion
Harker, Brian J
2012-01-01
We present a new computational approach to the inversion of solar photospheric Stokes polarization profiles, under the Milne-Eddington model, for vector magnetography. Our code, named GENESIS (GENEtic Stokes Inversion Strategy), employs multi-threaded parallel-processing techniques to harness the computing power of graphics processing units GPUs, along with algorithms designed to exploit the inherent parallelism of the Stokes inversion problem. Using a genetic algorithm (GA) engineered specifically for use with a GPU, we produce full-disc maps of the photospheric vector magnetic field from polarized spectral line observations recorded by the Synoptic Optical Long-term Investigations of the Sun (SOLIS) Vector Spectromagnetograph (VSM) instrument. We show the advantages of pairing a population-parallel genetic algorithm with data-parallel GPU-computing techniques, and present an overview of the Stokes inversion problem, including a description of our adaptation to the GPU-computing paradigm. Full-disc vector ma...
Stochastic reduced order models for inverse problems under uncertainty.
Warner, James E; Aquino, Wilkins; Grigoriu, Mircea D
2015-03-01
This work presents a novel methodology for solving inverse problems under uncertainty using stochastic reduced order models (SROMs). Given statistical information about an observed state variable in a system, unknown parameters are estimated probabilistically through the solution of a model-constrained, stochastic optimization problem. The point of departure and crux of the proposed framework is the representation of a random quantity using a SROM - a low dimensional, discrete approximation to a continuous random element that permits e cient and non-intrusive stochastic computations. Characterizing the uncertainties with SROMs transforms the stochastic optimization problem into a deterministic one. The non-intrusive nature of SROMs facilitates e cient gradient computations for random vector unknowns and relies entirely on calls to existing deterministic solvers. Furthermore, the method is naturally extended to handle multiple sources of uncertainty in cases where state variable data, system parameters, and boundary conditions are all considered random. The new and widely-applicable SROM framework is formulated for a general stochastic optimization problem in terms of an abstract objective function and constraining model. For demonstration purposes, however, we study its performance in the specific case of inverse identification of random material parameters in elastodynamics. We demonstrate the ability to efficiently recover random shear moduli given material displacement statistics as input data. We also show that the approach remains effective for the case where the loading in the problem is random as well.
Including geological information in the inverse problem of palaeothermal reconstruction
Trautner, S.; Nielsen, S. B.
2003-04-01
A reliable reconstruction of sediment thermal history is of central importance to the assessment of hydrocarbon potential and the understanding of basin evolution. However, only rarely do sedimentation history and borehole data in the form of present day temperatures and vitrinite reflectance constrain the past thermal evolution to a useful level of accuracy (Gallagher and Sambridge,1992; Nielsen,1998; Trautner and Nielsen,2003). This is reflected in the inverse solutions to the problem of determining heat flow history from borehole data: The recent heat flow is constrained by data while older values are governed by the chosen a prior heat flow. In this paper we reduce this problem by including geological information in the inverse problem. Through a careful analysis of geological and geophysical data the timing of the tectonic processes, which may influence heat flow, can be inferred. The heat flow history is then parameterised to allow for the temporal variations characteristic of the different tectonic events. The inversion scheme applies a Markov chain Monte Carlo (MCMC) approach (Nielsen and Gallagher, 1999; Ferrero and Gallagher,2002), which efficiently explores the model space and futhermore samples the posterior probability distribution of the model. The technique is demonstrated on wells in the northern North Sea with emphasis on the stretching event in Late Jurassic. The wells are characterised by maximum sediment temperature at the present day, which is the worst case for resolution of the past thermal history because vitrinite reflectance is determined mainly by the maximum temperature. Including geological information significantly improves the thermal resolution. Ferrero, C. and Gallagher,K.,2002. Stochastic thermal history modelling.1. Constraining heat flow histories and their uncertainty. Marine and Petroleum Geology, 19, 633-648. Gallagher,K. and Sambridge, M., 1992. The resolution of past heat flow in sedimentary basins from non-linear inversion
Stability of charge inversion, Thomson problem, and application to electrophoresis
Patra, Michael; Patriarca, Marco; Karttunen, Mikko
2003-03-01
We analyze charge inversion in colloidal systems at zero temperature using stability concepts, and connect this to the classical Thomson problem of arranging electrons on sphere. We show that for a finite microion charge, the globally stable, lowest-energy state of the complex formed by the colloid and the oppositely charged microions is always overcharged. This effect disappears in the continuous limit. Additionally, a layer of at least twice as many microions as required for charge neutrality is always locally stable. In an applied external electric field the stability of the microion cloud is reduced. Finally, this approach is applied to a system of two colloids at low but finite temperature.
Introduction to the 30th volume of Inverse Problems
Louis, Alfred K.
2014-01-01
Bill Symes in their big footsteps, and I consider it a privilege to thank all that have contributed to the success of the journal. In its 30 years of existence, the journal has evolved from a trimestral to monthly print publication, now paralleled by an electronic version that has led to publication speeds unheard of when the journal began. This timely publication is especially important for younger researchers, but equally for experienced ones, who in that respect still feel young. In addition, the scope has changed to focus more precisely on the core of inverse problems, characterized, for example, by data errors, incomplete information and so on. In the beginning, fields where questions were considered to lead to inverse problems were listed in the journal's scope to make it clear that the problems being discussed were inverse problems in character. With the development of the solution methods, we now see that inverse problems are fundamental to almost all areas of research. The journal now hosts a number of additional features. With Insights we provide a platform for authors to introduce themselves and their work group, and present their scientific results in a popular and non-specialist form. Insights are made freely available on the journal website to ensure that they are seen by a wider community, beyond the immediate readership of the journal. Special issues are devoted to fields that have matured in such a way that the readers of our journal can profit from their presentation when the time for writing text books has not yet come. In addition, the different approaches taken by different contributors to a special issue disclose the multiple aspects of that field. With Topical reviews we aim to present the new ideas and areas that are stimulating future research. We are thankful that highly acclaimed authors take the time to present the research at the forefront of their respective fields. It is always very enlightening to read these articles as they introduce
Pseudo almost periodic solutions to parabolic boundary value inverse problems
Institute of Scientific and Technical Information of China (English)
2008-01-01
We first define the pseudo almost periodic functions in a more general setting.Then we show the existence,uniqueness and stability of pseudo almost periodic solutions of parabolic inverse problems for a type of boundary value problems.
Perturbative methods for inverse problems on degenerate differential equations
Directory of Open Access Journals (Sweden)
Angelo Favini
2012-12-01
Full Text Available Pertubation results for linear relations satisfying a resolvent condition of weak parabolic type are established. Such results are applied to solve some inverse problems for degenerate differential equations, supplying a new method which avoids any fixed-point argument and essentially consists in reducing the original inverse problem to an auxiliary direct one.
Inverse Modelling Problems in Linear Algebra Undergraduate Courses
Martinez-Luaces, Victor E.
2013-01-01
This paper will offer an analysis from a theoretical point of view of mathematical modelling, applications and inverse problems of both causation and specification types. Inverse modelling problems give the opportunity to establish connections between theory and practice and to show this fact, a simple linear algebra example in two different…
Inverse Modelling Problems in Linear Algebra Undergraduate Courses
Martinez-Luaces, Victor E.
2013-01-01
This paper will offer an analysis from a theoretical point of view of mathematical modelling, applications and inverse problems of both causation and specification types. Inverse modelling problems give the opportunity to establish connections between theory and practice and to show this fact, a simple linear algebra example in two different…
Solutions of inverse problems for biodegradation of xenobiotic polymers
Watanabe, Masaji; Kawai, Fusako
2016-02-01
Mathematical techniques are applied to a microbial depolymerization process. A mathematical model for the transition of the weight distribution and the microbial population is described. Inverse problems for a molecular factor and a time factor of a degradation rate are derived. Numerical techniques to solve the inverse problems are illustrated, and numerical results are presented.
Electromagnetic tomography (EMT): image reconstruction based on the inverse problem
Institute of Scientific and Technical Information of China (English)
无
2003-01-01
Starting from Maxwell's equations for inhomogeneous media, nonlinear integral equations of the inverse problem of the electromagnetic tomography (EMT) are derived, whose kernel is the dyadic Green's function for the EMT sensor with a homogeneous medium in the object space. Then in terms of ill-posedness of the inverse problem, a Tikhonov-type regularization model is established based on a linearization-approximation of the nonlinear inverse problem. Finally, an iterative algorithm of image reconstruction based on the inverse problem and reconstruction images of some object flows for simplified sensor are given. Initial results of the image reconstruction show that the algorithm based on the inverse problem is superior to those based on the linear back-projection in the quality of image reconstruction.
Inverse Scattering Approach to Improving Pattern Recognition
Energy Technology Data Exchange (ETDEWEB)
Chapline, G; Fu, C
2005-02-15
The Helmholtz machine provides what may be the best existing model for how the mammalian brain recognizes patterns. Based on the observation that the ''wake-sleep'' algorithm for training a Helmholtz machine is similar to the problem of finding the potential for a multi-channel Schrodinger equation, we propose that the construction of a Schrodinger potential using inverse scattering methods can serve as a model for how the mammalian brain learns to extract essential information from sensory data. In particular, inverse scattering theory provides a conceptual framework for imagining how one might use EEG and MEG observations of brain-waves together with sensory feedback to improve human learning and pattern recognition. Longer term, implementation of inverse scattering algorithms on a digital or optical computer could be a step towards mimicking the seamless information fusion of the mammalian brain.
Inverse Coefficient Problems for Nonlinear Parabolic Differential Equations
Institute of Scientific and Technical Information of China (English)
Yun Hua OU; Alemdar HASANOV; Zhen Hai LIU
2008-01-01
This paper is devoted to a class of inverse problems for a nonlinear parabolic differential equation.The unknown coefficient of the equation depends on the gradient of the solution and belongs to a set of admissible coefficients.It is proved that the convergence of solutions for the corresponding direct problems continuously depends on the coefficient convergence.Based on this result the existence of a quasisolution of the inverse problem is obtained in the appropriate class of admissible coefficients.
Local regularization of linear inverse problems via variational filtering
Lamm, Patricia K.
2017-08-01
We develop local regularization methods for ill-posed linear inverse problems governed by general Fredholm integral operators. The methods are executed as filtering algorithms which are simple to implement and computationally efficient for a large class of problems. We establish a convergence theory and give convergence rates for such methods, and illustrate their computational speed in numerical tests for inverse problems in geomagnetic exploration and imaging.
An Efficient Pseudo-Inverse Solution to the Inverse Kinematic Problem for 6-Joint Manipulators
Directory of Open Access Journals (Sweden)
Stefano Chiaverini
1990-10-01
Full Text Available The use of the pseudo-inverse Jacobian matrix makes the solution of the inverse kinematic problem well-defined even at singular configurations of the robot arm, in the neighbourhood of a singularity, however, the computed solution often results in high joint velocities which may not be feasible to the real manipulator. Furthermore, the pseudo-inverse solution is computationally expensive, thus preventing real-time applications.
Galerkin approximation for inverse problems for nonautonomous nonlinear distributed systems
Banks, H. T.; Reich, Simeon; Rosen, I. G.
1988-01-01
An abstract framework and convergence theory is developed for Galerkin approximation for inverse problems involving the identification of nonautonomous nonlinear distributed parameter systems. A set of relatively easily verified conditions is provided which are sufficient to guarantee the existence of optimal solutions and their approximation by a sequence of solutions to a sequence of approximating finite dimensional identification problems. The approach is based on the theory of monotone operators in Banach spaces and is applicable to a reasonably broad class of nonlinear distributed systems. Operator theoretic and variational techniques are used to establish a fundamental convergence result. An example involving evolution systems with dynamics described by nonstationary quasilinear elliptic operators along with some applications are presented and discussed.
Index Theory-Based Algorithm for the Gradiometer Inverse Problem
2015-03-28
field generated by the positive eigenvector of the gradiometer tensor to the closeness of fit of the proposed inverse solution to the mass and...line field generated by the positive eigenvector of the gradiometer tensor to the closeness of fit of the proposed inverse solution to the mass and...2015). The inverse source problem for the gradiometer tensor can be stated generally as follows: given a gradiometer tensor field, extract
An Inverse Eigenvalue Problem for Jacobi Matrices
Directory of Open Access Journals (Sweden)
Zhengsheng Wang
2011-01-01
eigenvectors. The solvability of the problem is discussed, and some sufficient conditions for existence of the solution of this problem are proposed. Furthermore, a numerical algorithm and two examples are presented.
Stability analysis of the inverse transmembrane potential problem in electrocardiography
Burger, Martin; Mardal, Kent-André; Nielsen, Bjørn Fredrik
2010-10-01
In this paper we study some mathematical properties of an inverse problem arising in connection with electrocardiograms (ECGs). More specifically, we analyze the possibility for recovering the transmembrane potential in the heart from ECG recordings, a challenge currently investigated by a growing number of groups. Our approach is based on the bidomain model for the electrical activity in the myocardium, and leads to a parameter identification problem for elliptic partial differential equations (PDEs). It turns out that this challenge can be split into two subproblems: the task of recovering the potential at the heart surface from body surface recordings; the problem of computing the transmembrane potential inside the heart from the potential determined at the heart surface. Problem (1), which can be formulated as the Cauchy problem for an elliptic PDE, has been extensively studied and is well known to be severely ill-posed. The main purpose of this paper is to prove that problem (2) is stable and well posed if a suitable prior is available. Moreover, our theoretical findings are illuminated by a series of numerical experiments. Finally, we discuss some aspects of uniqueness related to the anisotropy in the heart.
REGULARIZED D-BAR METHOD FOR THE INVERSE CONDUCTIVITY PROBLEM
DEFF Research Database (Denmark)
Knudsen, Kim; Lassas, Matti; Mueller, Jennifer;
2009-01-01
A strategy for regularizing the inversion procedure for the two-dimensional D-bar reconstruction algorithm based on the global uniqueness proof of Nachman [Ann. Math. 143 (1996)] for the ill-posed inverse conductivity problem is presented. The strategy utilizes truncation of the boundary integral...
Inverse problem of Ocean Acoustic Tomography (OAT) - A numerical experiment
Digital Repository Service at National Institute of Oceanography (India)
Murty, T.V.R.; Somayajulu, Y.K.; Mahadevan, R.; Murty, C.S.
layers, or grids developEd. by solving the forward problem of the acoustic model enable build the generalized inverse operator (GIO) that operates on the travel time perturbation data. Resolution matrices obtained through SVD helped to examine...
Modeling and inverse problems in the presence of uncertainty
Banks, H T; Thompson, W Clayton
2014-01-01
Modeling and Inverse Problems in the Presence of Uncertainty collects recent research-including the authors' own substantial projects-on uncertainty propagation and quantification. It covers two sources of uncertainty: where uncertainty is present primarily due to measurement errors and where uncertainty is present due to the modeling formulation itself. After a useful review of relevant probability and statistical concepts, the book summarizes mathematical and statistical aspects of inverse problem methodology, including ordinary, weighted, and generalized least-squares formulations. It then
Inverse Coefficient Problems for Nonlinear Elliptic Variational Inequalities
Institute of Scientific and Technical Information of China (English)
Run-sheng Yang; Yun-hua Ou
2011-01-01
This paper is devoted to a class of inverse coefficient problems for nonlinear elliptic variational inequalities. The unknown coefficient of elliptic variational inequalities depends on the gradient of the solution and belongs to a set of admissible coefficients. It is shown that the nonlinear elliptic variational inequalities is unique solvable for the given class of coefficients. The existence of quasisolutions of the inverse problems is obtained.
Analysis of the Gibbs Sampler for Hierarchical Inverse Problems
Agapiou, Sergios; Bardsley, Johnathan M.; Papaspiliopoulos, Omiros; Stuart, Andrew M.
2014-01-01
Many inverse problems arising in applications come from continuum models where the unknown parameter is a field. In practice the unknown field is discretized resulting in a problem in $\\mathbb{R}^N$, with an understanding that refining the discretization, that is increasing $N$, will often be desirable. In the context of Bayesian inversion this situation suggests the importance of two issues: (i) defining hyper-parameters in such a way that they are interpretable in the continuum limit $N \\to...
THE INVERSE PROBLEM OF CENTROSYMMETRIC MATRICES WITH A SUBMATRIX CONSTRAINT
Institute of Scientific and Technical Information of China (English)
Zhen-yun Peng; Xi-yan Hu; Lei Zhang
2004-01-01
By using Moore-Penrose generalized inverse and the general singular value decomposition of matrices, this paper establishes the necessary and sufficient conditions for the existence of and the expressions for the centrosymmetric solutions with a submatrix constraint of matrix inverse problem AX = B. In addition, in the solution set of corresponding problem, the expression of the optimal approximation solution to a given matrix is derived.
Basis set expansion for inverse problems in plasma diagnostic analysis
Jones, B.; Ruiz, C. L.
2013-07-01
A basis set expansion method [V. Dribinski, A. Ossadtchi, V. A. Mandelshtam, and H. Reisler, Rev. Sci. Instrum. 73, 2634 (2002)], 10.1063/1.1482156 is applied to recover physical information about plasma radiation sources from instrument data, which has been forward transformed due to the nature of the measurement technique. This method provides a general approach for inverse problems, and we discuss two specific examples relevant to diagnosing fast z pinches on the 20-25 MA Z machine [M. E. Savage, L. F. Bennett, D. E. Bliss, W. T. Clark, R. S. Coats, J. M. Elizondo, K. R. LeChien, H. C. Harjes, J. M. Lehr, J. E. Maenchen, D. H. McDaniel, M. F. Pasik, T. D. Pointon, A. C. Owen, D. B. Seidel, D. L. Smith, B. S. Stoltzfus, K. W. Struve, W. A. Stygar, L. K. Warne, J. R. Woodworth, C. W. Mendel, K. R. Prestwich, R. W. Shoup, D. L. Johnson, J. P. Corley, K. C. Hodge, T. C. Wagoner, and P. E. Wakeland, in Proceedings of the Pulsed Power Plasma Sciences Conference (IEEE, 2007), p. 979]. First, Abel inversion of time-gated, self-emission x-ray images from a wire array implosion is studied. Second, we present an approach for unfolding neutron time-of-flight measurements from a deuterium gas puff z pinch to recover information about emission time history and energy distribution. Through these examples, we discuss how noise in the measured data limits the practical resolution of the inversion, and how the method handles discontinuities in the source function and artifacts in the projected image. We add to the method a propagation of errors calculation for estimating uncertainties in the inverted solution.
Basis set expansion for inverse problems in plasma diagnostic analysis
Energy Technology Data Exchange (ETDEWEB)
Jones, B.; Ruiz, C. L. [Sandia National Laboratories, PO Box 5800, Albuquerque, New Mexico 87185 (United States)
2013-07-15
A basis set expansion method [V. Dribinski, A. Ossadtchi, V. A. Mandelshtam, and H. Reisler, Rev. Sci. Instrum. 73, 2634 (2002)] is applied to recover physical information about plasma radiation sources from instrument data, which has been forward transformed due to the nature of the measurement technique. This method provides a general approach for inverse problems, and we discuss two specific examples relevant to diagnosing fast z pinches on the 20–25 MA Z machine [M. E. Savage, L. F. Bennett, D. E. Bliss, W. T. Clark, R. S. Coats, J. M. Elizondo, K. R. LeChien, H. C. Harjes, J. M. Lehr, J. E. Maenchen, D. H. McDaniel, M. F. Pasik, T. D. Pointon, A. C. Owen, D. B. Seidel, D. L. Smith, B. S. Stoltzfus, K. W. Struve, W. A. Stygar, L. K. Warne, J. R. Woodworth, C. W. Mendel, K. R. Prestwich, R. W. Shoup, D. L. Johnson, J. P. Corley, K. C. Hodge, T. C. Wagoner, and P. E. Wakeland, in Proceedings of the Pulsed Power Plasma Sciences Conference (IEEE, 2007), p. 979]. First, Abel inversion of time-gated, self-emission x-ray images from a wire array implosion is studied. Second, we present an approach for unfolding neutron time-of-flight measurements from a deuterium gas puff z pinch to recover information about emission time history and energy distribution. Through these examples, we discuss how noise in the measured data limits the practical resolution of the inversion, and how the method handles discontinuities in the source function and artifacts in the projected image. We add to the method a propagation of errors calculation for estimating uncertainties in the inverted solution.
Piecewise polynomial solutions to linear inverse problems
DEFF Research Database (Denmark)
Hansen, Per Christian; Mosegaard, K.
1996-01-01
We have presented a new algorithm PP-TSVD that computes piecewise polynomial solutions to ill-posed problems, without a priori knowledge about the positions of the break points. In particular, we can compute piecewise constant functions that describe layered models. Such solutions are useful, e.g.......g., in seismological problems, and the algorithm can also be used as a preprocessor for other methods where break points/discontinuities must be incorporated explicitly....
Inverse problems in vision and 3D tomography
Mohamad-Djafari, Ali
2013-01-01
The concept of an inverse problem is a familiar one to most scientists and engineers, particularly in the field of signal and image processing, imaging systems (medical, geophysical, industrial non-destructive testing, etc.) and computer vision. In imaging systems, the aim is not just to estimate unobserved images, but also their geometric characteristics from observed quantities that are linked to these unobserved quantities through the forward problem. This book focuses on imagery and vision problems that can be clearly written in terms of an inverse problem where an estimate for the image a
Numerical methods for forward and inverse problems in optical imaging
Gao, Hao
The main objective of this work is to develop efficient and accurate numerical algorithms for mathematical problems in optical imaging: forward modeling and inverse problems. Radiative transfer equation (RTE) can be regarded as the gold standard of modeling in vivo photon migration, however an efficient solver of RTE is extremely computationally challenging. In this work we develop a fast multigrid solver for steady-state or frequency-domain RTE on 2D and 3D structured and unstructured meshes with vacuum or reflection boundary condition. The error estimate and convergence analysis of the algorithm is given. The subsequent effort is devoted to quantitatively improve the reconstruction from ill-posed problems, such as multilevel approach with L1+TV regularization for bioluminescence tomography, multilevel regularization for diffuse optical tomography, linear complex-source method for fluorescence tomography, and Bregman method for quantitative photoacoustic tomography. Most of the developed methods are general in the sense that they are not limited to a particular reconstruction problem and can be combined in a synergetic way.
Inverse problem of bottom slope design for aerator devices
Institute of Scientific and Technical Information of China (English)
吴建华; 樊博; 许唯临
2013-01-01
Air entrainment is an effective approach to protect release works from cavitation damage. The traditional method of aera-tor device designs is that, for given flow conditions, the geometries of the aerator device are designed and then the effects are experi-mentally tested for cavitation damage control. The present paper proposes an inverse problem method of determining the bottom slopes in front of and behind an aerator if the requirements of air entrainment, flow conditions and some of aerator geometric para-meters are given. An RBF neural network model is developed and the relevant bottom slopes are calculated in different conditions of flow and geometry on the basis of the data of 19 aerator devices from different discharge tunnels with safe operation. The case study shows that the methodology provides an effective way to design aerator devices under given target conditions.
Fractal-Based Methods and Inverse Problems for Differential Equations: Current State of the Art
Kunze, Herb E.; Davide La Torre; Franklin Mendivil; Manuel Ruiz Galán; Rachad Zaki
2014-01-01
We illustrate, in this short survey, the current state of the art of fractal-based techniques and their application to the solution of inverse problems for ordinary and partial differential equations. We review several methods based on the Collage Theorem and its extensions. We also discuss two innovative applications: the first one is related to a vibrating string model while the second one considers a collage-based approach for solving inverse problems for partial differential equations on ...
Reconstruction Methods for Inverse Problems with Partial Data
DEFF Research Database (Denmark)
Hoffmann, Kristoffer
This thesis presents a theoretical and numerical analysis of a general mathematical formulation of hybrid inverse problems in impedance tomography. This includes problems from several existing hybrid imaging modalities such as Current Density Impedance Imaging, Magnetic Resonance Electrical...... Impedance Tomography, and Ultrasound Modulated Electrical Impedance Tomography. After giving an introduction to hybrid inverse problems in impedance tomography and the mathematical tools that facilitate the related analysis, we explain in detail the stability properties associated with the classification...... of a linearised hybrid inverse problem. This is done using pseudo-differential calculus and theory for overdetermined boundary value problem. Using microlocal analysis we then present novel results on the propagation of singularities, which give a precise description of the distinct features of solutions...
Ensemble methods for large scale inverse problems
Heemink, A.W.; Umer Altaf, M.; Barbu, A.L.; Verlaan, M.
2013-01-01
Variational data assimilation, also sometimes simply called the ‘adjoint method’, is used very often for large scale model calibration problems. Using the available data, the uncertain parameters in the model are identified by minimizing a certain cost function that measures the difference between t
IPDO-2007: Inverse Problems, Design and Optimization Symposium
2007-08-01
108 INVERSE APPROACHES IN IMPROVEMENT OF AIR POLUTION PLUME DISPERSION MODELS FOR REGULATORY APPLICATIONS 517 109 USING OF THE IOSO NM SOFTWARE FOR...Dulikravich, G.S., Orlande, H.R.B., Tanaka, M. and Colaco, M.J.), Miami Beach, FL, April 16-18, 2007. 5. Inverse Approaches in Improvement of Air Pollution...A. Woodbury (USA) Prof. Anatoly G. Yagola (Russia) 5.4 SPONSORS AND PROMOTERS OF IPDO-2007 AFOSR/Numerical Mathematics (United States Air Force
Stabilizing inverse problems by internal data
Kuchment, Peter
2012-07-30
Several newly developing hybrid imaging methods (e.g., those combining electrical impedance or optical imaging with acoustics) enable one to obtain some auxiliary interior information (usually some combination of the electrical conductivity and the current) about the parameters of the tissues. This information, in turn, happens to stabilize the exponentially unstable and thus low-resolution optical and electrical impedance tomography. Various known instances of this effect have been studied individually. We show that there is a simple general technique (covering all known cases) that shows what kinds of interior data stabilize the reconstruction, and why. Namely, we show when the linearized problem becomes an elliptic pseudo-differential one, and thus stable. Stability here is meant as the problem being Fredholm, so the local uniqueness is not shown and probably does not hold in such generality. © 2012 IOP Publishing Ltd.
Stabilizing Inverse Problems by Internal Data
Kuchment, Peter
2011-01-01
Several newly developing hybrid imaging methods (e.g., those combining electrical impedance or optical imaging with acoustics) enable one to obtain some auxiliary interior information (usually some combination of the electrical conductivity and the current) about the parameters of the tissues. This information, in turn, happens to stabilize the exponentially unstable and thus low resolution optical and electrical impedance tomography. Various known instances of this effect have been studied individually. We show that there is a simple general technique (covering all known cases) that shows what kind of interior data stabilizes the reconstruction, and why. Namely, we show when the linearized problem becomes elliptic pseudo-differential one, and thus stable. Stability here is meant as the problem being Fredholm, so the local uniqueness is not shown and probably does not hold in such generality.
Transdimensional seismic inversion using the Hamiltonian Monte-Carlo approach
Sen, M. K.; Biswas, R.
2014-12-01
In an inverse problem, the number of model parameters is often a choice dictated by computational convenience. In a transdimensional inverse problem, the number of model parameters is treated as an additional variable that we solve for. The Reversible jump Markov Chain Monte Carlo (RJMCMC) is generally employed for model exploration and uncertainty quantification in such problems. A typical RJMCMC is computationally expensive and therefore, its application has so far been limited to problems with a small number of model parameters, where the forward modeling can be done rapidly. Here we report a practical transdimesional seismic inversion algorithm where the model perturbations are generated according to the birth-death approach. However, we determine the model acceptance by a Hamiltonian approach that introduces a new momentum variable. This results in an update rule that makes use of the gradient information together with the Metropolis criterion. The algorithm can either be used for model exploration (sampling at a constant temperature) or model exploitation using annealing. We apply this technique to 1D waveform inversion problem of seismic reflection data where at each location we make use of several hundred model parameters. Forward modelng is carried out using reflectivity layer matrices in the frequency wavenumber domain and the objective function is evaluated in delaytime-ray parameter domain. Only at a few selected sufrace locations (CMP gathers) we carry out detailed uncertainty analysis ; at all other places we determine the best fit model. We are able to estimate geologically meaningful results that are correlated very well with collocated well logs at a few selected locations. We also explore several proposal distributions diifferent from the standard birth-date appproach for trial model generation that can be computationally fast.
The inverse moment problem for convex polytopes
Gravin, Nick; Pasechnik, Dmitrii; Robins, Sinai
2011-01-01
The goal of this paper is to present a general and novel approach for the reconstruction of any convex d-dimensional polytope P, from knowledge of its moments. In particular, we show that the vertices of an N-vertex polytope in R^d can be reconstructed from the knowledge of O(DN) axial moments (w.r.t. to an unknown polynomial measure od degree D) in d+1 distinct generic directions. Our approach is based on the collection of moment formulas due to Brion, Lawrence, Khovanskii-Pukhikov, and Barvinok that arise in the discrete geometry of polytopes, and what variously known as Prony's method, or Vandermonde factorization of finite rank Hankel matrices.
Inverse Problems for Nonlinear Delay Systems
2011-03-15
Ba82]. For nonlinear delay systems such as those discussed here, approximation in the context of a linear semigroup framework as presented [BBu1, BBu2...linear part generates a linear semigroup as in [BBu1, BBu2, BKap]. One then uses the linear semigroup in a vari- ation of parameters implicit...BBu2, BKap] (for the linear semigroup ) plus a Gronwall inequality. An alternative (and more general) approach given in [Ba82] eschews use of the Trotter
One-dimensional inverse problems of mathematical physics
Lavrent'ev, M M; Yakhno, V G; Schulenberger, J R
1986-01-01
This monograph deals with the inverse problems of determining a variable coefficient and right side for hyperbolic and parabolic equations on the basis of known solutions at fixed points of space for all times. The problems are one-dimensional in nature since the desired coefficient of the equation is a function of only one coordinate, while the desired right side is a function only of time. The authors use methods based on the spectral theory of ordinary differential operators of second order and also methods which make it possible to reduce the investigation of the inverse problems to the in
Minimax theory for a class of nonlinear statistical inverse problems
Ray, Kolyan; Schmidt-Hieber, Johannes
2016-06-01
We study a class of statistical inverse problems with nonlinear pointwise operators motivated by concrete statistical applications. A two-step procedure is proposed, where the first step smoothes the data and inverts the nonlinearity. This reduces the initial nonlinear problem to a linear inverse problem with deterministic noise, which is then solved in a second step. The noise reduction step is based on wavelet thresholding and is shown to be minimax optimal (up to logarithmic factors) in a pointwise function-dependent sense. Our analysis is based on a modified notion of Hölder smoothness scales that are natural in this setting.
Inverse problems basics, theory and applications in geophysics
Richter, Mathias
2016-01-01
The overall goal of the book is to provide access to the regularized solution of inverse problems relevant in geophysics without requiring more mathematical knowledge than is taught in undergraduate math courses for scientists and engineers. From abstract analysis only the concept of functions as vectors is needed. Function spaces are introduced informally in the course of the text, when needed. Additionally, a more detailed, but still condensed introduction is given in Appendix B. A second goal is to elaborate the single steps to be taken when solving an inverse problem: discretization, regularization and practical solution of the regularized optimization problem. These steps are shown in detail for model problems from the fields of inverse gravimetry and seismic tomography. The intended audience is mathematicians, physicists and engineers having a good working knowledge of linear algebra and analysis at the upper undergraduate level.
The Inverse Source Problem for Maxwell’s Equations
2006-10-01
of applied biomedical engineering and also as a mathematical problem (see for example [3, 22, 15, 7, 14, 9, 6, 121 where we have emphasized...BLEISTEIN AND J. COHEN, Nonuniqueness in the inverse source problem in acoustics and electromagnetics, Journal of Mathematical Physics, 18 (1977), pp. 194
Unfolding in particle physics: a window on solving inverse problems
Directory of Open Access Journals (Sweden)
Spanò Francesco
2013-07-01
Full Text Available Unfolding is the ensemble of techniques aimed at resolving inverse, ill-posed problems. A pedagogical introduction to the origin and main problems related to unfolding is presented and used as the the stepping stone towards the illustration of some of the most common techniques that are currently used in particle physics experiments.
AN INVERSE MAXIMUM CAPACITY PATH PROBLEM WITH LOWER BOUND CONSTRAINTS
Institute of Scientific and Technical Information of China (English)
杨超; 陈学旗
2002-01-01
The computational complexity of inverse mimimum capacity path problem with lower bound on capacity of maximum capacity path is examined, and it is proved that solution of this problem is NP-complete. A strong polynomial algorithm for a local optimal solution is provided.
On a class of inverse electrostatic and elasticity problems
Artemev, Andrei; Parnovski, Leonid; Polterovich, Iosif
2012-01-01
We study the inverse electrostatic and elasticity problems associated with Poisson and Navier equations. The uniqueness of solutions of these problems is proved for piecewise constant electric charge and internal stress distributions having a checkered structure: they are constant on rectangular blocks. Such distributions appear naturally in practical applications. We also discuss computational challenges arising in the numerical implementation of our method.
Uncertainty quantification and weak approximation of an elliptic inverse problem
Dashti, Masoumeh
2011-01-01
We consider the inverse problem of determining the permeability from the pressure in a Darcy model of flow in a porous medium. Mathematically the problem is to find the diffusion coefficient for a linear uniformly elliptic partial differential equation in divergence form, in a bounded domain in dimension $d \\le 3$, from measurements of the solution in the interior. We adopt a Bayesian approach to the problem. We place a prior random field measure on the log permeability, specified through the Karhunen-Lo\\`eve expansion of its draws. We consider Gaussian measures constructed this way, and study the regularity of functions drawn from them. We also study the Lipschitz properties of the observation operator mapping the log permeability to the observations. Combining these regularity and continuity estimates, we show that the posterior measure is well-defined on a suitable Banach space. Furthermore the posterior measure is shown to be Lipschitz with respect to the data in the Hellinger metric, giving rise to a for...
Approaching the Island of Inversion: 34P
Energy Technology Data Exchange (ETDEWEB)
Bender, P.C.; Hoffman, C.R.; Wiedeking, M.; Allmond, J.M.; Bernstein, L.A.; Burke, J.T.; Bleuel, D.L.; Clark, R.M.; Fallon, P.; Goldblum, B.L.; Hinners, T.A.; Jeppesen, H.B.; Lee, Sangjin; Lee, I.Y.; Lesher, S.R.; Machiavelli, A.O.; McMahan, M.A.; Morris, D.; Perry, M.; Phair, L.; Scielzo, N.D.; Tabor, S.L.; Tripathi, Vandana; Volya, A.
2011-06-14
Yrast states in 34P were investigated using the 18O(18O,pn) reaction at energies of 20, 24, 25, 30, and 44 MeV at Florida State University and at Lawrence Berkeley National Laboratory. The level scheme was expanded, ray angular distributions were measured, and lifetimes were inferred with the Doppler-shift attenuation method by detecting decay protons in coincidence with one or more rays. The results provide a clearer picture of the evolution of structure approaching the 'Island of Inversion', particularly how the 1 and 2 particle-hole (ph) states fall in energy with increasing neutro number approaching inversion. However, the agreement of the lowest few states with pure sd shell model predictions shows that the level scheme of 34P is not itself inverted. Rather, the accumulated evidence indicates that the 1-ph states start at 2.3 MeV. A good candidate for the lowest 2-ph state lies at 6236 keV, just below the neutron separation energy of 6291 keV. Shell model calculations made using a small modification of the WBP interaction reproduce the negative-parity, 1-ph states rather well.
Inverse problems in geographical economics: parameter identification in the spatial Solow model.
Engbers, Ralf; Burger, Martin; Capasso, Vincenzo
2014-11-13
The identification of production functions from data is an important task in the modelling of economic growth. In this paper, we consider a non-parametric approach to this identification problem in the context of the spatial Solow model which allows for rather general production functions, in particular convex-concave ones that have recently been proposed as reasonable shapes. We formulate the inverse problem and apply Tikhonov regularization. The inverse problem is discretized by finite elements and solved iteratively via a preconditioned gradient descent approach. Numerical results for the reconstruction of the production function are given and analysed at the end of this paper.
Geometric MCMC for infinite-dimensional inverse problems
Beskos, Alexandros; Girolami, Mark; Lan, Shiwei; Farrell, Patrick E.; Stuart, Andrew M.
2017-04-01
Bayesian inverse problems often involve sampling posterior distributions on infinite-dimensional function spaces. Traditional Markov chain Monte Carlo (MCMC) algorithms are characterized by deteriorating mixing times upon mesh-refinement, when the finite-dimensional approximations become more accurate. Such methods are typically forced to reduce step-sizes as the discretization gets finer, and thus are expensive as a function of dimension. Recently, a new class of MCMC methods with mesh-independent convergence times has emerged. However, few of them take into account the geometry of the posterior informed by the data. At the same time, recently developed geometric MCMC algorithms have been found to be powerful in exploring complicated distributions that deviate significantly from elliptic Gaussian laws, but are in general computationally intractable for models defined in infinite dimensions. In this work, we combine geometric methods on a finite-dimensional subspace with mesh-independent infinite-dimensional approaches. Our objective is to speed up MCMC mixing times, without significantly increasing the computational cost per step (for instance, in comparison with the vanilla preconditioned Crank-Nicolson (pCN) method). This is achieved by using ideas from geometric MCMC to probe the complex structure of an intrinsic finite-dimensional subspace where most data information concentrates, while retaining robust mixing times as the dimension grows by using pCN-like methods in the complementary subspace. The resulting algorithms are demonstrated in the context of three challenging inverse problems arising in subsurface flow, heat conduction and incompressible flow control. The algorithms exhibit up to two orders of magnitude improvement in sampling efficiency when compared with the pCN method.
A nonlinear approach of elastic reflection waveform inversion
Guo, Qiang
2016-09-06
Elastic full waveform inversion (EFWI) embodies the original intention of waveform inversion at its inception as it is a better representation of the mostly solid Earth. However, compared with the acoustic P-wave assumption, EFWI for P- and S-wave velocities using multi-component data admitted mixed results. Full waveform inversion (FWI) is a highly nonlinear problem and this nonlinearity only increases under the elastic assumption. Reflection waveform inversion (RWI) can mitigate the nonlinearity by relying on transmissions from reflections focused on inverting low wavenumber components of the model. In our elastic endeavor, we split the P- and S-wave velocities into low wavenumber and perturbation components and propose a nonlinear approach to invert for both of them. The new optimization problem is built on an objective function that depends on both background and perturbation models. We utilize an equivalent stress source based on the model perturbation to generate reflection instead of demigrating from an image, which is applied in conventional RWI. Application on a slice of an ocean-bottom data shows that our method can efficiently update the low wavenumber parts of the model, but more so, obtain perturbations that can be added to the low wavenumbers for a high resolution output.
The inverse problem based on a full dispersive wave equation
Institute of Scientific and Technical Information of China (English)
Gegentana Bao; Naranmandula Bao
2012-01-01
The inverse problem for harmonic waves and wave packets was studied based on a full dispersive wave equation. First, a full dispersive wave equation which describes wave propagation in nondissipative microstructured linear solids is established based on the Mindlin theory, and the dispersion characteristics are discussed. Second, based on the full dispersive wave equation, an inverse problem for determining the four unknown coefficients of wave equa- tion is posed in terms of the frequencies and corresponding wave numbers of four different harmonic waves, and the inverse problem is demonstrated with rigorous mathematical theory. Research proves that the coefficients of wave equation related to material properties can be uniquely determined in cases of normal and anomalous dispersions by measuring the frequen- cies and corresponding wave numbers of four different harmonic waves which propagate in a nondissipative microstructured linear solids.
On numerical methods for direct and inverse problems in electromagnetism
Zemanova, Viera
2009-01-01
This thesis is devoted to the study of processes in the propagation of electromagnetic fields. We do not aim at one particular problem, actually very different kinds of topics are analyzed here. We deal with direct problems as well as with inverse ones, low frequency electromagnetism is discussed and consequently the wave propagation problem in high frequency domain is studied. Study of electromagnetic materials and their behavior is of a huge interest for the technological world. Its impo...
Kouri, Donald J.; Vijay, Amrendra; Zhang, Haiyan; Zhang, Jingfeng; Hoffman, David K.
2007-05-01
A method and system for solving the inverse acoustic scattering problem using an iterative approach with consideration of half-off-shell transition matrix elements (near-field) information, where the Volterra inverse series correctly predicts the first two moments of the interaction, while the Fredholm inverse series is correct only for the first moment and that the Volterra approach provides a method for exactly obtaining interactions which can be written as a sum of delta functions.
A Survey on Inverse Problems for Applied Sciences
Directory of Open Access Journals (Sweden)
Fatih Yaman
2013-01-01
Full Text Available The aim of this paper is to introduce inversion-based engineering applications and to investigate some of the important ones from mathematical point of view. To do this we employ acoustic, electromagnetic, and elastic waves for presenting different types of inverse problems. More specifically, we first study location, shape, and boundary parameter reconstruction algorithms for the inaccessible targets in acoustics. The inverse problems for the time-dependent differential equations of isotropic and anisotropic elasticity are reviewed in the following section of the paper. These problems were the objects of the study by many authors in the last several decades. The physical interpretations for almost all of these problems are given, and the geophysical applications for some of them are described. In our last section, an introduction with many links into the literature is given for modern algorithms which combine techniques from classical inverse problems with stochastic tools into ensemble methods both for data assimilation as well as for forecasting.
Solution of inverse problems with limited forward solver evaluations: a Bayesian perspective
Bilionis, I.; Zabaras, N.
2014-01-01
Solving inverse problems based on computationally demanding forward models is ubiquitously difficult since one is necessarily limited to just a few observations of the response surface. The usual practice is to replace the response surface with a surrogate. However, this approach induces additional uncertainties on the posterior distributions. The main contribution of this work is the reformulation of the Bayesian solution of the inverse problem when the expensive forward model is replaced by the surrogate. We derive three approximations of the reformulated solution with increasing complexity and fidelity. We demonstrate numerically that the proposed approximations capture the uncertainty of the solution of the inverse problem induced by the fact that the forward model is replaced by a finite number of simulations. We demonstrate our approach in two different problems: locating the contamination source of a diffusive process and inferring the permeability field of an oil reservoir based on measurements of the oil-cut curves.
Parameter Identification Of Multilayer Thermal Insulation By Inverse Problems
Nenarokomov, Aleksey V.; Alifanov, Oleg M.; Gonzalez, Vivaldo M.
2012-07-01
The purpose of this paper is to introduce an iterative regularization method in the research of radiative and thermal properties of materials with further applications in the design of Thermal Control Systems (TCS) of spacecrafts. In this paper the radiative and thermal properties (heat capacity, emissivity and thermal conductance) of a multilayered thermal-insulating blanket (MLI), which is a screen-vacuum thermal insulation as a part of the (TCS) for perspective spacecrafts, are estimated. Properties of the materials under study are determined in the result of temperature and heat flux measurement data processing based on the solution of the Inverse Heat Transfer Problem (IHTP) technique. Given are physical and mathematical models of heat transfer processes in a specimen of the multilayered thermal-insulating blanket located in the experimental facility. A mathematical formulation of the IHTP, based on sensitivity function approach, is presented too. The practical testing was performed for specimen of the real MLI. This paper consists of recent researches, which developed the approach suggested at [1].
A time domain sampling method for inverse acoustic scattering problems
Guo, Yukun; Hömberg, Dietmar; Hu, Guanghui; Li, Jingzhi; Liu, Hongyu
2016-06-01
This work concerns the inverse scattering problems of imaging unknown/inaccessible scatterers by transient acoustic near-field measurements. Based on the analysis of the migration method, we propose efficient and effective sampling schemes for imaging small and extended scatterers from knowledge of time-dependent scattered data due to incident impulsive point sources. Though the inverse scattering problems are known to be nonlinear and ill-posed, the proposed imaging algorithms are totally "direct" involving only integral calculations on the measurement surface. Theoretical justifications are presented and numerical experiments are conducted to demonstrate the effectiveness and robustness of our methods. In particular, the proposed static imaging functionals enhance the performance of the total focusing method (TFM) and the dynamic imaging functionals show analogous behavior to the time reversal inversion but without solving time-dependent wave equations.
Inverse problem for multi-body interaction of nonlinear waves
Marruzzo, Alessia; Antenucci, Fabrizio; Pagnani, Andrea; Leuzzi, Luca
2016-01-01
The inverse problem is studied in multi-body systems with nonlinear dynamics representing, e.g., phase-locked wave systems, standard multimode and random lasers. Using a general model for four-body interacting complex-valued variables we test two methods based on pseudolikelihood, respectively with regularization and with decimation, to determine the coupling constants from sets of measured configurations. We test statistical inference predictions for increasing number of sampled configurations and for an externally tunable {\\em temperature}-like parameter mimicing real data noise and helping minimization procedures. Analyzed models with phasors and rotors are generalizations of problems of real-valued spherical problems (e.g., density fluctuations), discrete spins (Ising and vectorial Potts) or finite number of states (standard Potts): inference methods presented here can, then, be straightforward applied to a large class of inverse problems.
A new method of solving the coefficient inverse problem
Institute of Scientific and Technical Information of China (English)
2007-01-01
This paper is concerned with the new method for solving the coefficient inverse problem in the reproducing kernel space. It is different from the previous studies. This method gives accurate results and shows that it is valid by the numerical example.
Hidden information in ill-posed inverse problems
Kahrobaei, S.; Mansoori, M.; Joosten, G.J.P.; Van den Hof, P.M.J.; Jansen, J.D.
2014-01-01
It is well known that parameter updating of large-scale numerical reservoir flow models (a.k.a. ‘computer assisted history matching’) is an ill-posed inverse problem. Typically the number of uncertain parameters in a reservoir flow model is very large whereas the available information for estimating
A comparative analysis of algorithms for the magnetoencephalography inverse problem
Energy Technology Data Exchange (ETDEWEB)
Sorrentino, A [CNR-INFM LAMIA, Genova (Italy); Pascarella, A; Piana, M [Dipartimento di Informatica, Universita di Verona, Ca Vignal 2, Strada le Grazie 15, 37134, Verona (Italy); Campi, C [Dipartimento di Matematica, Universita di Genova, via Dodecaneso 35, 16146, Genova (Italy)], E-mail: sorrentino@fisica.unige.it
2008-11-01
We present a comparison of three methods for the solution of the magnetoencephalography inverse problem. The methods are: an eigenspace projected beamformer, an algorithm implementing multiple signal classification with recursively applied projection and a particle filter for Bayesian tracking. Synthetic data with neurophysiological significance are analyzed by the three methods to recover position and amplitude time course of the active sources.
Solving the Inverse-Square Problem with Complex Variables
Gauthier, N.
2005-01-01
The equation of motion for a mass that moves under the influence of a central, inverse-square force is formulated and solved as a problem in complex variables. To find the solution, the constancy of angular momentum is first established using complex variables. Next, the complex position coordinate and complex velocity of the particle are assumed…
Hidden information in ill-posed inverse problems
Kahrobaei, S.; Mansoori, M.; Joosten, G.J.P.; Van den Hof, P.M.J.; Jansen, J.D.
2014-01-01
It is well known that parameter updating of large-scale numerical reservoir flow models (a.k.a. ‘computer assisted history matching’) is an ill-posed inverse problem. Typically the number of uncertain parameters in a reservoir flow model is very large whereas the available information for estimating
A mathematical framework for inverse wave problems in heterogeneous media
Blazek, K.D.; Stolk, C.; Symes, W.W.
2013-01-01
This paper provides a theoretical foundation for some common formulations of inverse problems in wave propagation, based on hyperbolic systems of linear integro-differential equations with bounded and measurable coefficients. The coefficients of these time-dependent partial differential equations re
Dimensionality Reduction and Uncertainty Quantification for Inverse Problems
van Leeuwen, Tristan
2015-01-01
Many inverse problems in science and engineering involve multi-experiment data and thus require a large number of forward simulations. Dimensionality reduction techniques aim at reducing the number of forward solves by (randomly) subsampling the data. In the special case of non-linear least-squares
Data-Driven Model Order Reduction for Bayesian Inverse Problems
Cui, Tiangang
2014-01-06
One of the major challenges in using MCMC for the solution of inverse problems is the repeated evaluation of computationally expensive numerical models. We develop a data-driven projection- based model order reduction technique to reduce the computational cost of numerical PDE evaluations in this context.
Inverse Eigenvalue Problems for Two Special Acyclic Matrices
Directory of Open Access Journals (Sweden)
Debashish Sharma
2016-03-01
Full Text Available In this paper, we study two inverse eigenvalue problems (IEPs of constructing two special acyclic matrices. The first problem involves the reconstruction of matrices whose graph is a path, from given information on one eigenvector of the required matrix and one eigenvalue of each of its leading principal submatrices. The second problem involves reconstruction of matrices whose graph is a broom, the eigen data being the maximum and minimum eigenvalues of each of the leading principal submatrices of the required matrix. In order to solve the problems, we use the recurrence relations among leading principal minors and the property of simplicity of the extremal eigenvalues of acyclic matrices.
A regularized GMRES method for inverse blackbody radiation problem
Directory of Open Access Journals (Sweden)
Wu Jieer
2013-01-01
Full Text Available The inverse blackbody radiation problem is focused on determining temperature distribution of a blackbody from measured total radiated power spectrum. This problem consists of solving a first kind of Fredholm integral equation and many numerical methods have been proposed. In this paper, a regularized GMRES method is presented to solve the linear ill-posed problem caused by the discretization of such an integral equation. This method projects the orignal problem onto a lower dimensional subspaces by the Arnoldi process. Tikhonov regularization combined with GCV criterion is applied to stabilize the numerical iteration process. Three numerical examples indicate the effectiveness of the regularized GMRES method.
An inverse Sturm–Liouville problem with a fractional derivative
Jin, Bangti
2012-05-01
In this paper, we numerically investigate an inverse problem of recovering the potential term in a fractional Sturm-Liouville problem from one spectrum. The qualitative behaviors of the eigenvalues and eigenfunctions are discussed, and numerical reconstructions of the potential with a Newton method from finite spectral data are presented. Surprisingly, it allows very satisfactory reconstructions for both smooth and discontinuous potentials, provided that the order . α∈. (1,. 2) of fractional derivative is sufficiently away from 2. © 2012 Elsevier Inc.
Forward and inverse problems in fundamental and applied magnetohydrodynamics
2012-01-01
This Minireview summarizes the recent efforts to solve forward and inverse problems as they occur in different branches of fundamental and applied magnetohydrodynamics. As for the forward problem, the main focus is on the numerical treatment of induction processes, including self-excitation of magnetic fields in non-spherical domains and/or under the influence of non-homogeneous material parameters. As an important application of the developed numerical schemes, the functioning of the von-K\\'...
Two numerical methods for an inverse problem for the 2-D Helmholtz equation
Gryazin, Y A; Lucas, T R
2003-01-01
Two solution methods for the inverse problem for the 2-D Helmholtz equation are developed, tested, and compared. The proposed approaches are based on a marching finite-difference scheme which requires the solution of an overdetermined system at each step. The preconditioned conjugate gradient method is used for rapid solutions of these systems and an efficient preconditioner has been developed for this class of problems. Underlying target applications include the imaging of land mines, unexploded ordinance, and pollutant plumes in environmental cleanup sites, each formulated as an inverse problem for a 2-D Helmholtz equation. The images represent the electromagnetic properties of the respective underground regions. Extensive numerical results are presented.
Ivanov, J.; Miller, R.D.; Xia, J.; Steeples, D.
2005-01-01
This paper is the second of a set of two papers in which we study the inverse refraction problem. The first paper, "Types of Geophysical Nonuniqueness through Minimization," studies and classifies the types of nonuniqueness that exist when solving inverse problems depending on the participation of a priori information required to obtain reliable solutions of inverse geophysical problems. In view of the classification developed, in this paper we study the type of nonuniqueness associated with the inverse refraction problem. An approach for obtaining a realistic solution to the inverse refraction problem is offered in a third paper that is in preparation. The nonuniqueness of the inverse refraction problem is examined by using a simple three-layer model. Like many other inverse geophysical problems, the inverse refraction problem does not have a unique solution. Conventionally, nonuniqueness is considered to be a result of insufficient data and/or error in the data, for any fixed number of model parameters. This study illustrates that even for overdetermined and error free data, nonlinear inverse refraction problems exhibit exact-data nonuniqueness, which further complicates the problem of nonuniqueness. By evaluating the nonuniqueness of the inverse refraction problem, this paper targets the improvement of refraction inversion algorithms, and as a result, the achievement of more realistic solutions. The nonuniqueness of the inverse refraction problem is examined initially by using a simple three-layer model. The observations and conclusions of the three-layer model nonuniqueness study are used to evaluate the nonuniqueness of more complicated n-layer models and multi-parameter cell models such as in refraction tomography. For any fixed number of model parameters, the inverse refraction problem exhibits continuous ranges of exact-data nonuniqueness. Such an unfavorable type of nonuniqueness can be uniquely solved only by providing abundant a priori information
From Bayes to Tarantola: New insights to understand uncertainty in inverse problems
Fernández-Martínez, J. L.; Fernández-Muñiz, Z.; Pallero, J. L. G.; Pedruelo-González, L. M.
2013-11-01
Anyone working on inverse problems is aware of their ill-posed character. In the case of inverse problems, this concept (ill-posed) proposed by J. Hadamard in 1902, admits revision since it is somehow related to their ill-conditioning and the use of local optimization methods to find their solution. A more general and interesting approach regarding risk analysis and epistemological decision making would consist in analyzing the existence of families of equivalent model parameters that are compatible with the prior information and predict the observed data within the same error bounds. Otherwise said, the ill-posed character of discrete inverse problems (ill-conditioning) originates that their solution is uncertain. Traditionally nonlinear inverse problems in discrete form have been solved via local optimization methods with regularization, but linear analysis techniques failed to account for the uncertainty in the solution that it is adopted. As a result of this fact uncertainty analysis in nonlinear inverse problems has been approached in a probabilistic framework (Bayesian approach), but these methods are hindered by the curse of dimensionality and by the high computational cost needed to solve the corresponding forward problems. Global optimization techniques are very attractive, but most of the times are heuristic and have the same limitations than Monte Carlo methods. New research is needed to provide uncertainty estimates, especially in the case of high dimensional nonlinear inverse problems with very costly forward problems. After the discredit of deterministic methods and some initial years of Bayesian fever, now the pendulum seems to return back, because practitioners are aware that the uncertainty analysis in high dimensional nonlinear inverse problems cannot (and should not be) solved via random sampling methodologies. The main reason is that the uncertainty “space” of nonlinear inverse problems has a mathematical structure that is embedded in the
Effect of head shape variations among individuals on the EEG/MEG forward and inverse problems.
von Ellenrieder, Nicolás; Muravchik, Carlos H; Wagner, Michael; Nehorai, Arye
2009-03-01
We study the effect of the head shape variations on the EEG/magnetoencephalography (MEG) forward and inverse problems. We build a random head model such that each sample represents the head shape of a different individual and solve the forward problem assuming this random head model, using a polynomial chaos expansion. The random solution of the forward problem is then used to quantify the effect of the geometry when the inverse problem is solved with a standard head model. The results derived with this approach are valid for a continuous family of head models, rather than just for a set of cases. The random model consists of three random surfaces that define layers of different electric conductivity, and we built an example based on a set of 30 deterministic models from adults. Our results show that for a dipolar source model, the effect of the head shape variations on the EEG/MEG inverse problem due to the random head model is slightly larger than the effect of the electronic noise present in the sensors. The variations in the EEG inverse problem solutions are due to the variations in the shape of the volume conductor, while the variations in the MEG inverse problem solutions, larger than the EEG ones, are caused mainly by the variations of the absolute position of the sources in a coordinate system based on anatomical landmarks, in which the magnetometers have a fixed position.
Solving Large-Scale Inverse Magnetostatic Problems using the Adjoint Method
Bruckner, Florian; Wautischer, Gregor; Huber, Christian; Vogler, Christoph; Hinze, Michael; Suess, Dieter
2016-01-01
An efficient algorithm for the reconstruction of the magnetization state within magnetic components is presented. The occurring inverse magnetostatic problem is solved by means of an adjoint approach, based on the Fredkin-Koehler method for the solution of the forward problem. Due to the use of hybrid FEM-BEM coupling combined with matrix compression techniques the resulting algorithm is well suited for large-scale problems. Furthermore the reconstruction of the magnetization state within a permanent magnet is demonstrated.
SIAM conference on inverse problems: Geophysical applications. Final technical report
Energy Technology Data Exchange (ETDEWEB)
NONE
1995-12-31
This conference was the second in a series devoted to a particular area of inverse problems. The theme of this series is to discuss problems of major scientific importance in a specific area from a mathematical perspective. The theme of this symposium was geophysical applications. In putting together the program we tried to include a wide range of mathematical scientists and to interpret geophysics in as broad a sense as possible. Our speaker came from industry, government laboratories, and diverse departments in academia. We managed to attract a geographically diverse audience with participation from five continents. There were talks devoted to seismology, hydrology, determination of the earth`s interior on a global scale as well as oceanographic and atmospheric inverse problems.
Iterative total variation schemes for nonlinear inverse problems
Bachmayr, Markus; Burger, Martin
2009-10-01
In this paper we discuss the construction, analysis and implementation of iterative schemes for the solution of inverse problems based on total variation regularization. Via different approximations of the nonlinearity we derive three different schemes resembling three well-known methods for nonlinear inverse problems in Hilbert spaces, namely iterated Tikhonov, Levenberg-Marquardt and Landweber. These methods can be set up such that all arising subproblems are convex optimization problems, analogous to those appearing in image denoising or deblurring. We provide a detailed convergence analysis and appropriate stopping rules in the presence of data noise. Moreover, we discuss the implementation of the schemes and the application to distributed parameter estimation in elliptic partial differential equations.
Frechet derivatives for shallow water ocean acoustic inverse problems
Odom, Robert I.
2003-04-01
For any inverse problem, finding a model fitting the data is only half the problem. Most inverse problems of interest in ocean acoustics yield nonunique model solutions, and involve inevitable trade-offs between model and data resolution and variance. Problems of uniqueness and resolution and variance trade-offs can be addressed by examining the Frechet derivatives of the model-data functional with respect to the model variables. Tarantola [Inverse Problem Theory (Elsevier, Amsterdam, 1987), p. 613] published analytical formulas for the basic derivatives, e.g., derivatives of pressure with respect to elastic moduli and density. Other derivatives of interest, such as the derivative of transmission loss with respect to attenuation, can be easily constructed using the chain rule. For a range independent medium the analytical formulas involve only the Green's function and the vertical derivative of the Green's function for the medium. A crucial advantage of the analytical formulas for the Frechet derivatives over numerical differencing is that they can be computed with a single pass of any program which supplies the Green's function. Various derivatives of interest in shallow water ocean acoustics are presented and illustrated by an application to the sensitivity of measured pressure to shallow water sediment properties. [Work supported by ONR.
Integral geometry and inverse problems for hyperbolic equations
Romanov, V G
1974-01-01
There are currently many practical situations in which one wishes to determine the coefficients in an ordinary or partial differential equation from known functionals of its solution. These are often called "inverse problems of mathematical physics" and may be contrasted with problems in which an equation is given and one looks for its solution under initial and boundary conditions. Although inverse problems are often ill-posed in the classical sense, their practical importance is such that they may be considered among the pressing problems of current mathematical re search. A. N. Tihonov showed [82], [83] that there is a broad class of inverse problems for which a particular non-classical definition of well-posed ness is appropriate. This new definition requires that a solution be unique in a class of solutions belonging to a given subset M of a function space. The existence of a solution in this set is assumed a priori for some set of data. The classical requirement of continuous dependence of the solutio...
Obtaining sparse distributions in 2D inverse problems
Reci, A.; Sederman, A. J.; Gladden, L. F.
2017-08-01
The mathematics of inverse problems has relevance across numerous estimation problems in science and engineering. L1 regularization has attracted recent attention in reconstructing the system properties in the case of sparse inverse problems; i.e., when the true property sought is not adequately described by a continuous distribution, in particular in Compressed Sensing image reconstruction. In this work, we focus on the application of L1 regularization to a class of inverse problems; relaxation-relaxation, T1-T2, and diffusion-relaxation, D-T2, correlation experiments in NMR, which have found widespread applications in a number of areas including probing surface interactions in catalysis and characterizing fluid composition and pore structures in rocks. We introduce a robust algorithm for solving the L1 regularization problem and provide a guide to implementing it, including the choice of the amount of regularization used and the assignment of error estimates. We then show experimentally that L1 regularization has significant advantages over both the Non-Negative Least Squares (NNLS) algorithm and Tikhonov regularization. It is shown that the L1 regularization algorithm stably recovers a distribution at a signal to noise ratio direct spectroscopic discrimination is impossible, and hence measurement of chemical composition within porous media, such as catalysts or rocks, is possible while still being stable to high levels of noise.
Seismic inverse scattering in the downward continuation approach
Stolk, C.C.; Hoop, de M.V.
2006-01-01
Seismic data are commonly modeled by a linearization around a smooth background medium in combination with a high frequency approximation. The perturbation of the medium coefficient is assumed to contain the discontinuities. This leads to two inverse problems, first the linearized inverse problem fo
Statistical mechanics of the inverse Ising problem and the optimal objective function
Berg, Johannes
2017-08-01
The inverse Ising problem seeks to reconstruct the parameters of an Ising Hamiltonian on the basis of spin configurations sampled from the Boltzmann measure. Over the last decade, many applications of the inverse Ising problem have arisen, driven by the advent of large-scale data across different scientific disciplines. Recently, strategies to solve the inverse Ising problem based on convex optimisation have proven to be very successful. These approaches maximise particular objective functions with respect to the model parameters. Examples are the pseudolikelihood method and interaction screening. In this paper, we establish a link between approaches to the inverse Ising problem based on convex optimisation and the statistical physics of disordered systems. We characterise the performance of an arbitrary objective function and calculate the objective function which optimally reconstructs the model parameters. We evaluate the optimal objective function within a replica-symmetric ansatz and compare the results of the optimal objective function with other reconstruction methods. Apart from giving a theoretical underpinning to solving the inverse Ising problem by convex optimisation, the optimal objective function outperforms state-of-the-art methods, albeit by a small margin.
Wang, Mengyu; Brigham, John C.
2017-03-01
A computationally efficient gradient-based optimization approach for inverse material characterization from incomplete system response measurements that can utilize a generally applicable parameterization (e.g., finite element-type parameterization) is presented and evaluated. The key to this inverse characterization algorithm is the use of a direct inversion strategy with Gappy proper orthogonal decomposition (POD) response field estimation to initialize the inverse solution estimate prior to gradient-based optimization. Gappy POD is used to estimate the complete (i.e., all components over the entire spatial domain) system response field from incomplete (e.g., partial spatial distribution) measurements obtained from some type of system testing along with some amount of a priori information regarding the potential distribution of the unknown material property. The estimated complete system response is used within a physics-based direct inversion procedure with a finite element-type parameterization to estimate the spatial distribution of the desired unknown material property with minimal computational expense. Then, this estimated spatial distribution of the unknown material property is used to initialize a gradient-based optimization approach, which uses the adjoint method for computationally efficient gradient calculations, to produce the final estimate of the material property distribution. The three-step [(1) Gappy POD, (2) direct inversion, and (3) gradient-based optimization] inverse characterization approach is evaluated through simulated test problems based on the characterization of elastic modulus distributions with localized variations (e.g., inclusions) within simple structures. Overall, this inverse characterization approach is shown to efficiently and consistently provide accurate inverse characterization estimates for material property distributions from incomplete response field measurements. Moreover, the solution procedure is shown to be capable
Solving Inverse Kinematics – A New Approach to the Extended Jacobian Technique
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M. Šoch
2005-01-01
Full Text Available This paper presents a brief summary of current numerical algorithms for solving the Inverse Kinematics problem. Then a new approach based on the Extended Jacobian technique is compared with the current Jacobian Inversion method. The presented method is intended for use in the field of computer graphics for animation of articulated structures.
Institute of Scientific and Technical Information of China (English)
WANG Yi-bo; YANG Hai-tian; WU Rui-feng
2005-01-01
By modeling direct transient heat conduction problems via finite element method (FEM) and precise integral algorithm, a new approach is presented to solve transient inverse heat conduction problems with multi-variables. Firstly, the spatial space and temporal domain are discretized by FEM and precise integral algorithm respectively. Then, the high accuracy semi-analytical solution of direct problem can be got. Finally, based on the solution, the computing model of inverse problem and expression of sensitivity analysis are established. Single variable and variables combined identifications including thermal parameters, boundary conditions and source-related terms etc. are given to validate the approach proposed in 1-D and 2-D cases. The effects of noise data and initial guess on the results are investigated. The numerical examples show the effectiveness of this approach.
Forward and inverse problems in fundamental and applied magnetohydrodynamics
Giesecke, Andre; Stefani, Frank; Wondrak, Thomas; Xu, Mingtian
2013-03-01
This minireview summarizes the recent efforts to solve forward and inverse problems as they occur in different branches of fundamental and applied magnetohydrodynamics. For the forward problem, the main focus is on the numerical treatment of induction processes, including self-excitation of magnetic fields in non-spherical domains and/or under the influence of non-homogeneous material parameters. As an important application of the developed numerical schemes, the functioning of the von-Kármán-sodium (VKS) dynamo experiment is shown to depend crucially on the presence of soft-iron impellers. As for the inverse problem, the main focus is on the mathematical background and some initial practical applications of contactless inductive flow tomography (CIFT), in which flow induced magnetic field perturbations are utilized to reconstruct the velocity field. The promises of CIFT for flow field monitoring in the continuous casting of steel are substantiated by results obtained at a test rig with a low-melting liquid metal. While CIFT is presently restricted to flows with low magnetic Reynolds numbers, some selected problems from non-linear inverse dynamo theory, with possible applications to geo- and astrophysics, are also discussed.
General bounds for electrode mislocation on the EEG inverse problem.
Beltrachini, L; von Ellenrieder, N; Muravchik, C H
2011-07-01
We analyze the effect of electrode mislocation on the electroencephalography (EEG) inverse problem using the Cramér-Rao bound (CRB) for single dipolar source parameters. We adopt a realistic head shape model, and solve the forward problem using the Boundary Element Method; the use of the CRB allows us to obtain general results which do not depend on the algorithm used for solving the inverse problem. We consider two possible causes for the electrode mislocation, errors in the measurement of the electrode positions and an imperfect registration between the electrodes and the scalp surfaces. For 120 electrodes placed in the scalp according to the 10-20 standard, and errors on the electrode location with a standard deviation of 5mm, the lower bound on the standard deviation in the source depth estimation is approximately 1mm in the worst case. Therefore, we conclude that errors in the electrode location may be tolerated since their effect on the EEG inverse problem are negligible from a practical point of view. Copyright © 2010 Elsevier Ireland Ltd. All rights reserved.
Wu, Sheng-Jhih; Chu, Moody T.
2017-08-01
An inverse eigenvalue problem usually entails two constraints, one conditioned upon the spectrum and the other on the structure. This paper investigates the problem where triple constraints of eigenvalues, singular values, and diagonal entries are imposed simultaneously. An approach combining an eclectic mix of skills from differential geometry, optimization theory, and analytic gradient flow is employed to prove the solvability of such a problem. The result generalizes the classical Mirsky, Sing-Thompson, and Weyl-Horn theorems concerning the respective majorization relationships between any two of the arrays of main diagonal entries, eigenvalues, and singular values. The existence theory fills a gap in the classical matrix theory. The problem might find applications in wireless communication and quantum information science. The technique employed can be implemented as a first-step numerical method for constructing the matrix. With slight modification, the approach might be used to explore similar types of inverse problems where the prescribed entries are at general locations.
Inverse minimum spanning tree problem and reverse shortest-path problem with discrete values
Institute of Scientific and Technical Information of China (English)
LIU Longcheng; HE Yong
2006-01-01
In this paper, we consider two network improvement problems with given discrete values: the inverse minimum spanning tree problem and the reverse shortest-path problem, where the decrements of the weight of the edges are given discrete values. First,for the three models of the inverse minimum spanning tree problem (the sum-type, the bottleneck-type and the constrained bottlenecktype), we present their respective strongly polynomial algorithms. Then, we show that the reverse shortest-path problem is strongly NP-complete.
Aghasi, Alireza; Miller, Eric L; Ramsburg, C Andrew; Abriola, Linda M
2013-01-01
This paper presents a new joint inversion approach to shape-based inverse problems. Given two sets of data from distinct physical models, the main objective is to obtain a unified characterization of inclusions within the spatial domain of the physical properties to be reconstructed. Although our proposed method generally applies to many types of inversion problems, the main motivation here is to characterize subsurface contaminant source-zones by processing down gradient hydrological data and cross-gradient electrical resistance tomography (ERT) observations. Inspired by Newton's method for multi-objective optimization, we present an iterative inversion scheme that suggests taking descent steps that can simultaneously reduce both data-model misfit terms. Such an approach, however, requires solving a non-smooth convex problem at every iteration, which is computationally expensive for a pixel-based inversion over the whole domain. Instead, we employ a parametric level set (PaLS) technique that substantially re...
Inverse problem of HIV cell dynamics using Genetic Algorithms
González, J. A.; Guzmán, F. S.
2017-01-01
In order to describe the cell dynamics of T-cells in a patient infected with HIV, we use a flavour of Perelson's model. This is a non-linear system of Ordinary Differential Equations that describes the evolution of healthy, latently infected, infected T-cell concentrations and the free viral cells. Different parameters in the equations give different dynamics. Considering the concentration of these types of cells is known for a particular patient, the inverse problem consists in estimating the parameters in the model. We solve this inverse problem using a Genetic Algorithm (GA) that minimizes the error between the solutions of the model and the data from the patient. These errors depend on the parameters of the GA, like mutation rate and population, although a detailed analysis of this dependence will be described elsewhere.
A new approach for inversion of large random matrices in massive MIMO systems.
Directory of Open Access Journals (Sweden)
Muhammad Ali Raza Anjum
Full Text Available We report a novel approach for inversion of large random matrices in massive Multiple-Input Multiple Output (MIMO systems. It is based on the concept of inverse vectors in which an inverse vector is defined for each column of the principal matrix. Such an inverse vector has to satisfy two constraints. Firstly, it has to be in the null-space of all the remaining columns. We call it the null-space problem. Secondly, it has to form a projection of value equal to one in the direction of selected column. We term it as the normalization problem. The process essentially decomposes the inversion problem and distributes it over columns. Each column can be thought of as a node in the network or a particle in a swarm seeking its own solution, the inverse vector, which lightens the computational load on it. Another benefit of this approach is its applicability to all three cases pertaining to a linear system: the fully-determined, the over-determined, and the under-determined case. It eliminates the need of forming the generalized inverse for the last two cases by providing a new way to solve the least squares problem and the Moore and Penrose's pseudoinverse problem. The approach makes no assumption regarding the size, structure or sparsity of the matrix. This makes it fully applicable to much in vogue large random matrices arising in massive MIMO systems. Also, the null-space problem opens the door for a plethora of methods available in literature for null-space computation to enter the realm of matrix inversion. There is even a flexibility of finding an exact or approximate inverse depending on the null-space method employed. We employ the Householder's null-space method for exact solution and present a complete exposition of the new approach. A detailed comparison with well-established matrix inversion methods in literature is also given.
A new approach for inversion of large random matrices in massive MIMO systems.
Anjum, Muhammad Ali Raza; Ahmed, Muhammad Mansoor
2014-01-01
We report a novel approach for inversion of large random matrices in massive Multiple-Input Multiple Output (MIMO) systems. It is based on the concept of inverse vectors in which an inverse vector is defined for each column of the principal matrix. Such an inverse vector has to satisfy two constraints. Firstly, it has to be in the null-space of all the remaining columns. We call it the null-space problem. Secondly, it has to form a projection of value equal to one in the direction of selected column. We term it as the normalization problem. The process essentially decomposes the inversion problem and distributes it over columns. Each column can be thought of as a node in the network or a particle in a swarm seeking its own solution, the inverse vector, which lightens the computational load on it. Another benefit of this approach is its applicability to all three cases pertaining to a linear system: the fully-determined, the over-determined, and the under-determined case. It eliminates the need of forming the generalized inverse for the last two cases by providing a new way to solve the least squares problem and the Moore and Penrose's pseudoinverse problem. The approach makes no assumption regarding the size, structure or sparsity of the matrix. This makes it fully applicable to much in vogue large random matrices arising in massive MIMO systems. Also, the null-space problem opens the door for a plethora of methods available in literature for null-space computation to enter the realm of matrix inversion. There is even a flexibility of finding an exact or approximate inverse depending on the null-space method employed. We employ the Householder's null-space method for exact solution and present a complete exposition of the new approach. A detailed comparison with well-established matrix inversion methods in literature is also given.
Inverse Problem in the Surface EMG: A Feasibility Study
2007-11-02
the arm containing the studied muscle is modelised (figure 1). The multi- electrode recording system is composed of 16 electrodes regularly...nature of this study (feasibility of the inverse problem in SEMG), the modelisation was made with a few simplifying hypotheses in mind to facilitate the...implementation of the localisation algorithm. This modelisation is nevertheless inspired by previous works [6], and the shapes of the synthetic MUAPs
Inverse Problems for Matrix Exponential in System Identification: System Aliasing
Yue, Zuogong; Thunberg, Johan; Goncalves, Jorge
2016-01-01
This note addresses identification of the $A$-matrix in continuous time linear dynamical systems on state-space form. If this matrix is partially known or known to have a sparse structure, such knowledge can be used to simplify the identification. We begin by introducing some general conditions for solvability of the inverse problems for matrix exponential. Next, we introduce "system aliasing" as an issue in the identification of slow sampled systems. Such aliasing give rise to non-unique mat...
Explicit inverse distance weighting mesh motion for coupled problems
Witteveen, J.A.S.; Bijl, H.
2009-01-01
An explicit mesh motion algorithm based on inverse distance weighting interpolation is presented. The explicit formulation leads to a fast mesh motion algorithm and an easy implementation. In addition, the proposed point-by-point method is robust and flexible in case of large deformations, hanging nodes, and parallelization. Mesh quality results and CPU time comparisons are presented for triangular and hexahedral unstructured meshes in an airfoil flutter fluid-structure interaction problem.
Explicit solution for an infinite dimensional generalized inverse eigenvalue problem
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Kazem Ghanbari
2001-01-01
Full Text Available We study a generalized inverse eigenvalue problem (GIEP, Ax=λBx, in which A is a semi-infinite Jacobi matrix with positive off-diagonal entries ci>0, and B= diag (b0,b1,…, where bi≠0 for i=0,1,…. We give an explicit solution by establishing an appropriate spectral function with respect to a given set of spectral data.
Liu, Gao-Lian
1991-01-01
Advances in inverse design and optimization theory in engineering fields in China are presented. Two original approaches, the image-space approach and the variational approach, are discussed in terms of turbomachine aerodynamic inverse design. Other areas of research in turbomachine aerodynamic inverse design include the improved mean-streamline (stream surface) method and optimization theory based on optimal control. Among the additional engineering fields discussed are the following: the inverse problem of heat conduction, free-surface flow, variational cogeneration of optimal grid and flow field, and optimal meshing theory of gears.
Solving the structural inverse gravity problem by the modified gradient methods
Martyshko, P. S.; Akimova, E. N.; Misilov, V. E.
2016-09-01
New methods for solving the three-dimensional inverse gravity problem in the class of contact surfaces are described. Based on the approach previously suggested by the authors, new algorithms are developed. Application of these algorithms significantly reduces the number of the iterations and computing time compared to the previous ones. The algorithms have been numerically implemented on the multicore processor. The example of solving the structural inverse gravity problem for a model of four-layer medium (with the use of gravity field measurements) is constructed.
THE INVERSE PROBLEM OF A REPRODUCTION MODEL OF NATIONAL INCOME
Directory of Open Access Journals (Sweden)
Laipanova Z. M.
2016-02-01
Full Text Available In practice, there were developed and tested some mathematical models of balance relationships (balance model, economic growth, expanding economy, labour market, theories of consumption, production, competitive equilibrium models of the economy in conditions of imperfect competition and others. The basis of these models were based on linear algebra, mathematical analysis, mathematical programming, differential equations, optimization methods, optimal control theory, probability theory, stochastic processes, operations research, game theory, statistical analysis. The inverse problem in various models of mathematical Economics was considered quite rare. These tasks were sufficiently investigated in the study of physical processes. As shown by the analysis of the theoretical and applied studies of economic processes, they represent considerable interest for practice. Therefore, the considered in the study inverse problems of the mathematical model, as it is shown by the already introduced results of other mathematical models, are of considerable interest in applied and theoretical research. In this article, the authors have formulated and investigated an inverse problem for a model of economic growth. For its solution the authors propose to build a system of algebraic equations, using a reproduction model of national income; then, using methods of quadratic programming, to find the best average quadratic estimates of the model parameter
The Neuroelectromagnetic Inverse Problem and the Zero Dipole Localization Error
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Rolando Grave de Peralta
2009-01-01
Full Text Available A tomography of neural sources could be constructed from EEG/MEG recordings once the neuroelectromagnetic inverse problem (NIP is solved. Unfortunately the NIP lacks a unique solution and therefore additional constraints are needed to achieve uniqueness. Researchers are then confronted with the dilemma of choosing one solution on the basis of the advantages publicized by their authors. This study aims to help researchers to better guide their choices by clarifying what is hidden behind inverse solutions oversold by their apparently optimal properties to localize single sources. Here, we introduce an inverse solution (ANA attaining perfect localization of single sources to illustrate how spurious sources emerge and destroy the reconstruction of simultaneously active sources. Although ANA is probably the simplest and robust alternative for data generated by a single dominant source plus noise, the main contribution of this manuscript is to show that zero localization error of single sources is a trivial and largely uninformative property unable to predict the performance of an inverse solution in presence of simultaneously active sources. We recommend as the most logical strategy for solving the NIP the incorporation of sound additional a priori information about neural generators that supplements the information contained in the data.
From inverse problems in mathematical physiology to quantitative differential diagnoses.
Directory of Open Access Journals (Sweden)
Sven Zenker
2007-11-01
Full Text Available The improved capacity to acquire quantitative data in a clinical setting has generally failed to improve outcomes in acutely ill patients, suggesting a need for advances in computer-supported data interpretation and decision making. In particular, the application of mathematical models of experimentally elucidated physiological mechanisms could augment the interpretation of quantitative, patient-specific information and help to better target therapy. Yet, such models are typically complex and nonlinear, a reality that often precludes the identification of unique parameters and states of the model that best represent available data. Hypothesizing that this non-uniqueness can convey useful information, we implemented a simplified simulation of a common differential diagnostic process (hypotension in an acute care setting, using a combination of a mathematical model of the cardiovascular system, a stochastic measurement model, and Bayesian inference techniques to quantify parameter and state uncertainty. The output of this procedure is a probability density function on the space of model parameters and initial conditions for a particular patient, based on prior population information together with patient-specific clinical observations. We show that multimodal posterior probability density functions arise naturally, even when unimodal and uninformative priors are used. The peaks of these densities correspond to clinically relevant differential diagnoses and can, in the simplified simulation setting, be constrained to a single diagnosis by assimilating additional observations from dynamical interventions (e.g., fluid challenge. We conclude that the ill-posedness of the inverse problem in quantitative physiology is not merely a technical obstacle, but rather reflects clinical reality and, when addressed adequately in the solution process, provides a novel link between mathematically described physiological knowledge and the clinical concept of
From inverse problems in mathematical physiology to quantitative differential diagnoses.
Zenker, Sven; Rubin, Jonathan; Clermont, Gilles
2007-11-01
The improved capacity to acquire quantitative data in a clinical setting has generally failed to improve outcomes in acutely ill patients, suggesting a need for advances in computer-supported data interpretation and decision making. In particular, the application of mathematical models of experimentally elucidated physiological mechanisms could augment the interpretation of quantitative, patient-specific information and help to better target therapy. Yet, such models are typically complex and nonlinear, a reality that often precludes the identification of unique parameters and states of the model that best represent available data. Hypothesizing that this non-uniqueness can convey useful information, we implemented a simplified simulation of a common differential diagnostic process (hypotension in an acute care setting), using a combination of a mathematical model of the cardiovascular system, a stochastic measurement model, and Bayesian inference techniques to quantify parameter and state uncertainty. The output of this procedure is a probability density function on the space of model parameters and initial conditions for a particular patient, based on prior population information together with patient-specific clinical observations. We show that multimodal posterior probability density functions arise naturally, even when unimodal and uninformative priors are used. The peaks of these densities correspond to clinically relevant differential diagnoses and can, in the simplified simulation setting, be constrained to a single diagnosis by assimilating additional observations from dynamical interventions (e.g., fluid challenge). We conclude that the ill-posedness of the inverse problem in quantitative physiology is not merely a technical obstacle, but rather reflects clinical reality and, when addressed adequately in the solution process, provides a novel link between mathematically described physiological knowledge and the clinical concept of differential diagnoses
Source localization in electromyography using the inverse potential problem
van den Doel, Kees; Ascher, Uri M.; Pai, Dinesh K.
2011-02-01
We describe an efficient method for reconstructing the activity in human muscles from an array of voltage sensors on the skin surface. MRI is used to obtain morphometric data which are segmented into muscle tissue, fat, bone and skin, from which a finite element model for volume conduction is constructed. The inverse problem of finding the current sources in the muscles is solved using a careful regularization technique which adds a priori information, yielding physically reasonable solutions from among those that satisfy the basic potential problem. Several regularization functionals are considered and numerical experiments on a 2D test model are performed to determine which performs best. The resulting scheme leads to numerical difficulties when applied to large-scale 3D problems. We clarify the nature of these difficulties and provide a method to overcome them, which is shown to perform well in the large-scale problem setting.
Inverse problem for in vivo NMR spatial localization
Energy Technology Data Exchange (ETDEWEB)
Hasenfeld, A.C.
1985-11-01
The basic physical problem of NMR spatial localization is considered. To study diseased sites, one must solve the problem of adequately localizing the NMR signal. We formulate this as an inverse problem. As the NMR Bloch equations determine the motion of nuclear spins in applied magnetic fields, a theoretical study is undertaken to answer the question of how to design magnetic field configurations to achieve these localized excited spin populations. Because of physical constraints in the production of the relevant radiofrequency fields, the problem factors into a temporal one and a spatial one. We formulate the temporal problem as a nonlinear transformation, called the Bloch Transform, from the rf input to the magnetization response. In trying to invert this transformation, both linear (for the Fourier Transform) and nonlinear (for the Bloch Transform) modes of radiofrequency excitation are constructed. The spatial problem is essentially a statics problem for the Maxwell equations of electromagnetism, as the wavelengths of the radiation considered are on the order of ten meters, and so propagation effects are negligible. In the general case, analytic solutions are unavailable, and so the methods of computer simulation are used to map the rf field spatial profiles. Numerical experiments are also performed to verify the theoretical analysis, and experimental confirmation of the theory is carried out on the 0.5 Tesla IBM/Oxford Imaging Spectrometer at the LBL NMR Medical Imaging Facility. While no explicit inverse is constructed to ''solve'' this problem, the combined theoretical/numerical analysis is validated experimentally, justifying the approximations made. 56 refs., 31 figs.
ROBUST PARTIAL INVERSE NETWORK FLOW PROBLEMS%强部分逆网络流问题
Institute of Scientific and Technical Information of China (English)
杨晓光
2001-01-01
In this paper,a new model for inverse network flow problems,robust partial inverse problem is presented. For a given partial solution,the robust partial inverse problem is to modify the coefficients optimally such that all full solutions containing the partial solution become optimal under new coefficients. It has been shown that the robust partial inverse spanning tree problem can be formulated as a combinatorial linear program,while the robust partial inverse minimum cut problem and the robust partial inverse assignment problem can be solved by combinatorial strongly polynomial algorithms.
Inverse problem of pulsed eddy current field of ferromagnetic plates
Chen, Xing-Le; Lei, Yin-Zhao
2015-03-01
To determine the wall thickness, conductivity and permeability of a ferromagnetic plate, an inverse problem is established with measured values and calculated values of time-domain induced voltage in pulsed eddy current testing on the plate. From time-domain analytical expressions of the partial derivatives of induced voltage with respect to parameters, it is deduced that the partial derivatives are approximately linearly dependent. Then the constraints of these parameters are obtained by solving a partial linear differential equation. It is indicated that only the product of conductivity and wall thickness, and the product of relative permeability and wall thickness can be determined accurately through the inverse problem with time-domain induced voltage. In the practical testing, supposing the conductivity of the ferromagnetic plate under test is a fixed value, and then the relative variation of wall thickness between two testing points can be calculated via the ratio of the corresponding inversion results of the product of conductivity and wall thickness. Finally, this method for wall thickness measurement is verified by the experiment results of a carbon steel plate. Project supported by the National Defense Basic Technology Research Program of China (Grant No. Z132013T001).
Inverse problem for multi-body interaction of nonlinear waves.
Marruzzo, Alessia; Tyagi, Payal; Antenucci, Fabrizio; Pagnani, Andrea; Leuzzi, Luca
2017-06-14
The inverse problem is studied in multi-body systems with nonlinear dynamics representing, e.g., phase-locked wave systems, standard multimode and random lasers. Using a general model for four-body interacting complex-valued variables we test two methods based on pseudolikelihood, respectively with regularization and with decimation, to determine the coupling constants from sets of measured configurations. We test statistical inference predictions for increasing number of sampled configurations and for an externally tunable temperature-like parameter mimicing real data noise and helping minimization procedures. Analyzed models with phasors and rotors are generalizations of problems of real-valued spherical problems (e.g., density fluctuations), discrete spins (Ising and vectorial Potts) or finite number of states (standard Potts): inference methods presented here can, then, be straightforward applied to a large class of inverse problems. The high versatility of the exposed techniques also concerns the number of expected interactions: results are presented for different graph topologies, ranging from sparse to dense graphs.
The physical and mathematical aspects of inverse problems in radiation detection and applications
Energy Technology Data Exchange (ETDEWEB)
Hussein, Esam M.A., E-mail: hussein@unb.ca [Laboratory for Threat Material Detection, Department of Mechanical Engineering, University of New Brunswick, Fredericton, NB, E3B 5A3 (Canada)
2012-07-15
The inverse problem is the problem of converting detectable measurements into useful quantifiable indications. It is the problem of spectrum unfolding, image reconstruction, identifying a threat material, or devising a radiotherapy plan. The solution of an inverse problem requires a forward model that relates the quantities of interest to measurements. This paper explores the physical issues associated with formulating a radiation-transport forward model best suited for inversion, and the mathematical challenges associated with the solution of the corresponding inverse problem.
Quadratic function approaching method for magnetotelluric soundingdata inversion
Energy Technology Data Exchange (ETDEWEB)
Liangjun, Yan; Wenbao, Hu; Zhang, Keni
2004-04-05
The quadratic function approaching method (QFAM) is introduced for magnetotelluric sounding (MT) data inversion. The method takes the advantage of that quadratic function has single extreme value, which avoids leading to an inversion solution for local minimum and ensures the solution for global minimization of an objective function. The method does not need calculation of sensitivity matrix and not require a strict initial earth model. Examples for synthetic data and field measurement data indicate that the proposed inversion method is effective.
Network connections that evolve to circumvent the inverse optics problem.
Ng, Cherlyn; Sundararajan, Janani; Hogan, Michael; Purves, Dale
2013-01-01
A fundamental problem in vision science is how useful perceptions and behaviors arise in the absence of information about the physical sources of retinal stimuli (the inverse optics problem). Psychophysical studies show that human observers contend with this problem by using the frequency of occurrence of stimulus patterns in cumulative experience to generate percepts. To begin to understand the neural mechanisms underlying this strategy, we examined the connectivity of simple neural networks evolved to respond according to the cumulative rank of stimulus luminance values. Evolved similarities with the connectivity of early level visual neurons suggests that biological visual circuitry uses the same mechanisms as a means of creating useful perceptions and behaviors without information about the real world.
Network connections that evolve to circumvent the inverse optics problem.
Directory of Open Access Journals (Sweden)
Cherlyn Ng
Full Text Available A fundamental problem in vision science is how useful perceptions and behaviors arise in the absence of information about the physical sources of retinal stimuli (the inverse optics problem. Psychophysical studies show that human observers contend with this problem by using the frequency of occurrence of stimulus patterns in cumulative experience to generate percepts. To begin to understand the neural mechanisms underlying this strategy, we examined the connectivity of simple neural networks evolved to respond according to the cumulative rank of stimulus luminance values. Evolved similarities with the connectivity of early level visual neurons suggests that biological visual circuitry uses the same mechanisms as a means of creating useful perceptions and behaviors without information about the real world.
Inverse problem for porosity estimation during solidification of TNT
Directory of Open Access Journals (Sweden)
Aldélio Bueno Caldeira
2016-08-01
Full Text Available In the present study, the porosity formed during the solidification process is estimated by an inverse problem technique based on particle swarm optimization. The effective heat capacity method is adopted to model the heat transfer problem. The transient-diffusive heat transfer equation is solved numerically by the finite volume method with an explicit scheme, employing the central difference interpolation function. The solution of the direct problem is compared to reference solutions. The model is applied to trinitrotoluene (TNT solidification process. The results show that the proposed procedure was able to estimate the porosity for different Stefan numbers. The analysis of the heat flux in the mold is indicated to predict the porosity formation during the casting process.
Corrado, Cesare; Gerbeau, Jean-Frédéric; Moireau, Philippe
2015-02-01
This work addresses the inverse problem of electrocardiography from a new perspective, by combining electrical and mechanical measurements. Our strategy relies on the definition of a model of the electromechanical contraction which is registered on ECG data but also on measured mechanical displacements of the heart tissue typically extracted from medical images. In this respect, we establish in this work the convergence of a sequential estimator which combines for such coupled problems various state of the art sequential data assimilation methods in a unified consistent and efficient framework. Indeed, we aggregate a Luenberger observer for the mechanical state and a Reduced-Order Unscented Kalman Filter applied on the parameters to be identified and a POD projection of the electrical state. Then using synthetic data we show the benefits of our approach for the estimation of the electrical state of the ventricles along the heart beat compared with more classical strategies which only consider an electrophysiological model with ECG measurements. Our numerical results actually show that the mechanical measurements improve the identifiability of the electrical problem allowing to reconstruct the electrical state of the coupled system more precisely. Therefore, this work is intended to be a first proof of concept, with theoretical justifications and numerical investigations, of the advantage of using available multi-modal observations for the estimation and identification of an electromechanical model of the heart.
Forward and inverse problems in fundamental and applied magnetohydrodynamics
Giesecke, Andre; Wondrak, Thomas; Xu, Mingtian
2012-01-01
This Minireview summarizes the recent efforts to solve forward and inverse problems as they occur in different branches of fundamental and applied magnetohydrodynamics. As for the forward problem, the main focus is on the numerical treatment of induction processes, including self-excitation of magnetic fields in non-spherical domains and/or under the influence of non-homogeneous material parameters. As an important application of the developed numerical schemes, the functioning of the von-K\\'{a}rm\\'{a}n-sodium (VKS) dynamo experiment is shown to depend crucially on the presence of soft-iron impellers. As for the inverse problem, the main focus is on the mathematical background and some first practical applications of the Contactless Inductive Flow Tomography (CIFT), in which flow induced magnetic field perturbations are utilized for the reconstruction of the velocity field. The promises of CIFT for flow field monitoring in the continuous casting of steel are substantiated by results obtained at a test rig wit...
Inverse Problem Solution in Landmines Detection Based on Active Thermography
Directory of Open Access Journals (Sweden)
B. Szymanik
2014-12-01
Full Text Available Landmines still affect numerous territories in the whole world and pose a serious threat, mostly to civilians. Widely used non-metallic landmines are undetectable using metal detector. Therefore, there is an urging need to improve methods of detecting such objects. In the present study we introduce relatively new method of landmines' detection: active infrared thermography with microwave excitation. In this paper we present the optimization based method of solving inverse problem for microwave heating. This technique will be used in the reconstruction of detected landmines geometric and material properties.
THE INVERSE PROBLEM OF OPTIMAL REGULATORS AND ITS AP PLICATION
Institute of Scientific and Technical Information of China (English)
无
2000-01-01
This paper presents a new solution to the inverse problem of linear optimal regulators to minimize a cost function and meet the requirements of relative stability in the presence of a constant but unknown disturbance. A state feedback matrix is developed using Lyapunov's second method. Moreover, the relationships between the state feedback matrix and the cost function are obtained, and a formula to solve the weighting matrices is suggest ed. The developed method is applied successfully to design the horizontal loops in the inertial navigation system.
Solution of inverse localization problem associated to multistatic radar system
Directory of Open Access Journals (Sweden)
Boutkhil M.
2016-01-01
Full Text Available This work deals with the problem of inverse localization by a target with the aim to retrieve the position of the target, given the intensity and phase of the electromagnetic waves scattered by this object. Assuming the surface cross section to be known as well as the intensity and phase of the scattered waves, the target position was reconstructed through the echo signals scattered of each bistatic. We develop in the same time a multistatic ambiguity function trough bistatic ambiguity function to investigate several fundamental aspects that determine multistatic radar performance. We used a multistatic radar constructed of two bistatic radars, two transmitters and one receiver.
Bayesian inference for inverse problems occurring in uncertainty analysis
Fu, Shuai; Celeux, Gilles; Bousquet, Nicolas; Couplet, Mathieu
2012-01-01
The inverse problem considered here is to estimate the distribution of a non-observed random variable $X$ from some noisy observed data $Y$ linked to $X$ through a time-consuming physical model $H$. Bayesian inference is considered to take into account prior expert knowledge on $X$ in a small sample size setting. A Metropolis-Hastings within Gibbs algorithm is proposed to compute the posterior distribution of the parameters of $X$ through a data augmentation process. Since calls to $H$ are qu...
A direct sampling method to an inverse medium scattering problem
Ito, Kazufumi
2012-01-10
In this work we present a novel sampling method for time harmonic inverse medium scattering problems. It provides a simple tool to directly estimate the shape of the unknown scatterers (inhomogeneous media), and it is applicable even when the measured data are only available for one or two incident directions. A mathematical derivation is provided for its validation. Two- and three-dimensional numerical simulations are presented, which show that the method is accurate even with a few sets of scattered field data, computationally efficient, and very robust with respect to noises in the data. © 2012 IOP Publishing Ltd.
On Inverse Topology Problem for Laplace Operators on Graphs
Directory of Open Access Journals (Sweden)
Yu. Yu. Ershova
2014-12-01
Full Text Available Laplacian operators on finite compact metric graphs are considered under the assumption that matching conditions at graph vertices are of $\\delta$ type. Under one additional assumption, the inverse topology problem is treated. Using the apparatus of boundary triples, we generalize and extend existing results on necessary conditions of isospectrality of two Laplacians defined on different graphs. A result is also given covering the case of Schrodinger operators.
On a dense minimizer of empirical risk in inverse problems
Directory of Open Access Journals (Sweden)
Jacek Podlewski
2016-01-01
Full Text Available Properties of estimators of a functional parameter in an inverse problem setup are studied. We focus on estimators obtained through dense minimization (as opposed to minimization over \\(\\delta\\-nets of suitably defined empirical risk. At the cost of imposition of a sort of local finite-dimensionality assumption, we fill some gaps in the proofs of results published by Klemelä and Mammen [Ann. Statist. 38 (2010, 482-511]. We also give examples of functional classes that satisfy the modified assumptions.
A New Inexact Inverse Subspace Iteration for Generalized Eigenvalue Problems
Directory of Open Access Journals (Sweden)
Fatemeh Mohammad
2014-05-01
Full Text Available In this paper, we represent an inexact inverse subspace iteration method for computing a few eigenpairs of the generalized eigenvalue problem $Ax = \\lambda Bx$[Q.~Ye and P.~Zhang, Inexact inverse subspace iteration for generalized eigenvalue problems, Linear Algebra and its Application, 434 (2011 1697-1715]. In particular, the linear convergence property of the inverse subspace iteration is preserved.
DEFF Research Database (Denmark)
Lange, Katrine; Frydendall, Jan; Cordua, Knud Skou;
2012-01-01
The frequency matching method defines a closed form expression for a complex prior that quantifies the higher order statistics of a proposed solution model to an inverse problem. While existing solution methods to inverse problems are capable of sampling the solution space while taking into account...... solution model to an inverse problem by using a priori information based on multiple point statistics learned from training images. We demonstrate the applicability of the suggested method on a synthetic tomographic crosshole inverse problem....
Forecasting wind-driven wildfires using an inverse modelling approach
Directory of Open Access Journals (Sweden)
O. Rios
2013-12-01
Full Text Available A technology able to rapidly forecast wildlfire dynamics would lead to a paradigm shift in the response to emergencies, providing the Fire Service with essential information about the on-going fire. The article at hand presents and explores a novel methodology to forecast wildfire dynamics in wind-driven conditions, using real time data assimilation and inverse modelling. The forecasting algorithm combines Rothermel's rate of spread theory with a perimeter expansion model based on Huygens principle and solves the optimisation problem with a tangent linear approach and a forward automatic differentiation. Its potential is investigated using synthetic data and evaluated in different wildfire scenarios. The results show the high capacity of the method to quickly predict the location of the fire front with a positive lead time (ahead of the event. This work opens the door to further advances framework and more sophisticated models while keeping the computational time suitable for operativeness.
Review on solving the inverse problem in EEG source analysis
Directory of Open Access Journals (Sweden)
Fabri Simon G
2008-11-01
Full Text Available Abstract In this primer, we give a review of the inverse problem for EEG source localization. This is intended for the researchers new in the field to get insight in the state-of-the-art techniques used to find approximate solutions of the brain sources giving rise to a scalp potential recording. Furthermore, a review of the performance results of the different techniques is provided to compare these different inverse solutions. The authors also include the results of a Monte-Carlo analysis which they performed to compare four non parametric algorithms and hence contribute to what is presently recorded in the literature. An extensive list of references to the work of other researchers is also provided. This paper starts off with a mathematical description of the inverse problem and proceeds to discuss the two main categories of methods which were developed to solve the EEG inverse problem, mainly the non parametric and parametric methods. The main difference between the two is to whether a fixed number of dipoles is assumed a priori or not. Various techniques falling within these categories are described including minimum norm estimates and their generalizations, LORETA, sLORETA, VARETA, S-MAP, ST-MAP, Backus-Gilbert, LAURA, Shrinking LORETA FOCUSS (SLF, SSLOFO and ALF for non parametric methods and beamforming techniques, BESA, subspace techniques such as MUSIC and methods derived from it, FINES, simulated annealing and computational intelligence algorithms for parametric methods. From a review of the performance of these techniques as documented in the literature, one could conclude that in most cases the LORETA solution gives satisfactory results. In situations involving clusters of dipoles, higher resolution algorithms such as MUSIC or FINES are however preferred. Imposing reliable biophysical and psychological constraints, as done by LAURA has given superior results. The Monte-Carlo analysis performed, comparing WMN, LORETA, sLORETA and SLF
Cardiac electromechanics and the forward/inverse problems of electrocardiology.
Buist, M; Smith, N P; Pullan, A J
2005-01-01
The mechanical motion of the heart plays a role in determining the waveforms observed in an ECG. This study is designed to ascertain, from a theoretical perspective, the influence of this motion. This is achieved through an analysis of a detailed forward model including a full bidomain description and a strongly coupled model of cardiac electromechanics. Simulations were run on identical problems with and without the inclusion of mechanical deformation and the results were analyzed with a view towards the inverse problem of electrocardiology. Initial results have shown the QRS complex to be largely invariant under deformation, but significant changes in T wave morphology have been observed. Further analysis has revealed that it is the effect of the cell-level mechanics on repolarization that is primarily responsible for these changes as opposed to the tissue deformation.
Comparison of optimal design methods in inverse problems
Banks, H. T.; Holm, K.; Kappel, F.
2011-07-01
Typical optimal design methods for inverse or parameter estimation problems are designed to choose optimal sampling distributions through minimization of a specific cost function related to the resulting error in parameter estimates. It is hoped that the inverse problem will produce parameter estimates with increased accuracy using data collected according to the optimal sampling distribution. Here we formulate the classical optimal design problem in the context of general optimization problems over distributions of sampling times. We present a new Prohorov metric-based theoretical framework that permits one to treat succinctly and rigorously any optimal design criteria based on the Fisher information matrix. A fundamental approximation theory is also included in this framework. A new optimal design, SE-optimal design (standard error optimal design), is then introduced in the context of this framework. We compare this new design criterion with the more traditional D-optimal and E-optimal designs. The optimal sampling distributions from each design are used to compute and compare standard errors; the standard errors for parameters are computed using asymptotic theory or bootstrapping and the optimal mesh. We use three examples to illustrate ideas: the Verhulst-Pearl logistic population model (Banks H T and Tran H T 2009 Mathematical and Experimental Modeling of Physical and Biological Processes (Boca Raton, FL: Chapman and Hall/CRC)), the standard harmonic oscillator model (Banks H T and Tran H T 2009) and a popular glucose regulation model (Bergman R N, Ider Y Z, Bowden C R and Cobelli C 1979 Am. J. Physiol. 236 E667-77 De Gaetano A and Arino O 2000 J. Math. Biol. 40 136-68 Toffolo G, Bergman R N, Finegood D T, Bowden C R and Cobelli C 1980 Diabetes 29 979-90).
Abstract Cauchy problems three approaches
Melnikova, Irina V
2001-01-01
Although the theory of well-posed Cauchy problems is reasonably understood, ill-posed problems-involved in a numerous mathematical models in physics, engineering, and finance- can be approached in a variety of ways. Historically, there have been three major strategies for dealing with such problems: semigroup, abstract distribution, and regularization methods. Semigroup and distribution methods restore well-posedness, in a modern weak sense. Regularization methods provide approximate solutions to ill-posed problems. Although these approaches were extensively developed over the last decades by many researchers, nowhere could one find a comprehensive treatment of all three approaches.Abstract Cauchy Problems: Three Approaches provides an innovative, self-contained account of these methods and, furthermore, demonstrates and studies some of the profound connections between them. The authors discuss the application of different methods not only to the Cauchy problem that is not well-posed in the classical sense, b...
Solving Large-Scale Inverse Magnetostatic Problems using the Adjoint Method
Bruckner, Florian; Abert, Claas; Wautischer, Gregor; Huber, Christian; Vogler, Christoph; Hinze, Michael; Suess, Dieter
2017-01-01
An efficient algorithm for the reconstruction of the magnetization state within magnetic components is presented. The occurring inverse magnetostatic problem is solved by means of an adjoint approach, based on the Fredkin-Koehler method for the solution of the forward problem. Due to the use of hybrid FEM-BEM coupling combined with matrix compression techniques the resulting algorithm is well suited for large-scale problems. Furthermore the reconstruction of the magnetization state within a permanent magnet as well as an optimal design application are demonstrated.
Solving Large-Scale Inverse Magnetostatic Problems using the Adjoint Method
Bruckner, Florian; Abert, Claas; Wautischer, Gregor; Huber, Christian; Vogler, Christoph; Hinze, Michael; Suess, Dieter
2017-01-01
An efficient algorithm for the reconstruction of the magnetization state within magnetic components is presented. The occurring inverse magnetostatic problem is solved by means of an adjoint approach, based on the Fredkin-Koehler method for the solution of the forward problem. Due to the use of hybrid FEM-BEM coupling combined with matrix compression techniques the resulting algorithm is well suited for large-scale problems. Furthermore the reconstruction of the magnetization state within a permanent magnet as well as an optimal design application are demonstrated. PMID:28098851
A numerical study of the inverse problem of breast infrared thermography modeling
Jiang, Li; Zhan, Wang; Loew, Murray H.
2010-03-01
Infrared thermography has been shown to be a useful adjunctive tool for breast cancer detection. Previous thermography modeling techniques generally dealt with the "forward problem", i.e., to estimate the breast thermogram from known properties of breast tissues. The present study aims to deal with the so-called "inverse problem", namely to estimate the thermal properties of the breast tissues from the observed surface temperature distribution. By comparison, the inverse problem is a more direct way of interpreting a breast thermogram for specific physiological and/or pathological information. In tumor detection, for example, it is particularly important to estimate the tumor-induced thermal contrast, even though the corresponding non-tumor thermal background usually is unknown due to the difficulty of measuring the individual thermal properties. Inverse problem solving is technically challenging due to its ill-posed nature, which is evident primarily by its sensitivity to imaging noise. Taking advantage of our previously developed forward-problemsolving techniques with comprehensive thermal-elastic modeling, we examine here the feasibility of solving the inverse problem of the breast thermography. The approach is based on a presumed spatial constraint applied to three major thermal properties, i.e., thermal conductivity, blood perfusion, and metabolic heat generation, for each breast tissue type. Our results indicate that the proposed inverse-problem-solving scheme can be numerically stable under imaging noise of SNR ranging 32 ~ 40 dB, and that the proposed techniques can be effectively used to improve the estimation to the tumor-induced thermal contrast, especially for smaller and deeper tumors.
Solving an Inverse Sturm-Liouville Problem by a Lie-Group Method
Directory of Open Access Journals (Sweden)
Liu Chein-Shan
2008-01-01
Full Text Available Abstract Solving an inverse Sturm-Liouville problem requires a mathematical process to determine unknown function in the Sturm-Liouville operator from given data in addition to the boundary values. In this paper, we identify a Sturm-Liouville potential function by using the data of one eigenfunction and its corresponding eigenvalue, and identify a spatial-dependent unknown function of a Sturm-Liouville differential operator. The method we employ is to transform the inverse Sturm-Liouville problem into a parameter identification problem of a heat conduction equation. Then a Lie-group estimation method is developed to estimate the coefficients in a system of ordinary differential equations discretized from the heat conduction equation. Numerical tests confirm the accuracy and efficiency of present approach. Definite and random disturbances are also considered when comparing the present method with that by using a technique of numerical differentiation.
Solving an Inverse Sturm-Liouville Problem by a Lie-Group Method
Directory of Open Access Journals (Sweden)
Chein-Shan Liu
2008-03-01
Full Text Available Solving an inverse Sturm-Liouville problem requires a mathematical process to determine unknown function in the Sturm-Liouville operator from given data in addition to the boundary values. In this paper, we identify a Sturm-Liouville potential function by using the data of one eigenfunction and its corresponding eigenvalue, and identify a spatial-dependent unknown function of a Sturm-Liouville differential operator. The method we employ is to transform the inverse Sturm-Liouville problem into a parameter identification problem of a heat conduction equation. Then a Lie-group estimation method is developed to estimate the coefficients in a system of ordinary differential equations discretized from the heat conduction equation. Numerical tests confirm the accuracy and efficiency of present approach. Definite and random disturbances are also considered when comparing the present method with that by using a technique of numerical differentiation.
An inverse problem for a mathematical model of aquaponic agriculture
Bobak, Carly; Kunze, Herb
2017-01-01
Aquaponic agriculture is a sustainable ecosystem that relies on a symbiotic relationship between fish and macrophytes. While the practice has been growing in popularity, relatively little mathematical models exist which aim to study the system processes. In this paper, we present a system of ODEs which aims to mathematically model the population and concetrations dynamics present in an aquaponic environment. Values of the parameters in the system are estimated from the literature so that simulated results can be presented to illustrate the nature of the solutions to the system. As well, a brief sensitivity analysis is performed in order to identify redundant parameters and highlight those which may need more reliable estimates. Specifically, an inverse problem with manufactured data for fish and plants is presented to demonstrate the ability of the collage theorem to recover parameter estimates.
The inverse problem of estimating the gravitational time dilation
Energy Technology Data Exchange (ETDEWEB)
Gusev, A. V., E-mail: avg@sai.msu.ru; Litvinov, D. A.; Rudenko, V. N. [Moscow State University, Sternberg Astronomical Institute (Russian Federation)
2016-11-15
Precise testing of the gravitational time dilation effect suggests comparing the clocks at points with different gravitational potentials. Such a configuration arises when radio frequency standards are installed at orbital and ground stations. The ground-based standard is accessible directly, while the spaceborne one is accessible only via the electromagnetic signal exchange. Reconstructing the current frequency of the spaceborne standard is an ill-posed inverse problem whose solution depends significantly on the characteristics of the stochastic electromagnetic background. The solution for Gaussian noise is known, but the nature of the standards themselves is associated with nonstationary fluctuations of a wide class of distributions. A solution is proposed for a background of flicker fluctuations with a spectrum (1/f){sup γ}, where 1 < γ < 3, and stationary increments. The results include formulas for the error in reconstructing the frequency of the spaceborne standard and numerical estimates for the accuracy of measuring the relativistic redshift effect.
Detecting multi-spin interactions in the inverse Ising problem
Albert, Joseph; Swendsen, Robert H.
2017-10-01
While the usual goal in Monte Carlo (MC) simulations of Ising models is the efficient generation of spin configurations with Boltzmann probabilities, the inverse problem is to determine the coupling constants from a given set of spin configurations. Most recent work has been limited to local magnetic fields and pair-wise interactions. We have extended solutions to multi-spin interactions, using correlation function matching (CFM). A more serious limitation of previous work has been the uncertainty of whether a chosen set of interactions is capable of faithfully representing real data. We show how our confirmation testing method uses an additional MC simulation to detect significant interactions that might be missing in the assumed representation of the data.
Canonically Transformed Detectors Applied to the Classical Inverse Scattering Problem
Jung, C; Torres, J M
2005-01-01
The concept of measurement in classical scattering is interpreted as an overlap of a particle packet with some area in phase space that describes the detector. Considering that usually we record the passage of particles at some point in space, a common detector is described e.g. for one-dimensional systems as a narrow strip in phase space. We generalize this concept allowing this strip to be transformed by some, possibly non-linear, canonical transformation, introducing thus a canonically transformed detector. We show such detectors to be useful in the context of the inverse scattering problem in situations where recently discovered scattering echoes could not be seen without their help. More relevant applications in quantum systems are suggested.
Canonically Transformed Detectors Applied to the Classical Inverse Scattering Problem
Jung, C.; Seligman, T. H.; Torres, J. M.
The concept of measurement in classical scattering is interpreted as an overlap of a particle packet with some area in phase space that describes the detector. Considering that usually we record the passage of particles at some point in space, a common detector is described e.g. for one-dimensional systems as a narrow strip in phase space. We generalize this concept allowing this strip to be transformed by some, possibly non-linear, canonical transformation, introducing thus a canonically transformed detector. We show such detectors to be useful in the context of the inverse scattering problem in situations where recently discovered scattering echoes could not be seen without their help. More relevant applications in quantum systems are suggested.
Gross, Lutz; Altinay, Cihan; Fenwick, Joel; Smith, Troy
2014-05-01
The program package escript has been designed for solving mathematical modeling problems using python, see Gross et al. (2013). Its development and maintenance has been funded by the Australian Commonwealth to provide open source software infrastructure for the Australian Earth Science community (recent funding by the Australian Geophysical Observing System EIF (AGOS) and the AuScope Collaborative Research Infrastructure Scheme (CRIS)). The key concepts of escript are based on the terminology of spatial functions and partial differential equations (PDEs) - an approach providing abstraction from the underlying spatial discretization method (i.e. the finite element method (FEM)). This feature presents a programming environment to the user which is easy to use even for complex models. Due to the fact that implementations are independent from data structures simulations are easily portable across desktop computers and scalable compute clusters without modifications to the program code. escript has been successfully applied in a variety of applications including modeling mantel convection, melting processes, volcanic flow, earthquakes, faulting, multi-phase flow, block caving and mineralization (see Poulet et al. 2013). The recent escript release (see Gross et al. (2013)) provides an open framework for solving joint inversion problems for geophysical data sets (potential field, seismic and electro-magnetic). The strategy bases on the idea to formulate the inversion problem as an optimization problem with PDE constraints where the cost function is defined by the data defect and the regularization term for the rock properties, see Gross & Kemp (2013). This approach of first-optimize-then-discretize avoids the assemblage of the - in general- dense sensitivity matrix as used in conventional approaches where discrete programming techniques are applied to the discretized problem (first-discretize-then-optimize). In this paper we will discuss the mathematical framework for
An inverse heat transfer problem for optimization of the thermal process in machining
Indian Academy of Sciences (India)
M Gostimirovic; P Kovac; M Sekulic
2011-08-01
It is evident that machining process causes development of large quantities of thermal energy within a relatively narrow area of the cutting zone. The generated thermal energy and the problems of its evacuation from the cutting zone account for high temperatures in machining. These increased temperatures exert a pronounced negative effect on the tool and workpiece. This paper takes a different approach towards identiﬁcation of the thermal process in machining, using inverse heat transfer problem. Inverse heat transfer method allows the closest possible experimental and analytical approximation of thermal state for a machining process. Based on a temperature measured at any point within a workpiece, inverse method allows determination of a complete temperature ﬁeld in the cutting zone as well as the heat ﬂux distribution on the tool/workpiece interface. By knowing the heat ﬂux function, one deﬁnes criterium and method of optimization, the inverse heat transfer problem transforms into extreme case. Now, the task of optimization is to determine most favourable ratio between heat ﬂux parameters in order to preserve exploitation properties of the tool and workpiece.
Appraisal of geodynamic inversion results: a data mining approach
Baumann, T. S.
2016-11-01
Bayesian sampling based inversions require many thousands or even millions of forward models, depending on how nonlinear or non-unique the inverse problem is, and how many unknowns are involved. The result of such a probabilistic inversion is not a single `best-fit' model, but rather a probability distribution that is represented by the entire model ensemble. Often, a geophysical inverse problem is non-unique, and the corresponding posterior distribution is multimodal, meaning that the distribution consists of clusters with similar models that represent the observations equally well. In these cases, we would like to visualize the characteristic model properties within each of these clusters of models. However, even for a moderate number of inversion parameters, a manual appraisal for a large number of models is not feasible. This poses the question whether it is possible to extract end-member models that represent each of the best-fit regions including their uncertainties. Here, I show how a machine learning tool can be used to characterize end-member models, including their uncertainties, from a complete model ensemble that represents a posterior probability distribution. The model ensemble used here results from a nonlinear geodynamic inverse problem, where rheological properties of the lithosphere are constrained from multiple geophysical observations. It is demonstrated that by taking vertical cross-sections through the effective viscosity structure of each of the models, the entire model ensemble can be classified into four end-member model categories that have a similar effective viscosity structure. These classification results are helpful to explore the non-uniqueness of the inverse problem and can be used to compute representative data fits for each of the end-member models. Conversely, these insights also reveal how new observational constraints could reduce the non-uniqueness. The method is not limited to geodynamic applications and a generalized MATLAB
Dong, Li; Wijesinghe, Philip; Dantuono, James T.; Sampson, David D.; Munro, Peter R. T.; Kennedy, Brendan F.; Oberai, Assad A.
2016-03-01
Quantitative elasticity imaging, which retrieves elastic modulus maps from tissue, is preferred to qualitative strain imaging for acquiring system- and operator-independent images and longitudinal and multi-site diagnoses. Quantitative elasticity imaging has already been demonstrated in optical elastography by relating surface-acoustic and shear wave speed to Young's modulus via a simple algebraic relationship. Such approaches assume largely homogeneous samples and neglect the effect of boundary conditions. We present a general approach to quantitative elasticity imaging based upon the solution of the inverse elasticity problem using an iterative technique and apply it to compression optical coherence elastography. The inverse problem is one of finding the distribution of Young's modulus within a sample, that in response to an applied load, and a given displacement and traction boundary conditions, can produce a displacement field matching one measured in experiment. Key to our solution of the inverse elasticity problem is the use of the adjoint equations that allow the very efficient evaluation of the gradient of the objective function to be minimized with respect to the unknown values of Young's modulus within the sample. Although we present the approach for the case of linear elastic, isotropic, incompressible solids, this method can be employed for arbitrarily complex mechanical models. We present the details of the method and quantitative elastograms of phantoms and tissues. We demonstrate that by using the inverse approach, we can decouple the artefacts produced by mechanical tissue heterogeneity from the true distribution of Young's modulus, which are often evident in techniques that employ first-order algebraic relationships.
Methodes entropiques appliquees au probleme inverse en magnetoencephalographie
Lapalme, Ervig
2005-07-01
This thesis is devoted to biomagnetic source localization using magnetoencephalography. This problem is known to have an infinite number of solutions. So methods are required to take into account anatomical and functional information on the solution. The work presented in this thesis uses the maximum entropy on the mean method to constrain the solution. This method originates from statistical mechanics and information theory. This thesis is divided into two main parts containing three chapters each. The first part reviews the magnetoencephalographic inverse problem: the theory needed to understand its context and the hypotheses for simplifying the problem. In the last chapter of this first part, the maximum entropy on the mean method is presented: its origins are explained and also how it is applied to our problem. The second part is the original work of this thesis presenting three articles; one of them already published and two others submitted for publication. In the first article, a biomagnetic source model is developed and applied in a theoretical con text but still demonstrating the efficiency of the method. In the second article, we go one step further towards a realistic modelization of the cerebral activation. The main priors are estimated using the magnetoencephalographic data. This method proved to be very efficient in realistic simulations. In the third article, the previous method is extended to deal with time signals thus exploiting the excellent time resolution offered by magnetoencephalography. Compared with our previous work, the temporal method is applied to real magnetoencephalographic data coming from a somatotopy experience and results agree with previous physiological knowledge about this kind of cognitive process.
Modeling Dynamic Programming Problems over Sequences and Trees with Inverse Coupled Rewrite Systems
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Robert Giegerich
2014-03-01
Full Text Available Dynamic programming is a classical algorithmic paradigm, which often allows the evaluation of a search space of exponential size in polynomial time. Recursive problem decomposition, tabulation of intermediate results for re-use, and Bellman’s Principle of Optimality are its well-understood ingredients. However, algorithms often lack abstraction and are difficult to implement, tedious to debug, and delicate to modify. The present article proposes a generic framework for specifying dynamic programming problems. This framework can handle all kinds of sequential inputs, as well as tree-structured data. Biosequence analysis, document processing, molecular structure analysis, comparison of objects assembled in a hierarchic fashion, and generally, all domains come under consideration where strings and ordered, rooted trees serve as natural data representations. The new approach introduces inverse coupled rewrite systems. They describe the solutions of combinatorial optimization problems as the inverse image of a term rewrite relation that reduces problem solutions to problem inputs. This specification leads to concise yet translucent specifications of dynamic programming algorithms. Their actual implementation may be challenging, but eventually, as we hope, it can be produced automatically. The present article demonstrates the scope of this new approach by describing a diverse set of dynamic programming problems which arise in the domain of computational biology, with examples in biosequence and molecular structure analysis.
Direct and inverse source problems for a space fractional advection dispersion equation
Aldoghaither, Abeer
2016-05-15
In this paper, direct and inverse problems for a space fractional advection dispersion equation on a finite domain are studied. The inverse problem consists in determining the source term from final observations. We first derive the analytic solution to the direct problem which we use to prove the uniqueness and the unstability of the inverse source problem using final measurements. Finally, we illustrate the results with a numerical example.
Inversion Approach For Thermal Data From A Convecting Hydrothermal System
Energy Technology Data Exchange (ETDEWEB)
Kasameyer, P.; Younker, L.; Hanson, J.
1985-01-01
Hydrothermal systems are often studied by collecting thermal gradient data and temperature depth curves. These data contain important information about the flow field, the evolution of the hydrothermal system, and the location and nature of the ultimate heat sources. Thermal data are conventionally interpreted by the ''forward'' method; the thermal field is calculated based on selected initial conditions and boundary conditions such as temperature and permeability distributions. If the calculated thermal field matches the data, the chosen conditions are inferred to be possibly correct. Because many sets of initial conditions may produce similar thermal fields, users of the ''forward'' method may inadvertently miss the correct set of initial conditions. Analytical methods for ''inverting'' data also allow the determination of all the possible solutions consistent with the definition of the problem. In this paper we suggest an approach for inverting thermal data from a hydrothermal system, and compare it to the more conventional approach. We illustrate the difference in the methods by comparing their application to the Salton Sea Geothermal Field by Lau (1980a) and Kasameyer, et al. (1984). In this particular example, the inverse method was used to draw conclusions about the age and total rate of fluid flow into the hydrothermal system.
Inverse problem of life cycle assessment (LCA: its application in designing for environment (DfE
Directory of Open Access Journals (Sweden)
Rybaczewska-Błażejowska Magdalena
2016-12-01
Full Text Available The inverse problem of life cycle assessment, used in designing for environment, is about determining the optimal values of environmental inputs that provide the required environmental impacts. The notion of the inverse problem of life cycle assessment is explained here using a case study of a coffee machine (abstract model SimaPro, based on models Sima and Pro described in SimaPro 8.1 software. The dependencies between input and output signals were defined by nonlinear functions of several variables. Next, linearization was used and coefficient aki was calculated. On the basis of 3 hypothetical experiments, recommendations have been made on the reduction of the value of the factors that are the most detrimental for the environment: the consumption of aluminium, electricity, and paper for coffee filters, for the analysed product. The results prove the high applicability and usefulness of the proposed approach during environmental evaluation and enhancement of products over the full product life cycle.
Ivanyshyn Yaman, Olha; Le Louër, Frédérique
2016-09-01
This paper deals with the material derivative analysis of the boundary integral operators arising from the scattering theory of time-harmonic electromagnetic waves and its application to inverse problems. We present new results using the Piola transform of the boundary parametrisation to transport the integral operators on a fixed reference boundary. The transported integral operators are infinitely differentiable with respect to the parametrisations and simplified expressions of the material derivatives are obtained. Using these results, we extend a nonlinear integral equations approach developed for solving acoustic inverse obstacle scattering problems to electromagnetism. The inverse problem is formulated as a pair of nonlinear and ill-posed integral equations for the unknown boundary representing the boundary condition and the measurements, for which the iteratively regularized Gauss-Newton method can be applied. The algorithm has the interesting feature that it avoids the numerous numerical solution of boundary value problems at each iteration step. Numerical experiments are presented in the special case of star-shaped obstacles.
HOMOTOPY SOLUTION OF THE INVERSE GENERALIZED EIGENVALUE PROBLEMS IN STRUCTURAL DYNAMICS
Institute of Scientific and Technical Information of China (English)
李书; 王波; 胡继忠
2004-01-01
The structural dynamics problems, such as structural design, parameter identification and model correction, are considered as a kind of the inverse generalized eigenvalue problems mathematically. The inverse eigenvalue problems are nonlinear. In general, they could be transformed into nonlinear equations to solve. The structural dynamics inverse problems were treated as quasi multiplicative inverse eigenalue problems which were solved by homotopy method for nonlinear equations. This method had no requirements for initial value essentially because of the homotopy path to solution. Numerical examples were presented to illustrate the homotopy method.
Energy Technology Data Exchange (ETDEWEB)
Haber, Eldad
2014-03-17
The focus of research was: Developing adaptive mesh for the solution of Maxwell's equations; Developing a parallel framework for time dependent inverse Maxwell's equations; Developing multilevel methods for optimization problems with inequal- ity constraints; A new inversion code for inverse Maxwell's equations in the 0th frequency (DC resistivity); A new inversion code for inverse Maxwell's equations in low frequency regime. Although the research concentrated on electromagnetic forward and in- verse problems the results of the research was applied to the problem of image registration.
Li, Xu; Chen, Zhigang; Gong, Jianmin; Taflove, Allen; Backman, Vadim
2004-06-01
Understanding light scattering by nonspherical particles is crucial in modeling the transport of light in realistic structures such as biological tissues. We report the application of novel analytical approaches based on modified Wentzel-Kramers-Brillouin and equiphase-sphere methods that facilitate accurate characterization of light scattering by a wide range of irregularly shaped dielectric particles. We also demonstrate that these approaches have the potential to address the inverse-scattering problem by means of a spectral analysis of the total scattering cross section of arbitrarily shaped particles.
Schuster, Thomas; Hofmann, Bernd; Kaltenbacher, Barbara
2012-10-01
Inverse problems can usually be modelled as operator equations in infinite-dimensional spaces with a forward operator acting between Hilbert or Banach spaces—a formulation which quite often also serves as the basis for defining and analyzing solution methods. The additional amount of structure and geometric interpretability provided by the concept of an inner product has rendered these methods amenable to a convergence analysis, a fact which has led to a rigorous and comprehensive study of regularization methods in Hilbert spaces over the last three decades. However, for numerous problems such as x-ray diffractometry, certain inverse scattering problems and a number of parameter identification problems in PDEs, the reasons for using a Hilbert space setting seem to be based on conventions rather than an appropriate and realistic model choice, so often a Banach space setting would be closer to reality. Furthermore, non-Hilbertian regularization and data fidelity terms incorporating a priori information on solution and noise, such as general Lp-norms, TV-type norms, or the Kullback-Leibler divergence, have recently become very popular. These facts have motivated intensive investigations on regularization methods in Banach spaces, a topic which has emerged as a highly active research field within the area of inverse problems. Meanwhile some of the most well-known regularization approaches, such as Tikhonov-type methods requiring the solution of extremal problems, and iterative ones like the Landweber method, the Gauss-Newton method, as well as the approximate inverse method, have been investigated for linear and nonlinear operator equations in Banach spaces. Convergence with rates has been proven and conditions on the solution smoothness and on the structure of nonlinearity have been formulated. Still, beyond the existing results a large number of challenging open questions have arisen, due to the more involved handling of general Banach spaces and the larger variety
Solving Inverse Detection Problems Using Passive Radiation Signatures
Energy Technology Data Exchange (ETDEWEB)
Favorite, Jeffrey A. [Los Alamos National Laboratory; Armstrong, Jerawan C. [Los Alamos National Laboratory; Vaquer, Pablo A. [Los Alamos National Laboratory
2012-08-15
The ability to reconstruct an unknown radioactive object based on its passive gamma-ray and neutron signatures is very important in homeland security applications. Often in the analysis of unknown radioactive objects, for simplicity or speed or because there is no other information, they are modeled as spherically symmetric regardless of their actual geometry. In these presentation we discuss the accuracy and implications of this approximation for decay gamma rays and for neutron-induced gamma rays. We discuss an extension of spherical raytracing (for uncollided fluxes) that allows it to be used when the exterior shielding is flat or cylindrical. We revisit some early results in boundary perturbation theory, showing that the Roussopolos estimate is the correct one to use when the quantity of interest is the flux or leakage on the boundary. We apply boundary perturbation theory to problems in which spherically symmetric systems are perturbed in asymmetric nonspherical ways. We apply mesh adaptive direct search (MADS) algorithms to object reconstructions. We present a benchmark test set that may be used to quantitatively evaluate inverse detection methods.
Forward and inverse problems of EEG dipole localization.
Musha, T; Okamoto, Y
1999-01-01
Mathematical procedures are discussed in detail of numerical solutions for obtaining scalp potentials from the electric sources. The finite-element method for an inhomogeneous volume conductor, the boundary-element method for a compartment model, and their hybrid for more general cases are discussed. Construction of the head model and typical estimation of electric conductivity of the compartment model is described, which can reduce errors in estimated dipole location caused by incorrect head geometry. The concept of reciprocity is explained, which is applied to understanding a relation between the electrode configuration and its sensitivity for various source conditions. Typical techniques for solving the inverse problem are reviewed for discrete source models. Methods of estimating accuracy of the dipole location in the presence of noise are discussed, together with some numerical examples. The dipolarity is a goodness-of-fit of the dipole approximation, and lowering of the dipolarity is related to inhomogeneous neuronal activity in the cortex. Finally, a criterion of determining the optimal number of model parameters is given in terms of AIC (Akaike Information Criterion), which is applied to decide the most probable number of equivalent dipoles.
Inverse modeling for heat conduction problem in human abdominal phantom.
Huang, Ming; Chen, Wenxi
2011-01-01
Noninvasive methods for deep body temperature measurement are based on the principle of heat equilibrium between the thermal sensor and the target location theoretically. However, the measurement position is not able to be definitely determined. In this study, a 2-dimensional mathematical model was built based upon some assumptions for the physiological condition of the human abdomen phantom. We evaluated the feasibility in estimating the internal organs temperature distribution from the readings of the temperature sensors arranged on the skin surface. It is a typical inverse heat conduction problem (IHCP), and is usually mathematically ill-posed. In this study, by integrating some physical and physiological a-priori information, we invoked the quasi-linear (QL) method to reconstruct the internal temperature distribution. The solutions of this method were improved by increasing the accuracy of the sensors and adjusting their arrangement on the outer surface, and eventually reached the state of converging at the best state accurately. This study suggests that QL method is able to reconstruct the internal temperature distribution in this phantom and might be worthy of a further study in an anatomical based model.
Goncharsky, Alexander V.; Romanov, Sergey Y.
2017-02-01
We develop efficient iterative methods for solving inverse problems of wave tomography in models incorporating both diffraction effects and attenuation. In the inverse problem the aim is to reconstruct the velocity structure and the function that characterizes the distribution of attenuation properties in the object studied. We prove mathematically and rigorously the differentiability of the residual functional in normed spaces, and derive the corresponding formula for the Fréchet derivative. The computation of the Fréchet derivative includes solving both the direct problem with the Neumann boundary condition and the reversed-time conjugate problem. We develop efficient methods for numerical computations where the approximate solution is found using the detector measurements of the wave field and its normal derivative. The wave field derivative values at detector locations are found by solving the exterior boundary value problem with the Dirichlet boundary conditions. We illustrate the efficiency of this approach by applying it to model problems. The algorithms developed are highly parallelizable and designed to be run on supercomputers. Among the most promising medical applications of our results is the development of ultrasonic tomographs for differential diagnosis of breast cancer.
Forecasting wind-driven wildfires using an inverse modelling approach
Directory of Open Access Journals (Sweden)
O. Rios
2014-06-01
Full Text Available A technology able to rapidly forecast wildfire dynamics would lead to a paradigm shift in the response to emergencies, providing the Fire Service with essential information about the ongoing fire. This paper presents and explores a novel methodology to forecast wildfire dynamics in wind-driven conditions, using real-time data assimilation and inverse modelling. The forecasting algorithm combines Rothermel's rate of spread theory with a perimeter expansion model based on Huygens principle and solves the optimisation problem with a tangent linear approach and forward automatic differentiation. Its potential is investigated using synthetic data and evaluated in different wildfire scenarios. The results show the capacity of the method to quickly predict the location of the fire front with a positive lead time (ahead of the event in the order of 10 min for a spatial scale of 100 m. The greatest strengths of our method are lightness, speed and flexibility. We specifically tailor the forecast to be efficient and computationally cheap so it can be used in mobile systems for field deployment and operativeness. Thus, we put emphasis on producing a positive lead time and the means to maximise it.
Yitembe, Bertrand Russel; Crevecoeur, Guillaume; Van Keer, Roger; Dupre, Luc
2011-05-01
The EEG is a neurological diagnostic tool with high temporal resolution. However, when solving the EEG inverse problem, its localization accuracy is limited because of noise in measurements and available uncertainties of the conductivity value in the forward model evaluations. This paper proposes the reduced conductivity dependence (RCD) method for decreasing the localization error in EEG source analysis by limiting the propagation of the uncertain conductivity values to the solutions of the inverse problem. We redefine the traditional EEG cost function, and in contrast to previous approaches, we introduce a selection procedure of the EEG potentials. The selected potentials are, as low as possible, affected by the uncertainties of the conductivity when solving the inverse problem. We validate the methodology on the widely used three-shell spherical head model with a single electrical dipole and multiple dipoles as source model. The proposed RCD method enhances the source localization accuracy with a factor ranging between 2 and 4, dependent on the dipole location and the noise in measurements. © 2011 IEEE
Lemes, N. H. T.; Borges, E.; Sousa, R. V.; Braga, J. P.
Important physical and chemical information can be extracted from scattering experiments data. This kind of problem is usually ill-posed in the sense that one of the three conditions, existence, uniqueness, and continuity, is not satisfied. For example, the inversion of intermolecular potential functions from scattering data, such as experimental cross section, is an ill-posed problem which can be modeled as a Fredholm integral equation. In this work, an inversion method based on recursive neural networks is proposed to solve this inverse quantum scattering problem within the Born approximation. As physical example, the repulsive component of the potential function for the interaction Ar-Ar is obtained from differential cross-section data. The sensitivity of the potential energy function to be inverted, in relation to the differential cross-section data, is also analyzed. The present approach is simple, general, and numerically stable.
Children's strategies to solving additive inverse problems: a preliminary analysis
Ding, Meixia; Auxter, Abbey E.
2017-03-01
Prior studies show that elementary school children generally "lack" formal understanding of inverse relations. This study goes beyond lack to explore what children might "have" in their existing conception. A total of 281 students, kindergarten to third grade, were recruited to respond to a questionnaire that involved both contextual and non-contextual tasks on inverse relations, requiring both computational and explanatory skills. Results showed that children demonstrated better performance in computation than explanation. However, many students' explanations indicated that they did not necessarily utilize inverse relations for computation. Rather, they appeared to possess partial understanding, as evidenced by their use of part-whole structure, which is a key to understanding inverse relations. A close inspection of children's solution strategies further revealed that the sophistication of children's conception of part-whole structure varied in representation use and unknown quantity recognition, which suggests rich opportunities to develop students' understanding of inverse relations in lower elementary classrooms.
Children's strategies to solving additive inverse problems: a preliminary analysis
Ding, Meixia; Auxter, Abbey E.
2017-01-01
Prior studies show that elementary school children generally "lack" formal understanding of inverse relations. This study goes beyond lack to explore what children might "have" in their existing conception. A total of 281 students, kindergarten to third grade, were recruited to respond to a questionnaire that involved both contextual and non-contextual tasks on inverse relations, requiring both computational and explanatory skills. Results showed that children demonstrated better performance in computation than explanation. However, many students' explanations indicated that they did not necessarily utilize inverse relations for computation. Rather, they appeared to possess partial understanding, as evidenced by their use of part-whole structure, which is a key to understanding inverse relations. A close inspection of children's solution strategies further revealed that the sophistication of children's conception of part-whole structure varied in representation use and unknown quantity recognition, which suggests rich opportunities to develop students' understanding of inverse relations in lower elementary classrooms.
Forward- vs. Inverse Problems in Modeling Seismic Attenuation
Morozov, I. B.
2015-12-01
Seismic attenuation is an important property of wave propagation used in numerous applications. However, the attenuation is also a complex phenomenon, and it is important to differentiate between its two typical uses: 1) in forward problems, to model the amplitudes and spectral contents of waves required for hazard assessment and geotechnical engineering, and 2) in inverse problems, to determine the physical properties of the subsurface. In the forward-problem sense, the attenuation is successfully characterized in terms of empirical parameters of geometric spreading, radiation patterns, scattering amplitudes, t-star, alpha, kappa, or Q. Arguably, the predicted energy losses can be correct even if the underlying attenuation model is phenomenological and not sufficiently based on physics. An example of such phenomenological model is the viscoelasticity based on the correspondence principle and the Q-factor assigned to the material. By contrast, when used to invert for in situ material properties, models addressing the specific physics are required. In many studies (including in this session), a Q-factor is interpreted as a property of a point within the subsurface; however this property is only phenomenological and may be physically insufficient or inconsistent. For example, the bulk or shear Q at the same point can be different when evaluated from different wave modes. The cases of frequency-dependent Q are particularly prone of ambiguities such as trade-off with the assumed background geometric spreading. To rigorously characterize the in situ material properties responsible for seismic-wave attenuation, it is insufficient to only focus on the seismic energy loss. Mechanical models of the material need to be considered. Such models can be constructed by using Lagrangian mechanics. These models should likely contain no Q but will be based on parameters of microstructure such as heterogeneity, fractures, or fluids. I illustrate several such models based on viscosity
Inverse Kinematics of a Humanoid Robot with Non-Spherical Hip: A Hybrid Algorithm Approach
2013-01-01
This paper describes an approach to solve the inverse kinematics problem of humanoid robots whose construction shows a small but non negligible offset at the hip which prevents any purely analytical solution to be developed. Knowing that a purely numerical solution is not feasible due to variable efficiency problems, the proposed one first neglects the offset presence in order to obtain an approximate “solution” by means of an analytical algorithm based on screw theory, a...
Parker, Peter A.; Geoffrey, Vining G.; Wilson, Sara R.; Szarka, John L., III; Johnson, Nels G.
2010-01-01
The calibration of measurement systems is a fundamental but under-studied problem within industrial statistics. The origins of this problem go back to basic chemical analysis based on NIST standards. In today's world these issues extend to mechanical, electrical, and materials engineering. Often, these new scenarios do not provide "gold standards" such as the standard weights provided by NIST. This paper considers the classic "forward regression followed by inverse regression" approach. In this approach the initial experiment treats the "standards" as the regressor and the observed values as the response to calibrate the instrument. The analyst then must invert the resulting regression model in order to use the instrument to make actual measurements in practice. This paper compares this classical approach to "reverse regression," which treats the standards as the response and the observed measurements as the regressor in the calibration experiment. Such an approach is intuitively appealing because it avoids the need for the inverse regression. However, it also violates some of the basic regression assumptions.
A regularizing iterative ensemble Kalman method for PDE-constrained inverse problems
Iglesias, Marco A.
2016-02-01
We introduce a derivative-free computational framework for approximating solutions to nonlinear PDE-constrained inverse problems. The general aim is to merge ideas from iterative regularization with ensemble Kalman methods from Bayesian inference to develop a derivative-free stable method easy to implement in applications where the PDE (forward) model is only accessible as a black box (e.g. with commercial software). The proposed regularizing ensemble Kalman method can be derived as an approximation of the regularizing Levenberg-Marquardt (LM) scheme (Hanke 1997 Inverse Problems 13 79-95) in which the derivative of the forward operator and its adjoint are replaced with empirical covariances from an ensemble of elements from the admissible space of solutions. The resulting ensemble method consists of an update formula that is applied to each ensemble member and that has a regularization parameter selected in a similar fashion to the one in the LM scheme. Moreover, an early termination of the scheme is proposed according to a discrepancy principle-type of criterion. The proposed method can be also viewed as a regularizing version of standard Kalman approaches which are often unstable unless ad hoc fixes, such as covariance localization, are implemented. The aim of this paper is to provide a detailed numerical investigation of the regularizing and convergence properties of the proposed regularizing ensemble Kalman scheme; the proof of these properties is an open problem. By means of numerical experiments, we investigate the conditions under which the proposed method inherits the regularizing properties of the LM scheme of (Hanke 1997 Inverse Problems 13 79-95) and is thus stable and suitable for its application in problems where the computation of the Fréchet derivative is not computationally feasible. More concretely, we study the effect of ensemble size, number of measurements, selection of initial ensemble and tunable parameters on the performance of the method
Inverse problems using artificial neural networks in long range atmospheric dispersion
Energy Technology Data Exchange (ETDEWEB)
Sharma, P.K.; Gera, B.; Ghosh, A.K. [Bhabha Atomic Research Centre, Trombay, Mumbai (India). Reactor Safety Div.
2011-05-15
Scalar dispersion in the atmosphere is an important area wherein different approaches are followed in development of good analytical models. The analyses based on Computational Fluid Dynamics (CFD) codes offer an opportunity of model development based on first principles of physics and hence such models have an edge over the existing models. Both forward and backward calculation methods are being developed for atmospheric dispersion around NPPs at BARC. Forward modeling methods, which describe the atmospheric transport from sources to receptors, use forward-running transport and dispersion models or computational fluid dynamics models which are run many times, and the resulting dispersion field is compared to observations from multiple sensors. Backward or inverse modeling methods use only one model run in the reverse direction from the receptors to estimate the upwind sources. Inverse modeling methods include adjoint and tangent linear models, Kalman filters, and variational data assimilation, and neural network. The present paper is aimed at developing a new approach where the identified specific signatures at receptor points form the basis for source estimation or inversions. This approach is expected to reduce the large transient data sets to reduced and meaningful data sets. In fact this reduces the inherently transient data set into a time independent mean data set. Forward computations were carried out with CFD code for various cases to generate a large set of data to train the Artificial Neural Network (ANN). Specific signature analysis was carried out to find the parameters of interest for ANN training like peak concentration, time to reach peak concentration and time to fall. The ANN was trained with data and source strength and locations were predicted from ANN. The inverse problem was performed using the ANN approach in long range atmospheric dispersion. An illustration of application of CFD code for atmospheric dispersion studies for a hypothetical
Zhang, Huaguang; Feng, Tao; Yang, Guang-Hong; Liang, Hongjing
2015-07-01
In this paper, the inverse optimal approach is employed to design distributed consensus protocols that guarantee consensus and global optimality with respect to some quadratic performance indexes for identical linear systems on a directed graph. The inverse optimal theory is developed by introducing the notion of partial stability. As a result, the necessary and sufficient conditions for inverse optimality are proposed. By means of the developed inverse optimal theory, the necessary and sufficient conditions are established for globally optimal cooperative control problems on directed graphs. Basic optimal cooperative design procedures are given based on asymptotic properties of the resulting optimal distributed consensus protocols, and the multiagent systems can reach desired consensus performance (convergence rate and damping rate) asymptotically. Finally, two examples are given to illustrate the effectiveness of the proposed methods.
Inverse transient heat conduction problems and identification of thermal parameters
Atchonouglo, K.; Banna, M.; Vallée, C.; Dupré, J.-C.
2008-04-01
This work deals with the estimation of polymers properties. An inverse analysis based on finite element method is applied to identify simultaneously the constants thermal conductivity and heat capacity per unit volume. The inverse method algorithm constructed is validated from simulated transient temperature recording taken at several locations on the surface of the solid. Transient temperature measures taped with infrared camera on polymers were used for identifying the thermal properties. The results show an excellent agreement between manufacturer and identified values.
Calculation of transport coefficient profiles in modulation experiments as an inverse problem
Escande, D F
2011-01-01
The calculation of transport profiles from experimental measurements belongs in the category of inverse problems which are known to come with issues of ill-conditioning or singularity. A reformulation of the calculation, the matricial approach, is proposed for periodically modulated experiments, within the context of the standard advection-diffusion model where these issues are related to the vanishing of the determinant of a 2x2 matrix. This sheds light on the accuracy of calculations with transport codes, and provides a path for a more precise assessment of the profiles and of the related uncertainty.
Directory of Open Access Journals (Sweden)
Slavica M. Perovich
2011-06-01
Full Text Available The subject of the theoretical analysis presented in this paper is an analytical approach to the temperature estimation, as an inverse problem, for different thermistors – linear resistances structures: series and parallel ones, by the STFT - Special Trans Functions Theory (S.M. Perovich. The mathematical formulae genesis of both cases is given. Some numerical and graphical simulations in MATHEMATICA program have been realized. The estimated temperature intervals for strongly determined values of the equivalent resistances of the nonlinear structures are given, as well.
Sommariva, Sara; Sorrentino, Alberto
2014-11-01
We discuss the use of a recent class of sequential Monte Carlo methods for solving inverse problems characterized by a semi-linear structure, i.e. where the data depend linearly on a subset of variables and nonlinearly on the remaining ones. In this type of problems, under proper Gaussian assumptions one can marginalize the linear variables. This means that the Monte Carlo procedure needs only to be applied to the nonlinear variables, while the linear ones can be treated analytically; as a result, the Monte Carlo variance and/or the computational cost decrease. We use this approach to solve the inverse problem of magnetoencephalography, with a multi-dipole model for the sources. Here, data depend nonlinearly on the number of sources and their locations, and depend linearly on their current vectors. The semi-analytic approach enables us to estimate the number of dipoles and their location from a whole time-series, rather than a single time point, while keeping a low computational cost.
Uniqueness and stability in an inverse problem for a Poisson’s equation
Institute of Scientific and Technical Information of China (English)
2007-01-01
Consider the Poisson’s equation（?）″（x）=-ev-（?）+e（?）-v-N（x）with the Diriehlet boundary data,and we mainly investigate the inverse problem of determining the unknown function N（x）from a parameter function family.Some uniqueness and stability results in the inverse problem are obtained.
DEFF Research Database (Denmark)
Lange, Katrine; Frydendall, Jan; Cordua, Knud Skou
2012-01-01
The frequency matching method defines a closed form expression for a complex prior that quantifies the higher order statistics of a proposed solution model to an inverse problem. While existing solution methods to inverse problems are capable of sampling the solution space while taking into accou...
Approximate Bayesian computation for machine learning, inverse problems and big data
Mohammad-Djafari, Ali
2017-06-01
This paper summarizes my tutorial talk in MaxEnt 2016 workshop. Starting from the basics of the Bayesian approach and simple example of low dimensional parameter estimation where almost all the computations can be done easily, we go very fast to high dimensional case. In those real world cases, even for the sample case of linear model with Gaussian prior, where the posterior law is also Gaussian, the cost of the computation of the posterior covariance becomes important and needs approximate and fast algorithms. Different approximation methods for model comparison and model selection in machine learning problems are presented in summary. Among the existing methods, we mention Laplace approximation, Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC) and Variational Bayesian Approximation (VBA) Methods. Finally, through two examples of inverse problems in imaging systems: X ray and Diffraction wave Computed Tomography (CT), we show how to handle the real great dimensional problems.
Modified Landweber Algorithm for Solving the Inverse Problem in EIT
Institute of Scientific and Technical Information of China (English)
WANGChao; WANGHuaxiang
2005-01-01
This paper analyses the Landweber iteration method and demonstrates that Landweber method is a modified of the generalized inverse constructed using the iteration solution. The phenomenon is explained that the image reconstructed using Landweber iteration algorithm through a large numbers of iteration steps is similar tothe minimum norm solution of the generalized inverse. A new reconstruction algorithm called the modified Landweber method is proposed, which divides the image reconstruction process into two steps， off-line pre-iteration and on-line one-step reconstruction. The reconstruction speed is markedly improved.
Inverse Eigenvalue Problems for a Structure with Linear Parameters
Institute of Scientific and Technical Information of China (English)
WU Liang-sheng; YANG Jia-hua; WEI Yuan-qian; MEN Hao; YANG Qing-kun; LIU Zhen-yu
2005-01-01
The inverse design method of a dynamic system with linear parameters has been studied. For some specified eigenvalues and eigenvectors, the design parameter vector which is often composed of whole or part of coefficients of spring and mass of the system can be obtained and the rigidity and mass matrices of an initially designed structure can be reconstructed through solving linear algebra equations. By using implicit function theorem, the conditions of existence and uniqueness of the solution are also deduced. The theory and method can be used for inverse vibration design of complex structure system.
Inverse problem for multivariate time series using dynamical latent variables
Zamparo, M.; Stramaglia, S.; Banavar, J. R.; Maritan, A.
2012-06-01
Factor analysis is a well known statistical method to describe the variability among observed variables in terms of a smaller number of unobserved latent variables called factors. While dealing with multivariate time series, the temporal correlation structure of data may be modeled by including correlations in latent factors, but a crucial choice is the covariance function to be implemented. We show that analyzing multivariate time series in terms of latent Gaussian processes, which are mutually independent but with each of them being characterized by exponentially decaying temporal correlations, leads to an efficient implementation of the expectation-maximization algorithm for the maximum likelihood estimation of parameters, due to the properties of block-tridiagonal matrices. The proposed approach solves an ambiguity known as the identifiability problem, which renders the solution of factor analysis determined only up to an orthogonal transformation. Samples with just two temporal points are sufficient for the parameter estimation: hence the proposed approach may be applied even in the absence of prior information about the correlation structure of latent variables by fitting the model to pairs of points with varying time delay. Our modeling allows one to make predictions of the future values of time series and we illustrate our method by applying it to an analysis of published gene expression data from cell culture HeLa.
Spectral inverse problem for q-deformed harmonic oscillator
Indian Academy of Sciences (India)
P K Bera; J Datta
2006-12-01
The supersymmetric quantization condition is used to study the wave functions of SWKB equivalent -deformed harmonic oscillator which are obtained by using only the knowledge of bound-state spectra of -deformed harmonic oscillator. We have also studied the nonuniqueness of the obtained interactions by this spectral inverse method.
Explicit inverse distance weighting mesh motion for coupled problems
Witteveen, J.A.S.; Bijl, H.
2009-01-01
An explicit mesh motion algorithm based on inverse distance weighting interpolation is presented. The explicit formulation leads to a fast mesh motion algorithm and an easy implementation. In addition, the proposed point-by-point method is robust and flexible in case of large deformations, hanging n
Agoshkov, Valery
2017-04-01
There are different approaches for modeling boundary conditions describing hydrophysical fields in water areas with "liquid" boundaries. Variational data assimilation may also be considered as one of such approaches. Development of computer equipment, together with an increase in the quantity and quality of data from the satellites and other monitoring tools proves that the development of this particular approach is perspective. The range of connected the problems is wide - different recording forms of boundary conditions, observational data assimilation procedures and used models of hydrodynamics are possible. In this work some inverse problems and corresponding variational data assimilation ones, connected with mathematical modeling of hydrophysical fields in water areas (seas and oceans) with "liquid" ("open") boundaries, are formulated and studied. Note that the surface of water area (which can also be considered as a "liquid" boundary) is not included in the set of "liquid" boundaries, in this case "liquid" boundaries are borders between the areas "water-water". In the work, mathematical model of hydrothermodynamics in the water areas with "liquid" ("open") part of the boundary, a generalized statement of the problem and the splitting method for time approximation are formulated. Also the problem of variational data assimilation and iterative algorithm for solving inverse problems mentioned above are formulated. The work is based on [1]. The work was partly supported by the Russian Science Foundation (project 14-11-00609, the general formulation of the inverse problems) and by the Russian Foundation for Basic Research (project 16-01-00548, the formulation of the problem and its study). [1] V.I. Agoshkov, Methods for solving inverse problems and variational data assimilation problems of observations in the problems of the large-scale dynamics of the oceans and seas, Institute of Numerical Mathematics, RAS, Moscow, 2016 (in Russian).
SIPPI: A Matlab toolbox for sampling the solution to inverse problems with complex prior information
DEFF Research Database (Denmark)
Hansen, Thomas Mejer; Cordua, Knud Skou; Caroline Looms, Majken
2013-01-01
for solving such probabilistically formulated inverse problems by sampling the a posteriori probability density function. In order to describe the a priori probability density function, we consider both simple Gaussian models and more complex (and realistic) a priori models based on higher order statistics....... These a priori models can be used with both linear and non-linear inverse problems. For linear inverse Gaussian problems we make use of least-squares and kriging-based methods to describe the a posteriori probability density function directly. For general non-linear (i.e. non-Gaussian) inverse problems, we make...... use of the extended Metropolis algorithm to sample the a posteriori probability density function. Together with the extended Metropolis algorithm, we use sequential Gibbs sampling that allow computationally efficient sampling of complex a priori models. The toolbox can be applied to any inverse...
A 2D forward and inverse code for streaming potential problems
Soueid Ahmed, A.; Jardani, A.; Revil, A.
2013-12-01
The self-potential method corresponds to the passive measurement of the electrical field in response to the occurrence of natural sources of current in the ground. One of these sources corresponds to the streaming current associated with the flow of the groundwater. We can therefore apply the self- potential method to recover non-intrusively some information regarding the groundwater flow. We first solve the forward problem starting with the solution of the groundwater flow problem, then computing the source current density, and finally solving a Poisson equation for the electrical potential. We use the finite-element method to solve the relevant partial differential equations. In order to reduce the number of (petrophysical) model parameters required to solve the forward problem, we introduced an effective charge density tensor of the pore water, which can be determined directly from the permeability tensor for neutral pore waters. The second aspect of our work concerns the inversion of the self-potential data using Tikhonov regularization with smoothness and weighting depth constraints. This approach accounts for the distribution of the electrical resistivity, which can be independently and approximately determined from electrical resistivity tomography. A numerical code, SP2DINV, has been implemented in Matlab to perform both the forward and inverse modeling. Three synthetic case studies are discussed.
SP2DINV: A 2D forward and inverse code for streaming potential problems
Soueid Ahmed, A.; Jardani, A.; Revil, A.; Dupont, J. P.
2013-09-01
The self-potential method corresponds to the passive measurement of the electrical field in response to the occurrence of natural sources of current in the ground. One of these sources corresponds to the streaming current associated with the flow of the ground water. We can therefore apply the self-potential method to recover non-intrusively some information regarding the ground water flow. We first solve the forward problem starting with the solution of the ground water flow problem, then computing the source current density, and finally solving a Poisson equation for the electrical potential. We use the finite-element method to solve the relevant partial differential equations. In order to reduce the number of (petrophysical) model parameters required to solve the forward problem, we introduced an effective charge density tensor of the pore water, which can be determined directly from the permeability tensor for neutral pore waters. The second aspect of our work concerns the inversion of the self-potential data using Tikhonov regularization with smoothness and weighting depth constraints. This approach accounts for the distribution of the electrical resistivity, which can be independently and approximately determined from electrical resistivity tomography. A numerical code, SP2DINV, has been implemented in Matlab to perform both the forward and inverse modeling. Three synthetic case studies are discussed.
A Hybrid Optimization Method for Solving Bayesian Inverse Problems under Uncertainty.
Directory of Open Access Journals (Sweden)
Kai Zhang
Full Text Available In this paper, we investigate the application of a new method, the Finite Difference and Stochastic Gradient (Hybrid method, for history matching in reservoir models. History matching is one of the processes of solving an inverse problem by calibrating reservoir models to dynamic behaviour of the reservoir in which an objective function is formulated based on a Bayesian approach for optimization. The goal of history matching is to identify the minimum value of an objective function that expresses the misfit between the predicted and measured data of a reservoir. To address the optimization problem, we present a novel application using a combination of the stochastic gradient and finite difference methods for solving inverse problems. The optimization is constrained by a linear equation that contains the reservoir parameters. We reformulate the reservoir model's parameters and dynamic data by operating the objective function, the approximate gradient of which can guarantee convergence. At each iteration step, we obtain the relatively 'important' elements of the gradient, which are subsequently substituted by the values from the Finite Difference method through comparing the magnitude of the components of the stochastic gradient, which forms a new gradient, and we subsequently iterate with the new gradient. Through the application of the Hybrid method, we efficiently and accurately optimize the objective function. We present a number numerical simulations in this paper that show that the method is accurate and computationally efficient.
A Hybrid Optimization Method for Solving Bayesian Inverse Problems under Uncertainty.
Zhang, Kai; Wang, Zengfei; Zhang, Liming; Yao, Jun; Yan, Xia
2015-01-01
In this paper, we investigate the application of a new method, the Finite Difference and Stochastic Gradient (Hybrid method), for history matching in reservoir models. History matching is one of the processes of solving an inverse problem by calibrating reservoir models to dynamic behaviour of the reservoir in which an objective function is formulated based on a Bayesian approach for optimization. The goal of history matching is to identify the minimum value of an objective function that expresses the misfit between the predicted and measured data of a reservoir. To address the optimization problem, we present a novel application using a combination of the stochastic gradient and finite difference methods for solving inverse problems. The optimization is constrained by a linear equation that contains the reservoir parameters. We reformulate the reservoir model's parameters and dynamic data by operating the objective function, the approximate gradient of which can guarantee convergence. At each iteration step, we obtain the relatively 'important' elements of the gradient, which are subsequently substituted by the values from the Finite Difference method through comparing the magnitude of the components of the stochastic gradient, which forms a new gradient, and we subsequently iterate with the new gradient. Through the application of the Hybrid method, we efficiently and accurately optimize the objective function. We present a number numerical simulations in this paper that show that the method is accurate and computationally efficient.
Higher order Nevanlinna functions and the inverse three spectra problem
Directory of Open Access Journals (Sweden)
Olga Boyko
2016-01-01
Full Text Available The three spectra problem of recovering the Sturm-Liouville equation by the spectrum of the Dirichlet-Dirichlet boundary value problem on \\([0,a]\\, the Dirichlet-Dirichlet problem on \\([0,a/2]\\ and the Neumann-Dirichlet problem on \\([a/2,a]\\ is considered. Sufficient conditions of solvability and of uniqueness of the solution to such a problem are found.
On t-local solvability of inverse scattering problems in two-dimensional layered media
Baev, A. V.
2015-06-01
The solvability of two-dimensional inverse scattering problems for the Klein-Gordon equation and the Dirac system in a time-local formulation is analyzed in the framework of the Galerkin method. A necessary and sufficient condition for the unique solvability of these problems is obtained in the form of an energy conservation law. It is shown that the inverse problems are solvable only in the class of potentials for which the stationary Navier-Stokes equation is solvable.
Lipschitz stability in an inverse problem for the Kuramoto-Sivashinsky equation
Baudouin, Lucie; Crépeau, Emmanuelle; Mercado, Alberto
2010-01-01
This paper presents an inverse problem for the Kuramoto-Sivashinsky (K-S) equation. The problem of retrieving the anti-diusion coefficient from a measurement of the solution is discussed. This measurement consists of the solution at some positive time and partial boundary data. Uniqueness and Lipschitz stability for this inverse problem are proven with the Bukhgeim-Klibanov method. The proof is based on a global Carleman inequality for the linearized K-S equation.
Solution of an inverse scattering problem for the acoustic wave equation in three-dimensional media
Baev, A. V.
2016-12-01
A three-dimensional inverse scattering problem for the acoustic wave equation is studied. The task is to determine the density and acoustic impedance of a medium. A necessary and sufficient condition for the unique solvability of this problem is established in the form of an energy conservation law. The interpretation of the solution to the inverse problem and the construction of medium images are discussed.
New progress in the inverse problem in the calculus of variations
2014-01-01
We present a new class of solutions for the inverse problem in the calculus of variations in arbitrary dimension $n$. This is the problem of determining the existence and uniqueness of Lagrangians for systems of $n$ second order ordinary differential equations. We also provide a number of new theorems concerning the inverse problem using exterior differential systems theory (EDS). Concentrating on the differential step of the EDS process, our new results provide a significant advance in the u...
Rizzuti, G.; Gisolf, A.
2017-03-01
We study a reconstruction algorithm for the general inverse scattering problem based on the estimate of not only medium properties, as in more conventional approaches, but also wavefields propagating inside the computational domain. This extended set of unknowns is justified as a way to prevent local minimum stagnation, which is a common issue for standard methods. At each iteration of the algorithm, (i) the model parameters are obtained by solution of a convex problem, formulated from a special bilinear relationship of the data with respect to properties and wavefields (where the wavefield is kept fixed), and (ii) a better estimate of the wavefield is calculated, based on the previously reconstructed properties. The resulting scheme is computationally convenient since step (i) can greatly benefit from parallelization and the wavefield update (ii) requires modeling only in the known background model, which can be sped up considerably by factorization-based direct methods. The inversion method is successfully tested on synthetic elastic datasets.
The forward and inverse problems in time-distance helioseismology
Jackiewicz, Jason; Gizon, Laurent; Birch, Aaron C.
2008-10-01
Time-distance helioseismology is a set of tools for peering into the solar interior. In this paper we discuss and provide examples of the steps that go into current high-resolution time-distance helioseismic analyses. These steps include observations (cross covariances, travel times), modeling of the seismic wavefield for a weakly inhomogeneous solar model, and inversion of the travel times. The discussion is framed in the context of studying quiet-Sun flows, although the extension to other solar perturbations is straightforward and analogous. The two-plus-one-dimensional (2+1D) inversion procedure implemented here produces maps of vector flows in the near-surface layers of the photosphere. We examine the flows obtained by compromising, or 'trading off', between different observation times, spatial resolutions, and noise levels. Also studied is the correlation of the flows at different depths and over different time intervals.
The forward and inverse problems in time-distance helioseismology
Energy Technology Data Exchange (ETDEWEB)
Jackiewicz, Jason; Gizon, Laurent [Max-Planck-Institut fuer Sonnensystemforschung, 37191 Katlenburg-Lindau (Germany); Birch, Aaron C [Colorado Research Associates, NWRA, 3380 Mitchell Lane, Boulder, CO 80301 (United States)], E-mail: jackiewicz@mps.mpg.de
2008-10-15
Time-distance helioseismology is a set of tools for peering into the solar interior. In this paper we discuss and provide examples of the steps that go into current high-resolution time-distance helioseismic analyses. These steps include observations (cross covariances, travel times), modeling of the seismic wavefield for a weakly inhomogeneous solar model, and inversion of the travel times. The discussion is framed in the context of studying quiet-Sun flows, although the extension to other solar perturbations is straightforward and analogous. The two-plus-one-dimensional (2+1D) inversion procedure implemented here produces maps of vector flows in the near-surface layers of the photosphere. We examine the flows obtained by compromising, or 'trading off', between different observation times, spatial resolutions, and noise levels. Also studied is the correlation of the flows at different depths and over different time intervals.
Directory of Open Access Journals (Sweden)
Morteza Ebrahimi
2012-01-01
Full Text Available The purpose of the present study is to provide a fast and accurate algorithm for identifying the medium temperature and the unknown radiation term from an overspecified condition on the boundary in an inverse problem of linear heat equation with nonlinear boundary condition. The design of the paper is to employ Taylor’s series expansion for linearize nonlinear term and then finite-difference approximation to discretize the problem domain. Owing to the application of the finite difference scheme, a large sparse system of linear algebraic equations is obtained. An approach of Monte Carlo method is employed to solve the linear system and estimate unknown radiation term. The Monte Carlo optimization is adopted to modify the estimated values. Results show that a good estimation on the radiation term can be obtained within a couple of minutes CPU time at pentium IV-2.4 GHz PC.
Energy Technology Data Exchange (ETDEWEB)
Hamman, E.; Zorgati, R.
1995-12-31
Eddy current non-destructive testing is used by EDF to detect flaws affecting conductive objects such as steam generator tubes. With a view to obtaining ever more accurate information on equipment integrity, thereby facilitating diagnosis, studies aimed at using measurements to reconstruct an image of the flaw have been proceeding now for about ten years. In this context, our approach to eddy current imaging is based on inverse problem formalism. The direct problem, involving a mathematical model linking measurements provided by a probe with variables characterizing the defect, is dealt with elsewhere. Using the model results, we study the possibility of inverting it, i.e. of reconstructing an image of the flaw from the measurements. We first give an overview of the different inversion techniques, representative of the state of the art and all based on linearization of the inverse problem by means of the Born approximation. The model error resulting from an excessive Born approximation nevertheless severely limits the quantity of the images which can be obtained. In order to counteract this often critical error and extend the eddy current imaging application field, we have to del with the non-linear inverse problem. A method derived from recent research is proposed and implemented to ensure consistency with the exact model. Based on an `optimization` type approach and provided with a convergence theorem, the method is highly efficient. (authors). 17 refs., 7 figs., 1 append.
Inverse problem of elastica of a variable-arc-length beam subjected to a concentrated load
Institute of Scientific and Technical Information of China (English)
Xiaowei Zhang; Jialing Yang; Keren Wang
2005-01-01
An inverse problem of elastica of a variable-arclength beam subjected to a concentrated load is investigated.The beam is fixed at one end, and can slide freely over a hinge support at the other end. The inverse problem is to determine the value of the load when the deflection of the action point of the load is given. Based on the elasitca equations and the elliptic integrals, a set of nonlinear equations for the inverse problem are derived, and an analytical solution by means of iterations and Quasi-Newton method is presented. From the results, the relationship between the loads and deflections of the loading point is obtained.
DEFF Research Database (Denmark)
Hansen, Thomas Mejer; Cordua, Knud Skou; Holm Jacobsen, Bo
2014-01-01
forward models, can be more than an order of magnitude larger than the measurement uncertainty. We also found that the modeling error is strongly linked to the spatial variability of the assumed velocity field, i.e., the a priori velocity model.We discovered some general tools by which the modeling error...... synthetic ground-penetrating radar crosshole tomographic inverse problems. Ignoring the modeling error can lead to severe artifacts, which erroneously appear to be well resolved in the solution of the inverse problem. Accounting for the modeling error leads to a solution of the inverse problem consistent...
Physics-based Inverse Problem to Deduce Marine Atmospheric Boundary Layer Parameters
2017-03-07
please find the Final Technical Report with SF 298 for Dr. Erin E. Hackett’s ONR grant entitled Physics -based Inverse Problem to Deduce Marine...From- To) 07/03/2017 Final Technica l Dec 2012- Dec 2016 4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER Physics -based Inverse Problem to Deduce Marine...19b. TELEPHONE NUMBER (Include area code) 843-349-4087 Standard Form 298 (Rev. 8/98) Prescribed by ANSI Std. Z39.18 Physics -Based Inverse Problem To
Free-energy functional method for inverse problem of self assembly
Torikai, Masashi
2015-04-01
A new theoretical approach is described for the inverse self-assembly problem, i.e., the reconstruction of the interparticle interaction from a given structure. This theory is based on the variational principle for the functional that is constructed from a free energy functional in combination with Percus's approach [J. Percus, Phys. Rev. Lett. 8, 462 (1962)]. In this theory, the interparticle interaction potential for the given structure is obtained as the function that maximizes the functional. As test cases, the interparticle potentials for two-dimensional crystals, such as square, honeycomb, and kagome lattices, are predicted by this theory. The formation of each target lattice from an initial random particle configuration in Monte Carlo simulations with the predicted interparticle interaction indicates that the theory is successfully applied to the test cases.
An inverse problem for a one-dimensional time-fractional diffusion problem
Jin, Bangti
2012-06-26
We study an inverse problem of recovering a spatially varying potential term in a one-dimensional time-fractional diffusion equation from the flux measurements taken at a single fixed time corresponding to a given set of input sources. The unique identifiability of the potential is shown for two cases, i.e. the flux at one end and the net flux, provided that the set of input sources forms a complete basis in L 2(0, 1). An algorithm of the quasi-Newton type is proposed for the efficient and accurate reconstruction of the coefficient from finite data, and the injectivity of the Jacobian is discussed. Numerical results for both exact and noisy data are presented. © 2012 IOP Publishing Ltd.
Hunziker, J.W.; Thorbecke, J.W.; Slob, E.C.
2015-01-01
We exploit the randomness of a genetic inversion algorithm to map the global minimum of the solution space of Controlled-Source Electromagnetic inversion problems. In this study, we focus on the information content that vertical electric or magnetic receivers could add to solve for anisotropic condu
An inverse-source problem for maximization of pore-fluid oscillation within poroelastic formations
Jeong, C.
2016-07-04
This paper discusses a mathematical and numerical modeling approach for identification of an unknown optimal loading time signal of a wave source, atop the ground surface, that can maximize the relative wave motion of a single-phase pore fluid within fluid-saturated porous permeable (poroelastic) rock formations, surrounded by non-permeable semi-infinite elastic solid rock formations, in a one-dimensional setting. The motivation stems from a set of field observations, following seismic events and vibrational tests, suggesting that shaking an oil reservoir is likely to improve oil production rates. This maximization problem is cast into an inverse-source problem, seeking an optimal loading signal that minimizes an objective functional – the reciprocal of kinetic energy in terms of relative pore-fluid wave motion within target poroelastic layers. We use the finite element method to obtain the solution of the governing wave physics of a multi-layered system, where the wave equations for the target poroelastic layers and the elastic wave equation for the surrounding non-permeable layers are coupled with each other. We use a partial-differential-equation-constrained-optimization framework (a state-adjoint-control problem approach) to tackle the minimization problem. The numerical results show that the numerical optimizer recovers optimal loading signals, whose dominant frequencies correspond to amplification frequencies, which can also be obtained by a frequency sweep, leading to larger amplitudes of relative pore-fluid wave motion within the target hydrocarbon formation than other signals.
THE INVERSE PROBLEM FOR PART SYMMETRIC MATRICES ON A SUBSPACE
Institute of Scientific and Technical Information of China (English)
Zhen-yun Peng; Xi-yan Hu; Lei Zhang
2003-01-01
In this paper, the following two problems are considered:Problem I. Given S ∈ Rn×p, X, B ∈ Rn×m, find A ∈ SRs,n such that AX = B, where SRs,n = {A ∈ Rn×n|xT(A - AT) = 0, for all x ∈ R(S)}.Problem Ⅱ. Given A* ∈ Rn×n, find A ∈ SE such that ‖A^-A*‖ = minA∈sE‖A-A*‖,where SE is the solution set of Problem Ⅰ.The necessary and sufficient conditions for the solvability of and the general form of the solutions of problem Ⅰ are given. For problem Ⅱ, the expression for the solution, a numerical algorithm and a numerical example are provided.
Pontes, P. C.; Naveira-Cotta, C. P.
2016-09-01
The theoretical analysis for the design of microreactors in biodiesel production is a complicated task due to the complex liquid-liquid flow and mass transfer processes, and the transesterification reaction that takes place within these microsystems. Thus, computational simulation is an important tool that aids in understanding the physical-chemical phenomenon and, consequently, in determining the suitable conditions that maximize the conversion of triglycerides during the biodiesel synthesis. A diffusive-convective-reactive coupled nonlinear mathematical model, that governs the mass transfer process during the transesterification reaction in parallel plates microreactors, under isothermal conditions, is here described. A hybrid numerical-analytical solution via the Generalized Integral Transform Technique (GITT) for this partial differential system is developed and the eigenfunction expansions convergence rates are extensively analyzed and illustrated. The heuristic method of Particle Swarm Optimization (PSO) is applied in the inverse analysis of the proposed direct problem, to estimate the reaction kinetics constants, which is a critical step in the design of such microsystems. The results present a good agreement with the limited experimental data in the literature, but indicate that the GITT methodology combined with the PSO approach provide a reliable computational algorithm for direct-inverse analysis in such reactive mass transfer problems.
The inverse problem of constructing a gravimetric geoid
Zlotnicki, V.; Parsons, B.; Wunsch, C.
1982-01-01
Computation of a single geoidal height from gravity acceleration data formally requires that the latter be known everywhere on the earth. A computational procedure based on linear inverse theory for estimating geoidal heights from incomplete sets of data is presented. The same scheme can be used to estimate gravity accelerations from altimetry-derived geoids. The systematic error owing to lack of data and the choice of a particular inverse operator is described by using resolution functions and their spherical harmonic expansions. An rms value of this error is also estimated by assuming a spectrum for the unknown geoid. The influence of the size of the data region, the spacing between data, the filtering applied to the data, and the model weighting function chosen are all quantified in a spherical geometry. The examples presented show that when low degree spherical harmonic coefficients are available - from satellite orbit analysis - a band-passed version of the geoid can be constructed from local gravity data, even with a relatively restricted data set.
The inverse problem of constructing a gravimetric geoid
Zlotnicki, V.; Parsons, B.; Wunsch, C.
1982-01-01
Computation of a single geoidal height from gravity acceleration data formally requires that the latter be known everywhere on the earth. A computational procedure based on linear inverse theory for estimating geoidal heights from incomplete sets of data is presented. The same scheme can be used to estimate gravity accelerations from altimetry-derived geoids. The systematic error owing to lack of data and the choice of a particular inverse operator is described by using resolution functions and their spherical harmonic expansions. An rms value of this error is also estimated by assuming a spectrum for the unknown geoid. The influence of the size of the data region, the spacing between data, the filtering applied to the data, and the model weighting function chosen are all quantified in a spherical geometry. The examples presented show that when low degree spherical harmonic coefficients are available - from satellite orbit analysis - a band-passed version of the geoid can be constructed from local gravity data, even with a relatively restricted data set.
Variational principles and optimal solutions of the inverse problems of creep bending of plates
Bormotin, K. S.; Oleinikov, A. I.
2012-09-01
It is shown that inverse problems of steady-state creep bending of plates in both the geometrically linear and nonlinear formulations can be represented in a variational formulation. Steady-state values of the obtained functionals corresponding to the solutions of the problems of inelastic deformation and elastic unloading are determined by applying a finite element procedure to the functionals. Optimal laws of creep deformation are formulated using the criterion of minimizing damage in the functionals of the inverse problems. The formulated problems are reduced to the problems solved by the finite element method using MSC.Marc software.
Solving the Monge-Amp\\`ere Equations for the Inverse Reflector Problem
Brix, Kolja; Platen, Andreas
2014-01-01
The inverse reflector problem arises in geometrical nonimaging optics: Given a light source and a target, the question is how to design a reflecting free-form surface such that a desired light density distribution is generated on the target, e.g., a projected image on a screen. This optical problem can mathematically be understood as a problem of optimal transport and equivalently be expressed by a secondary boundary value problem of the Monge-Amp\\`ere equation, which consists of a highly nonlinear partial differential equation of second order and constraints. In our approach the Monge-Amp\\`ere equation is numerically solved using a collocation method based on tensor-product B-splines, in which nested iteration techniques are applied to ensure the convergence of the nonlinear solver and to speed up the calculation. In the numerical method special care has to be taken for the constraint: It enters the discrete problem formulation via a Picard-type iteration. Numerical results are presented as well for benchmar...
Active Subspace Methods for Data-Intensive Inverse Problems
Energy Technology Data Exchange (ETDEWEB)
Wang, Qiqi [Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States)
2017-04-27
The project has developed theory and computational tools to exploit active subspaces to reduce the dimension in statistical calibration problems. This dimension reduction enables MCMC methods to calibrate otherwise intractable models. The same theoretical and computational tools can also reduce the measurement dimension for calibration problems that use large stores of data.
An Inverse Eigenvalue Problem for Damped Gyroscopic Second-Order Systems
Directory of Open Access Journals (Sweden)
Yongxin Yuan
2009-01-01
analytical mass and stiffness matrices, so that ( has a prescribed subset of eigenvalues and eigenvectors, is considered. Necessary and sufficient conditions under which this quadratic inverse eigenvalue problem is solvable are specified.
Error Analysis in the Joint Event Location/Seismic Calibration Inverse Problem
National Research Council Canada - National Science Library
Rodi, William L
2006-01-01
...: The analysis is being done in the context of the multiple-event inverse problem, in which the locations of multiple events are inferred jointly with travel-time corrections for the event-station paths...
Generalized Uncertainty Quantification for Linear Inverse Problems in X-ray Imaging
Energy Technology Data Exchange (ETDEWEB)
Fowler, Michael James [Clarkson Univ., Potsdam, NY (United States)
2014-04-25
In industrial and engineering applications, X-ray radiography has attained wide use as a data collection protocol for the assessment of material properties in cases where direct observation is not possible. The direct measurement of nuclear materials, particularly when they are under explosive or implosive loading, is not feasible, and radiography can serve as a useful tool for obtaining indirect measurements. In such experiments, high energy X-rays are pulsed through a scene containing material of interest, and a detector records a radiograph by measuring the radiation that is not attenuated in the scene. One approach to the analysis of these radiographs is to model the imaging system as an operator that acts upon the object being imaged to produce a radiograph. In this model, the goal is to solve an inverse problem to reconstruct the values of interest in the object, which are typically material properties such as density or areal density. The primary objective in this work is to provide quantitative solutions with uncertainty estimates for three separate applications in X-ray radiography: deconvolution, Abel inversion, and radiation spot shape reconstruction. For each problem, we introduce a new hierarchical Bayesian model for determining a posterior distribution on the unknowns and develop efficient Markov chain Monte Carlo (MCMC) methods for sampling from the posterior. A Poisson likelihood, based on a noise model for photon counts at the detector, is combined with a prior tailored to each application: an edge-localizing prior for deconvolution; a smoothing prior with non-negativity constraints for spot reconstruction; and a full covariance sampling prior based on a Wishart hyperprior for Abel inversion. After developing our methods in a general setting, we demonstrate each model on both synthetically generated datasets, including those from a well known radiation transport code, and real high energy radiographs taken at two U. S. Department of Energy
Improving landscape inference by integrating heterogeneous data in the inverse Ising problem
Barrat-Charlaix, Pierre; Figliuzzi, Matteo; Weigt, Martin
2016-11-01
The inverse Ising problem and its generalizations to Potts and continuous spin models have recently attracted much attention thanks to their successful applications in the statistical modeling of biological data. In the standard setting, the parameters of an Ising model (couplings and fields) are inferred using a sample of equilibrium configurations drawn from the Boltzmann distribution. However, in the context of biological applications, quantitative information for a limited number of microscopic spins configurations has recently become available. In this paper, we extend the usual setting of the inverse Ising model by developing an integrative approach combining the equilibrium sample with (possibly noisy) measurements of the energy performed for a number of arbitrary configurations. Using simulated data, we show that our integrative approach outperforms standard inference based only on the equilibrium sample or the energy measurements, including error correction of noisy energy measurements. As a biological proof-of-concept application, we show that mutational fitness landscapes in proteins can be better described when combining evolutionary sequence data with complementary structural information about mutant sequences.
The numerical solution of the boundary inverse problem for a parabolic equation
Vasil'ev, V. V.; Vasilyeva, M. V.; Kardashevsky, A. M.
2016-10-01
Boundary inverse problems occupy an important place among the inverse problems of mathematical physics. They are connected with the problems of diagnosis, when additional measurements on one of the borders or inside the computational domain are necessary to restore the boundary regime in the other border, inaccessible to direct measurements. The boundary inverse problems belong to a class of conditionally correct problems, and therefore, their numerical solution requires the development of special computational algorithms. The paper deals with the solution of the boundary inverse problem for one-dimensional second-order parabolic equations, consisting in the restoration of boundary regime according to measurements inside the computational domain. For the numerical solution of the inverse problem it is proposed to use an analogue of a computational algorithm, proposed and developed to meet the challenges of identification of the right side of the parabolic equations in the works P.N.Vabishchevich and his students based on a special decomposition of solving the problem at each temporal layer. We present and discuss the results of a computational experiment conducted on model problems with quasi-solutions, including with random errors in the input data.
The Use of Reciprocity in Atmospheric Source Inversion Problems
Energy Technology Data Exchange (ETDEWEB)
Nitao, J J
2004-10-13
The goal of the Event Reconstruction Project is to find the location and strength of atmospheric release points, both stationary and moving. Source inversion relies on observational data as input. The methodology is sufficiently general to allow various forms of data. In this report, the authors will focus primarily on concentration measurements obtained at point monitoring locations at various times. The algorithms being investigated in the Project are the MCMC (Markov Chain Monte Carlo), SMC (Sequential Monte Carlo) Methods, classical inversion methods, and hybrids of these. They refer the reader to the report by Johannesson et al. (2004) for explanations of these methods. These methods require computing the concentrations at all monitoring locations for a given ''proposed'' source characteristic (locations and strength history). It is anticipated that the largest portion of the CPU time will take place performing this computation. MCMC and SMC will require this computation to be done at least tens of thousands of times. Therefore, an efficient means of computing forward model predictions is important to making the inversion practical. In this report they show how Green's functions and reciprocal Green's functions can significantly accelerate forward model computations. First, instead of computing a plume for each possible source strength history, they can compute plumes from unit impulse sources only. By using linear superposition, they can obtain the response for any strength history. This response is given by the forward Green's function. Second, they may use the law of reciprocity. Suppose that they require the concentration at a single monitoring point x{sub m} due to a potential (unit impulse) source that is located at x{sub s}. instead of computing a plume with source location x{sub s}, they compute a ''reciprocal plume'' whose (unit impulse) source is at the monitoring locations x{sub m}. The
Genetic algorithms and smoothing filters in solving the geophysical inversion problem
Directory of Open Access Journals (Sweden)
Šešum Vesna
2002-01-01
Full Text Available The combination of genetic algorithms, smoothing filters and geophysical tomography is used in solving the geophysical inversion problem. This hybrid technique is developed to improve the results obtained by using genetic algorithm sonly. The application of smoothing filters can improve the performance of GA implementation for solving the geophysical inversion problem. Some test-examples and the obtained comparative results are presented.
Directory of Open Access Journals (Sweden)
Kiwoon Kwon
2015-01-01
measured data for the inverse problem. For anisotropic coefficient with anomaly with or without jumps from known or unknown background, nonuniqueness of the inverse problems is discussed and the relation to cloaking or illusion of the anomaly is explained. The uniqueness and nonuniqueness issues are discussed firstly for EIT and secondly for ISP in similar arguments. Arguing the relation between source-to-detector map and Dirichlet-to-Neumann map in DOT and the uniqueness and nonuniqueness of DOT are also explained.
Some Inverse Problems in Periodic Homogenization of Hamilton-Jacobi Equations
Luo, Songting; Tran, Hung V.; Yu, Yifeng
2016-09-01
We look at the effective Hamiltonian {overline{H}} associated with the Hamiltonian {H(p,x)=H(p)+V(x)} in the periodic homogenization theory. Our central goal is to understand the relation between {V} and {overline{H}}. We formulate some inverse problems concerning this relation. Such types of inverse problems are, in general, very challenging. In this paper, we discuss several special cases in both convex and nonconvex settings.
Directory of Open Access Journals (Sweden)
Abel Palafox
2014-01-01
Full Text Available We address a prototype inverse scattering problem in the interface of applied mathematics, statistics, and scientific computing. We pose the acoustic inverse scattering problem in a Bayesian inference perspective and simulate from the posterior distribution using MCMC. The PDE forward map is implemented using high performance computing methods. We implement a standard Bayesian model selection method to estimate an effective number of Fourier coefficients that may be retrieved from noisy data within a standard formulation.
Uniqueness of Inversion Problems Described by First-Kind Integral Equations
Institute of Scientific and Technical Information of China (English)
徐铁峰
2002-01-01
We propose a general method to prove the uniqueness of the inversion problems described by first-kind integral equations. The method depends on the analytical properties of the Fourier transform of the integral kernel and the finiteness of the total states (or probability, if normalized), the integration of the "local" density of states, which is a rather moderate condition and satisfied by many inversion problems arising from physics and engineering.
Uniqueness and local stability for the inverse scattering problem of determining the cavity
Institute of Scientific and Technical Information of China (English)
FENG; Lixin; MA; Fuming
2005-01-01
Considering a time-harmonic electromagnetic plane wave incident on an arbitrarily shaped open cavity embedded in infinite ground plane, the physical process is modelled by Maxwell's equations. We investigate the inverse problem of determining the shape of the open cavity from the information of the measured scattered field. Results on the uniqueness and the local stability of the inverse problem in the 2-dimensional TM (transverse magnetic) polarization are proved in this paper.
Review of the inverse scattering problem at fixed energy in quantum mechanics
Sabatier, P. C.
1972-01-01
Methods of solution of the inverse scattering problem at fixed energy in quantum mechanics are presented. Scattering experiments of a beam of particles at a nonrelativisitic energy by a target made up of particles are analyzed. The Schroedinger equation is used to develop the quantum mechanical description of the system and one of several functions depending on the relative distance of the particles. The inverse problem is the construction of the potentials from experimental measurements.
Moving Least Squares Method for a One-Dimensional Parabolic Inverse Problem
Directory of Open Access Journals (Sweden)
Baiyu Wang
2014-01-01
Full Text Available This paper investigates the numerical solution of a class of one-dimensional inverse parabolic problems using the moving least squares approximation; the inverse problem is the determination of an unknown source term depending on time. The collocation method is used for solving the equation; some numerical experiments are presented and discussed to illustrate the stability and high efficiency of the method.
Calculation Error of Numerical Solution for a Boundary—Value Inverse Heat Conduction Problem
Institute of Scientific and Technical Information of China (English)
LiXijing; HeQun; 等
1996-01-01
A one-dimensional linear inverse heat conduction problem is studied in this paper,This ill-posed problem is replaced by the perturbed problem with a non-localized boundary condition.After the derivation of its closed-from analytical solution,the calculation error can be determinde by the comparison between the numerical and exact solutions.
A Study of Inverse Problems Based on Two Kinds of Special Matrix Equations in Euclidean Space
Directory of Open Access Journals (Sweden)
Rui Huang
2014-01-01
Full Text Available Two special classes of symmetric coefficient matrices were defined based on characteristics matrix; meanwhile, the expressions of the solution to inverse problems are given and the conditions for the solvability of these problems are studied relying on researching. Finally, the optimal approximation solution of these problems is provided.
SQUIDs and inverse problem techniques in nondestructive evaluation of metals
Bruno, A C
2001-01-01
Superconducting Quantum Interference Devices coupled to gradiometers were used to defect flaws in metals. We detected flaws in aluminium samples carrying current, measuring fields at lift-off distances up to one order of magnitude larger than the size of the flaw. Configured as a susceptometer we detected surface-braking flaws in steel samples, measuring the distortion on the applied magnetic field. We also used spatial filtering techniques to enhance the visualization of the magnetic field due to the flaws. In order to assess its severity, we used the generalized inverse method and singular value decomposition to reconstruct small spherical inclusions in steel. In addition, finite elements and optimization techniques were used to image complex shaped flaws.
Energy Technology Data Exchange (ETDEWEB)
Manoli, Gabriele, E-mail: manoli@dmsa.unipd.it [Department of Mathematics, University of Padova, Via Trieste 63, 35121 Padova (Italy); Nicholas School of the Environment, Duke University, Durham, NC 27708 (United States); Rossi, Matteo [Department of Geosciences, University of Padova, Via Gradenigo 6, 35131 Padova (Italy); Pasetto, Damiano [Department of Mathematics, University of Padova, Via Trieste 63, 35121 Padova (Italy); Deiana, Rita [Dipartimento dei Beni Culturali, University of Padova, Piazza Capitaniato 7, 35139 Padova (Italy); Ferraris, Stefano [Interuniversity Department of Regional and Urban Studies and Planning, Politecnico and University of Torino, Viale Mattioli 39, 10125 Torino (Italy); Cassiani, Giorgio [Department of Geosciences, University of Padova, Via Gradenigo 6, 35131 Padova (Italy); Putti, Mario [Department of Mathematics, University of Padova, Via Trieste 63, 35121 Padova (Italy)
2015-02-15
The modeling of unsaturated groundwater flow is affected by a high degree of uncertainty related to both measurement and model errors. Geophysical methods such as Electrical Resistivity Tomography (ERT) can provide useful indirect information on the hydrological processes occurring in the vadose zone. In this paper, we propose and test an iterated particle filter method to solve the coupled hydrogeophysical inverse problem. We focus on an infiltration test monitored by time-lapse ERT and modeled using Richards equation. The goal is to identify hydrological model parameters from ERT electrical potential measurements. Traditional uncoupled inversion relies on the solution of two sequential inverse problems, the first one applied to the ERT measurements, the second one to Richards equation. This approach does not ensure an accurate quantitative description of the physical state, typically violating mass balance. To avoid one of these two inversions and incorporate in the process more physical simulation constraints, we cast the problem within the framework of a SIR (Sequential Importance Resampling) data assimilation approach that uses a Richards equation solver to model the hydrological dynamics and a forward ERT simulator combined with Archie's law to serve as measurement model. ERT observations are then used to update the state of the system as well as to estimate the model parameters and their posterior distribution. The limitations of the traditional sequential Bayesian approach are investigated and an innovative iterative approach is proposed to estimate the model parameters with high accuracy. The numerical properties of the developed algorithm are verified on both homogeneous and heterogeneous synthetic test cases based on a real-world field experiment.
Inverse modeling approach to allogenic karst system characterization.
Dörfliger, N; Fleury, P; Ladouche, B
2009-01-01
Allogenic karst systems function in a particular way that is influenced by the type of water infiltrating through river water losses, by karstification processes, and by water quality. Management of this system requires a good knowledge of its structure and functioning, for which a new methodology based on an inverse modeling approach appears to be well suited. This approach requires both spring and river inflow discharge measurements and a continuous record of chemical parameters in the river and at the spring. The inverse model calculates unit hydrographs and the impulse responses of fluxes from rainfall hydraulic head at the spring or rainfall flux data, the purpose of which is hydrograph separation. Hydrograph reconstruction is done using rainfall and river inflow data as model input and enables definition at each time step of the ratio of each component. Using chemical data, representing event and pre-event water, as input, it is possible to determine the origin of spring water (either fast flow through the epikarstic zone or slow flow through the saturated zone). This study made it possible to improve a conceptual model of allogenic karst system functioning. The methodology is used to study the Bas-Agly and the Cent Font karst systems, two allogenic karst systems in Southern France.
Computational experiment on the numerical solution of some inverse problems of mathematical physics
Vasil'ev, V. I.; Kardashevsky, A. M.; Sivtsev, PV
2016-11-01
In this article the computational experiment on the numerical solution of the most popular linear inverse problems for equations of mathematical physics are presented. The discretization of retrospective inverse problem for parabolic equation is performed using difference scheme with non-positive weight multiplier. Similar difference scheme is also used for the numerical solution of Cauchy problem for two-dimensional Laplace equation. The results of computational experiment, performed on model problems with exact solution, including ones with randomly perturbed input data are presented and discussed.
The algebraic method of the scattering inverse problem solution under untraditional statements
Popushnoj, M N
2001-01-01
The algebraic method of the scattering inverse problem solution under untraditional statements is proposed consistently in this review, in the framework of which some quantum theory od scattering charged particles problem were researched afterwards. The inverse problem of scattering theory of charged particles on the complex plane of the Coulomb coupling constant (CCC) is considered. A procedure of interaction potential restoration is established for the case when the energy, orbital moment quadrate and CCC are linearly dependent. The relation between one-parametric problems of the potential scattering of charged particles is investigated
Applying neural networks to the solution of forward and inverse heat conduction problems
Energy Technology Data Exchange (ETDEWEB)
Deng, S.; Hwang, Y. [Department of Weapon System Engineering, Chung Cheng Institute of Technology, National Defense University, No. 190, Sanyuan 1st St., Dashi Jen, Taoyuan 33509, Taiwan (Taiwan)
2006-12-15
This paper employs the continuous-time analogue Hopfield neural network to compute the temperature distribution in forward heat conduction problems and solves inverse heat conduction problems by using a back propagation neural (BPN) network to identify the unknown boundary conditions. The weak generalization capacity of BPN networks is improved by employing the Bayesian regularization algorithm. The feasibility of the proposed method is examined in a series of numerical simulations. The results show that the proposed neural network analysis method successfully solves forward heat conduction problems and is capable of predicting the unknown parameters in inverse problems with an acceptable error. (author)
The inverse maximum flow problem with lower and upper bounds for the flow
Directory of Open Access Journals (Sweden)
Deaconu Adrian
2008-01-01
Full Text Available The general inverse maximum flow problem (denoted GIMF is considered, where lower and upper bounds for the flow are changed so that a given feasible flow becomes a maximum flow and the distance (considering l1 norm between the initial vector of bounds and the modified vector is minimum. Strongly and weakly polynomial algorithms for solving this problem are proposed. In the paper it is also proved that the inverse maximum flow problem where only the upper bound for the flow is changed (IMF is a particular case of the GIMF problem.
On rational approximation methods for inverse source problems
Rundell, William
2011-02-01
The basis of most imaging methods is to detect hidden obstacles or inclusions within a body when one can only make measurements on an exterior surface. Such is the ubiquity of these problems, the underlying model can lead to a partial differential equation of any of the major types, but here we focus on the case of steady-state electrostatic or thermal imaging and consider boundary value problems for Laplace\\'s equation. Our inclusions are interior forces with compact support and our data consists of a single measurement of (say) voltage/current or temperature/heat flux on the external boundary. We propose an algorithm that under certain assumptions allows for the determination of the support set of these forces by solving a simpler "equivalent point source" problem, and which uses a Newton scheme to improve the corresponding initial approximation. © 2011 American Institute of Mathematical Sciences.
SIPPI: A Matlab toolbox for sampling the solution to inverse problems with complex prior information
DEFF Research Database (Denmark)
Hansen, Thomas Mejer; Cordua, Knud Skou; Looms, Majken Caroline
2013-01-01
We present an application of the SIPPI Matlab toolbox, to obtain a sample from the a posteriori probability density function for the classical tomographic inversion problem. We consider a number of different forward models, linear and non-linear, such as ray based forward models that rely...... on the high frequency approximation of the wave-equation and ‘fat’ ray based forward models relying on finite frequency theory. In order to sample the a posteriori probability density function we make use of both least squares based inversion, for linear Gaussian inverse problems, and the extended Metropolis...... sampler, for non-linear non-Gaussian inverse problems. To illustrate the applicability of the SIPPI toolbox to a tomographic field data set we use a cross-borehole traveltime data set from Arrenæs, Denmark. Both the computer code and the data are released in the public domain using open source and open...
Fast Inverse Nonlinear Fourier Transforms for Fiber Bragg Grating Design and Related Problems
Wahls, Sander
2016-01-01
The problem of constructing a fiber Bragg grating profile numerically such that the reflection coefficient of the grating matches a given specification is considered. The well-known analytic solution to this problem is given by a suitable inverse nonlinear Fourier transform (also known as inverse scattering transform) of the specificed reflection coefficient. Many different algorithms have been proposed to compute this inverse nonlinear Fourier transform numerically. The most efficient ones require $\\mathcal{O}(D^{2})$ floating point operations (flops) to generate $D$ samples of the grating profile. In this paper, two new fast inverse nonlinear Fourier transform algorithms that require only $\\mathcal{O}(D\\log^{2}D)$ flops are proposed. The merits of our algorithms are demonstrated in numerical examples, in which they are compared to a conventional layer peeling method, the Toeplitz inner bordering method and integral layer peeling. One of our two algorithms also extends to the design problem for fiber-assiste...
Adaptive eigenspace method for inverse scattering problems in the frequency domain
Grote, Marcus J.; Kray, Marie; Nahum, Uri
2017-02-01
A nonlinear optimization method is proposed for the solution of inverse scattering problems in the frequency domain, when the scattered field is governed by the Helmholtz equation. The time-harmonic inverse medium problem is formulated as a PDE-constrained optimization problem and solved by an inexact truncated Newton-type iteration. Instead of a grid-based discrete representation, the unknown wave speed is projected to a particular finite-dimensional basis of eigenfunctions, which is iteratively adapted during the optimization. Truncating the adaptive eigenspace (AE) basis at a (small and slowly increasing) finite number of eigenfunctions effectively introduces regularization into the inversion and thus avoids the need for standard Tikhonov-type regularization. Both analytical and numerical evidence underpins the accuracy of the AE representation. Numerical experiments demonstrate the efficiency and robustness to missing or noisy data of the resulting adaptive eigenspace inversion method.
Iterative Solutions to the Inverse Geometric Problem for Manipulators with no Closed Form Solution
Directory of Open Access Journals (Sweden)
Pål Johan From
2008-07-01
Full Text Available A set of new iterative solutions to the inverse geometric problem is presented. The approach is general and does not depend on intersecting axes or calculation of the Jacobian. The solution can be applied to any manipulator and is well suited for manipulators for which convergence is poor for conventional Jacobian-based iterative algorithms. For kinematically redundant manipulators, weights can be applied to each joint to introduce stiffness and for collision avoidance. The algorithm uses the unit quaternion to represent the position of each joint and calculates analytically the optimal position of the joint when only the respective joint is considered. This sub-problem is computationally very efficient due to the analytical solution. Several algorithms based on the solution of this sub-problem are presented. For difficult problems, for which the initial condition is far from a solution or the geometry of the manipulator makes the solution hard to reach, it is shown that the algorithm finds a solution fairly close to the solution in only a few iterations.
On the possibility of control restoration in some inverse problems of heat and mass transfer
Bilchenko, G. G.; Bilchenko, N. G.
2016-11-01
The hypersonic aircraft permeable surfaces effective heat protection problems are considered. The physic-chemical processes (the dissociation and the ionization) in laminar boundary layer of compressible gas are appreciated in mathematical model. The statements of direct problems of heat and mass transfer are given: according to preset given controls it is necessary to compute the boundary layer mathematical model parameters and determinate the local and total heat flows and friction forces and the power of blowing system. The A.A.Dorodnicyn's generalized integral relations method has been used as calculation basis. The optimal control - the blowing into boundary layer (for continuous functions) was constructed as the solution of direct problem in extreme statement with the use of this approach. The statement of inverse problems are given: the control laws ensuring the preset given local heat flow and local tangent friction are restored. The differences between the interpolation and the approximation statements are discussed. The possibility of unique control restoration is established and proved (in the stagnation point). The computational experiments results are presented.
Elastic reflection based waveform inversion with a nonlinear approach
Guo, Qiang
2017-08-16
Full waveform inversion (FWI) is a highly nonlinear problem due to the complex reflectivity of the Earth, and this nonlinearity only increases under the more expensive elastic assumption. In elastic media, we need a good initial P-wave velocity and even a better initial S-wave velocity models with accurate representation of the low model wavenumbers for FWI to converge. However, inverting for the low wavenumber components of P- and S-wave velocities using reflection waveform inversion (RWI) with an objective to fit the reflection shape, rather than produce reflections, may mitigate the limitations of FWI. Because FWI, performing as a migration operator, is in preference of the high wavenumber updates along reflectors. We propose a nonlinear elastic RWI that inverts for both the low wavenumber and perturbation components of the P- and S-wave velocities. To generate the full elastic reflection wavefields, we derive an equivalent stress source made up by the inverted model perturbations and incident wavefields. We update both the perturbation and propagation parts of the velocity models in a nested fashion. Applications on synthetic isotropic models and field data show that our method can efficiently update the low and high wavenumber parts of the models.
New technique for phase shift analysis multi-energy solution of inverse scattering problem
Cooper, S G; MacIntosh, R S; Kuznetsova, E V
1998-01-01
We demonstrate a new approach to the analysis of extensive multi-energy data. For the case of d + He-4, we produce a phase shift analysis covering for the energy range 3 to 11 MeV. The key idea is the use of iterative perturbative data-to-potential inversion which can produce potentials which reproduce the data simultaneously over a range of energies. It thus effectively regularizes the extraction of phase shifts from diverse, incomplete and possibly somewhat contradictory data sets. In doing so, it will provide guidance to experimentalists as to what further measurements should be made. This study is limited to vector spin observables and spin-orbit interactions. We discuss alternative ways in which the theory can be implemented and which provide insight into the ambiguity problems. We compare the extrapolation of these solutions to other energies. Majorana terms are presented for each potential component.
An inverse problem of thickness design for bilayer textile materials under low temperature
Xu, Dinghua; Cheng, Jianxin; Chen, Yuanbo; Ge, Meibao
2011-04-01
The human heat-moisture-comfort level is mainly determined by heat and moisture transfer characteristics in clothing. With respect to the model of steady-state heat and moisture transfer through parallel pore textiles, we propose an inverse problem of thickness design for bilayer textile material under low temperature in this paper. Adopting the idea of regularization method, we formulate the inverse problem solving into a function minimization problem. Combining the finite difference method for ordinary differential equations with direct search method of one-dimensional minimization problems, we derive three kinds of iteration algorithms of regularized solution for the inverse problem of thickness design. Numerical simulation is achieved to verify the efficiency of proposed methods.
The Inverse 1-Median Problem on Tree Networks with Variable Real Edge Lengths
Directory of Open Access Journals (Sweden)
Longshu Wu
2013-01-01
Full Text Available Location problems exist in the real world and they mainly deal with finding optimal locations for facilities in a network, such as net servers, hospitals, and shopping centers. The inverse location problem is also often met in practice and has been intensively investigated in the literature. As a typical inverse location problem, the inverse 1-median problem on tree networks with variable real edge lengths is discussed in this paper, which is to modify the edge lengths at minimum total cost such that a given vertex becomes a 1-median of the tree network with respect to the new edge lengths. First, this problem is shown to be solvable in linear time with variable nonnegative edge lengths. For the case when negative edge lengths are allowable, the NP-hardness is proved under Hamming distance, and strongly polynomial time algorithms are presented under l1 and l∞ norms, respectively.
Nassar, Mohamed K.; Ginn, Timothy R.
2014-08-01
We investigate the effect of computational error on the inversion of a density-dependent flow and transport model, using SEAWAT and UCODE-2005 in an inverse identification of hydraulic conductivity and dispersivity using head and concentration data from a 2-D laboratory experiment. We investigated inversions using three different solution schemes including variation of number of particles and time step length, in terms of the three aspects: the shape and smoothness of the objective function surface, the consequent impacts to the optimization, and the resulting Pareto analyses. This study demonstrates that the inversion is very sensitive to the choice of the forward model solution scheme. In particular, standard finite difference methods provide the smoothest objective function surface; however, this is obtained at the cost of numerical artifacts that can lead to erroneous warping of the objective function surface. Total variation diminishing (TVD) schemes limit these impacts at the cost of more computation time, while the hybrid method of characteristics (HMOC) approach with increased particle numbers and/or reduced time step gives both smoothed and accurate objective function surface. Use of the most accurate methods (TVD and HMOC) did lead to successful inversion of the two parameters; however, with distinct results for Pareto analyses. These results illuminate the sensitivity of the inversion to a number of aspects of the forward solution of the density-driven flow problem and reveal that parameter values may result that are erroneous but that counteract numerical errors in the solution.
INVERSE EIGENVALUE PROBLEM OF HERMITIAN GENERALIZED ANTI-HAMILTONIAN MATRICES%HGAH矩阵的逆特征值问题
Institute of Scientific and Technical Information of China (English)
张忠志; Liu Changrong
2004-01-01
In this paper, the inverse eigenvalue problem of Hermitian generalized anti-Hamiltonian matrices and relevant optimal approximate problem are considered. The necessary and sufficient conditions of the solvability for inverse eigenvalue problem and an expression of the general solution of the problem are derived. The solution of the relevant optimal approximate problem is given.
Optimal Experimental Design for Large-Scale Bayesian Inverse Problems
Ghattas, Omar
2014-01-06
We develop a Bayesian framework for the optimal experimental design of the shock tube experiments which are being carried out at the KAUST Clean Combustion Research Center. The unknown parameters are the pre-exponential parameters and the activation energies in the reaction rate expressions. The control parameters are the initial mixture composition and the temperature. The approach is based on first building a polynomial based surrogate model for the observables relevant to the shock tube experiments. Based on these surrogates, a novel MAP based approach is used to estimate the expected information gain in the proposed experiments, and to select the best experimental set-ups yielding the optimal expected information gains. The validity of the approach is tested using synthetic data generated by sampling the PC surrogate. We finally outline a methodology for validation using actual laboratory experiments, and extending experimental design methodology to the cases where the control parameters are noisy.
Inverse heat conduction problem in a phase change memory device
Battaglia, Jean-Luc; De, Indrayush; Sousa, Véronique
2017-01-01
An invers heat conduction problem is solved considering the thermal investigation of a phase change memory device using the scanning thermal microscopy. The heat transfer model rests on system identification for the probe thermal impedance and on a finite element method for the device thermal impedance. Unknown parameters in the model are then identified using a nonlinear least square algorithm that minimizes the quadratic gap between the measured probe temperature and the simulated one.
Hybrid modeling of spatial continuity for application to numerical inverse problems
Friedel, Michael J.; Iwashita, Fabio
2013-01-01
A novel two-step modeling approach is presented to obtain optimal starting values and geostatistical constraints for numerical inverse problems otherwise characterized by spatially-limited field data. First, a type of unsupervised neural network, called the self-organizing map (SOM), is trained to recognize nonlinear relations among environmental variables (covariates) occurring at various scales. The values of these variables are then estimated at random locations across the model domain by iterative minimization of SOM topographic error vectors. Cross-validation is used to ensure unbiasedness and compute prediction uncertainty for select subsets of the data. Second, analytical functions are fit to experimental variograms derived from original plus resampled SOM estimates producing model variograms. Sequential Gaussian simulation is used to evaluate spatial uncertainty associated with the analytical functions and probable range for constraining variables. The hybrid modeling of spatial continuity is demonstrated using spatially-limited hydrologic measurements at different scales in Brazil: (1) physical soil properties (sand, silt, clay, hydraulic conductivity) in the 42 km2 Vargem de Caldas basin; (2) well yield and electrical conductivity of groundwater in the 132 km2 fractured crystalline aquifer; and (3) specific capacity, hydraulic head, and major ions in a 100,000 km2 transboundary fractured-basalt aquifer. These results illustrate the benefits of exploiting nonlinear relations among sparse and disparate data sets for modeling spatial continuity, but the actual application of these spatial data to improve numerical inverse modeling requires testing.
An Improved Genetic Algorithm for Single-Machine Inverse Scheduling Problem
Directory of Open Access Journals (Sweden)
Jianhui Mou
2014-01-01
Full Text Available The goal of the scheduling is to arrange operations on suitable machines with optimal sequence for corresponding objectives. In order to meet market requirements, scheduling systems must own enough flexibility against uncertain events. These events can change production status or processing parameters, even causing the original schedule to no longer be optimal or even to be infeasible. Traditional scheduling strategies, however, cannot cope with these cases. Therefore, a new idea of scheduling called inverse scheduling has been proposed. In this paper, the inverse scheduling with weighted completion time (SMISP is considered in a single-machine shop environment. In this paper, an improved genetic algorithm (IGA with a local searching strategy is proposed. To improve the performance of IGA, efficient encoding scheme, fitness evaluation mechanism, feasible initialization methods, and a local search procedure have been employed in the paper. Because of the local improving method, the proposed IGA can balance its exploration ability and exploitation ability. We adopt 27 instances to verify the effectiveness of the proposed algorithm. The experimental results illustrated that the proposed algorithm can generate satisfactory solutions. This approach also has been applied to solve the scheduling problem in the real Chinese shipyard and can bring some benefits.
Reducing complexity of inverse problems using geostatistical priors
DEFF Research Database (Denmark)
Hansen, Thomas Mejer; Mosegaard, Klaus; Cordua, Knud Skou
a posterior sample, can be reduced significantly using informed priors based on geostatistical models. We discuss two approaches to include such geostatistically based prior information. One is based on a parametric description of the prior likelihood that applies to 2-point based statistical models...
Fujimoto, Ken'ichi; Tanaka, Yoshihiro; Abou Al-Ola, Omar M.; Yoshinaga, Tetsuya
2014-06-01
We propose a novel approach for solving box-constrained inverse problems in intensity-modulated radiation therapy (IMRT) treatment planning based on the idea of continuous dynamical methods and split-feasibility algorithms. Our method can compute a feasible solution without the second derivative of an objective function, which is required for gradient-based optimization algorithms. We prove theoretically that a double Kullback-Leibler divergence can be used as the Lyapunov function for the IMRT planning system.
Energy Technology Data Exchange (ETDEWEB)
Krukovsky, P.G. [Institute of Engineering Thermophysics, National Academy of Sciences of Ukraine, Kiev (Ukraine)
1997-12-31
The description of method and software FRIEND which provide a possibility of solution of inverse and inverse design problems on the basis of existing (base) CFD-software for solution of direct problems (in particular, heat-transfer and fluid-flow problems using software PHOENICS) are presented. FRIEND is an independent additional module that widens the operational capacities of the base software unified with this module. This unifying does not require any change or addition to the base software. Interfacing of FRIEND and the base software takes place through input and output files of the base software. A brief description of the computational technique applied for the inverse problem solution, same detailed information on the interfacing of FRIEND and CFD-software and solution results for testing inverse and inverse design problems, obtained using the tandem CFD-software PHOENICS and FRIEND, are presented. (author) 9 refs.
Inverse problem theory methods for data fitting and model parameter estimation
Tarantola, A
2002-01-01
Inverse Problem Theory is written for physicists, geophysicists and all scientists facing the problem of quantitative interpretation of experimental data. Although it contains a lot of mathematics, it is not intended as a mathematical book, but rather tries to explain how a method of acquisition of information can be applied to the actual world.The book provides a comprehensive, up-to-date description of the methods to be used for fitting experimental data, or to estimate model parameters, and to unify these methods into the Inverse Problem Theory. The first part of the book deals wi
An inverse problem of determining a nonlinear term in an ordinary differential equation
Kamimura, Yutaka
1998-01-01
An inverse problem for a nonlinear ordinary differential equation is discussed. We prove an existence theorem of a nonlinear term with which a boundary value problem admits a solution. This is an improvement of earlier work by A. Lorenzi. We also prove a uniqueness theorem of the nonlinear term.
Inverse Problems for a Parabolic Integrodifferential Equation in a Convolutional Weak Form
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Kairi Kasemets
2013-01-01
Full Text Available We deduce formulas for the Fréchet derivatives of cost functionals of several inverse problems for a parabolic integrodifferential equation in a weak formulation. The method consists in the application of an integrated convolutional form of the weak problem and all computations are implemented in regular Sobolev spaces.
On the inverse problem of blade design for centrifugal pumps and fans
Kruyt, Nicolaas P.; Westra, R.W.
2014-01-01
The inverse problem of blade design for centrifugal pumps and fans has been studied. The solution to this problem provides the geometry of rotor blades that realize specified performance characteristics, together with the corresponding flow field. Here a three-dimensional solution method is
A smoothing Newton method for a type of inverse semi-definite quadratic programming problem
Xiao, Xiantao; Zhang, Liwei; Zhang, Jianzhong
2009-01-01
We consider an inverse problem arising from the semi-definite quadratic programming (SDQP) problem. We represent this problem as a cone-constrained minimization problem and its dual (denoted ISDQD) is a semismoothly differentiable (SC1) convex programming problem with fewer variables than the original one. The Karush-Kuhn-Tucker conditions of the dual problem (ISDQD) can be formulated as a system of semismooth equations which involves the projection onto the cone of positive semi-definite matrices. A smoothing Newton method is given for getting a Karush-Kuhn-Tucker point of ISDQD. The proposed method needs to compute the directional derivative of the smoothing projector at the corresponding point and to solve one linear system per iteration. The quadratic convergence of the smoothing Newton method is proved under a suitable condition. Numerical experiments are reported to show that the smoothing Newton method is very effective for solving this type of inverse quadratic programming problems.
The geometry of discombinations and its applications to semi-inverse problems in anelasticity.
Yavari, Arash; Goriely, Alain
2014-09-08
The geometrical formulation of continuum mechanics provides us with a powerful approach to understand and solve problems in anelasticity where an elastic deformation is combined with a non-elastic component arising from defects, thermal stresses, growth effects or other effects leading to residual stresses. The central idea is to assume that the material manifold, prescribing the reference configuration for a body, has an intrinsic, non-Euclidean, geometrical structure. Residual stresses then naturally arise when this configuration is mapped into Euclidean space. Here, we consider the problem of discombinations (a new term that we introduce in this paper), that is, a combined distribution of fields of dislocations, disclinations and point defects. Given a discombination, we compute the geometrical characteristics of the material manifold (curvature, torsion, non-metricity), its Cartan's moving frames and structural equations. This identification provides a powerful algorithm to solve semi-inverse problems with non-elastic components. As an example, we calculate the residual stress field of a cylindrically symmetric distribution of discombinations in an infinite circular cylindrical bar made of an incompressible hyperelastic isotropic elastic solid.
Inverse problems: Fuzzy representation of uncertainty generates a regularization
Kreinovich, V.; Chang, Ching-Chuang; Reznik, L.; Solopchenko, G. N.
1992-01-01
In many applied problems (geophysics, medicine, and astronomy) we cannot directly measure the values x(t) of the desired physical quantity x in different moments of time, so we measure some related quantity y(t), and then we try to reconstruct the desired values x(t). This problem is often ill-posed in the sense that two essentially different functions x(t) are consistent with the same measurement results. So, in order to get a reasonable reconstruction, we must have some additional prior information about the desired function x(t). Methods that use this information to choose x(t) from the set of all possible solutions are called regularization methods. In some cases, we know the statistical characteristics both of x(t) and of the measurement errors, so we can apply statistical filtering methods (well-developed since the invention of a Wiener filter). In some situations, we know the properties of the desired process, e.g., we know that the derivative of x(t) is limited by some number delta, etc. In this case, we can apply standard regularization techniques (e.g., Tikhonov's regularization). In many cases, however, we have only uncertain knowledge about the values of x(t), about the rate with which the values of x(t) can change, and about the measurement errors. In these cases, usually one of the existing regularization methods is applied. There exist several heuristics that choose such a method. The problem with these heuristics is that they often lead to choosing different methods, and these methods lead to different functions x(t). Therefore, the results x(t) of applying these heuristic methods are often unreliable. We show that if we use fuzzy logic to describe this uncertainty, then we automatically arrive at a unique regularization method, whose parameters are uniquely determined by the experts knowledge. Although we start with the fuzzy description, but the resulting regularization turns out to be quite crisp.
Fast and accurate analytical model to solve inverse problem in SHM using Lamb wave propagation
Poddar, Banibrata; Giurgiutiu, Victor
2016-04-01
Lamb wave propagation is at the center of attention of researchers for structural health monitoring of thin walled structures. This is due to the fact that Lamb wave modes are natural modes of wave propagation in these structures with long travel distances and without much attenuation. This brings the prospect of monitoring large structure with few sensors/actuators. However the problem of damage detection and identification is an "inverse problem" where we do not have the luxury to know the exact mathematical model of the system. On top of that the problem is more challenging due to the confounding factors of statistical variation of the material and geometric properties. Typically this problem may also be ill posed. Due to all these complexities the direct solution of the problem of damage detection and identification in SHM is impossible. Therefore an indirect method using the solution of the "forward problem" is popular for solving the "inverse problem". This requires a fast forward problem solver. Due to the complexities involved with the forward problem of scattering of Lamb waves from damages researchers rely primarily on numerical techniques such as FEM, BEM, etc. But these methods are slow and practically impossible to be used in structural health monitoring. We have developed a fast and accurate analytical forward problem solver for this purpose. This solver, CMEP (complex modes expansion and vector projection), can simulate scattering of Lamb waves from all types of damages in thin walled structures fast and accurately to assist the inverse problem solver.
Institute of Scientific and Technical Information of China (English)
俞德孚; 陈庆东; 李文君
2003-01-01
Based on the theory and the practical experiences of linearity design of feasible design area and inverse solution of non-linear outer characteristic of suspension shock absorber, in accordance with non-linearity outer characteristic formed by open-up damping coefficient, full-open damping coefficient and smoothness to safety ratio of suspension shock absorber, a method and a research conclusion of the feasible design and inverse solution for the basic problems of designing and inverse solution of non-linear outer characteristic of suspension damping components are provided.
Material parameter identification and inverse problems in soft tissue biomechanics
Evans, Sam
2017-01-01
The articles in this book review hybrid experimental-computational methods applied to soft tissues which have been developed by worldwide specialists in the field. People developing computational models of soft tissues and organs will find solutions for calibrating the material parameters of their models; people performing tests on soft tissues will learn what to extract from the data and how to use these data for their models and people worried about the complexity of the biomechanical behavior of soft tissues will find relevant approaches to address this complexity.
Energy Technology Data Exchange (ETDEWEB)
Bakhos, Tania, E-mail: taniab@stanford.edu [Institute for Computational and Mathematical Engineering, Stanford University (United States); Saibaba, Arvind K. [Department of Electrical and Computer Engineering, Tufts University (United States); Kitanidis, Peter K. [Institute for Computational and Mathematical Engineering, Stanford University (United States); Department of Civil and Environmental Engineering, Stanford University (United States)
2015-10-15
We consider the problem of estimating parameters in large-scale weakly nonlinear inverse problems for which the underlying governing equations is a linear, time-dependent, parabolic partial differential equation. A major challenge in solving these inverse problems using Newton-type methods is the computational cost associated with solving the forward problem and with repeated construction of the Jacobian, which represents the sensitivity of the measurements to the unknown parameters. Forming the Jacobian can be prohibitively expensive because it requires repeated solutions of the forward and adjoint time-dependent parabolic partial differential equations corresponding to multiple sources and receivers. We propose an efficient method based on a Laplace transform-based exponential time integrator combined with a flexible Krylov subspace approach to solve the resulting shifted systems of equations efficiently. Our proposed solver speeds up the computation of the forward and adjoint problems, thus yielding significant speedup in total inversion time. We consider an application from Transient Hydraulic Tomography (THT), which is an imaging technique to estimate hydraulic parameters related to the subsurface from pressure measurements obtained by a series of pumping tests. The algorithms discussed are applied to a synthetic example taken from THT to demonstrate the resulting computational gains of this proposed method.
Shimelevich, M. I.; Obornev, E. A.; Obornev, I. E.; Rodionov, E. A.
2017-07-01
The iterative approximation neural network method for solving conditionally well-posed nonlinear inverse problems of geophysics is presented. The method is based on the neural network approximation of the inverse operator. The inverse problem is solved in the class of grid (block) models of the medium on a regularized parameterization grid. The construction principle of this grid relies on using the calculated values of the continuity modulus of the inverse operator and its modifications determining the degree of ambiguity of the solutions. The method provides approximate solutions of inverse problems with the maximal degree of detail given the specified degree of ambiguity with the total number of the sought parameters n × 103 of the medium. The a priori and a posteriori estimates of the degree of ambiguity of the approximated solutions are calculated. The work of the method is illustrated by the example of the three-dimensional (3D) inversion of the synthesized 2D areal geoelectrical (audio magnetotelluric sounding, AMTS) data corresponding to the schematic model of a kimberlite pipe.
Reification of galaxies: cognitive astrophysics and the multiwavelength inverse problem
Madore, Barry F.
2012-08-01
Lessons learned in the history and philosophy of science have generally had little immediate impact on how we as individual astronomers conduct our research. And yet we do share many common views on how we undertake basic research, and how we translate observations and theory into communicable knowledge. In this introductory talk I will illustrate how we as extragalactic astronomers have already violated some of the basic tenets of what constitutes ``science'' as seen from a philosophical point of view, and I will predict what the future of astronomy as a science may soon look like. Simple examples of how we are already ``cognitively closed'' to many immediate and tangible aspects of the Universe will be given and some solutions to this dilemma will be proposed. We may be at a point in time where more data is not necessarily the best solution to our problems. Discovering that familiar concepts and even certain objects may not exist in the traditional sense of the word could provide a motivation for broadening our way of conceptualizing the extragalactic Universe, more as a continuum of processes and phase transitions rather than an assembly of discrete objects. Once again the Universe may be ``forcing us to think''.
A Bayesian setting for an inverse problem in heat transfer
Ruggeri, Fabrizio
2014-01-06
In this work a Bayesian setting is developed to infer the thermal conductivity, an unknown parameter that appears into heat equation. Temperature data are available on the basis of cooling experiments. The realistic assumption that the boundary data are noisy is introduced, for a given prescribed initial condition. We show how to derive the global likelihood function for the forward boundary-initial condition problem, given the values of the temperature field plus Gaussian noise. We assume that the thermal conductivity parameter can be modelled a priori through a lognormal distributed random variable or by means of a space-dependent stationary lognormal random field. In both cases, given Gaussian priors for the time-dependent Dirichlet boundary values, we marginalize out analytically the joint posterior distribution of and the random boundary conditions, TL and TR, using the linearity of the heat equation. Synthetic data are used to carry out the inference. We exploit the concentration of the posterior distribution of , using the Laplace approximation and therefore avoiding costly MCMC computations.
Directory of Open Access Journals (Sweden)
Wolfgang Wagner
2009-11-01
Full Text Available Automated, image based methods for the retrieval of vegetation biophysical and biochemical variables are often hampered by the lack of a priori knowledge about land cover and phenology, which makes the retrieval a highly underdetermined problem. This study addresses this problem by presenting a novel approach, called CRASh, for the concurrent retrieval of leaf area index, leaf chlorophyll content, leaf water content and leaf dry matter content from high resolution solar reflective earth observation data. CRASh, which is based on the inversion of the combined PROSPECT+SAILh radiative transfer model (RTM, explores the benefits of combining semi-empirical and physically based approaches. The approach exploits novel ways to address the underdetermined problem in the context of an automated retrieval from mono-temporal high resolution data. To regularize the inverse problem in the variable domain, RTM inversion is coupled with an automated land cover classification. Model inversion is based on a two step lookup table (LUT approach: First, a range of possible solutions is selected from a previously calculated LUT based on the analogy between measured and simulated reflectance. The final solution is determined from this subset by minimizing the difference between the variables used to simulate the spectra contained in the reduced LUT and a first guess of the solution. This first guess of the variables is derived from predictive semi-empirical relationships between classical vegetation indices and the single variables. Additional spectral regularization is obtained by the use of hyperspectral data. Results show that estimates obtained with CRASh are significantly more accurate than those obtained with a tested conventional RTM inversion and semi-empirical approach. Accuracies obtained in this study are comparable to the results obtained by various authors for better constrained inversions that assume more a priori information. The completely automated
SOLVING A CLASS OF INVERSE QP PROBLEMS BY A SMOOTHING NEWTON METHOD
Institute of Scientific and Technical Information of China (English)
Xiantao Xiao; Liwei Zhang
2009-01-01
We consider an inverse quadratic programming (IQP) problem in which the parameters in the objective function of a given quadratic programming (QP) problem are adjusted as little as possible so that a known feasible solution becomes the optimal one. This problem can be formulated as a minimization problem with a positive semidefinite cone constraint and its dual (denoted IQD(A, b)) is a semismoothly differentiable (SC~1) convex program-ming problem with fewer variables than the original one. In this paper a smoothing New-ton method is used for getting a Karush-Kuhn-Tucker point of IQD(A, b). The proposed method needs to solve only one linear system per iteration and achieves quadratic conver-gence. Numerical experiments are reported to show that the smoothing Newton method is effective for solving this class of inverse quadratic programming problems.
Advanced model of eddy-current NDE inverse problem with sparse grid algorithm
Zhou, Liming; Sabbagh, Harold A.; Sabbagh, Elias H.; Murphy, R. Kim; Bernacchi, William
2017-02-01
In model-based inverse problem, some unknown parameters need to be estimated. These parameters are used not only to characterize the physical properties of cracks, but also to describe the position of the probes (such as lift off and angles) in the calibration. After considering the effect of the position of the probes in the inverse problem, the accuracy of the inverse result will be improved. With increasing the number of the parameters in the inverse problems, the burden of calculations will increase exponentially in the traditional full grid method. The sparse grid algorithm, which was introduced by Sergey A. Smolyak, was used in our work. With this algorithm, we obtain a powerful interpolation method that requires significantly fewer support nodes than conventional interpolation on a full grid. In this work, we combined sparse grid toolbox TASMANIAN, which is produced by Oak Ridge National Laboratory, and professional eddy-current NDE software, VIC-3D R◯, to solve a specific inverse problem. An advanced model based on our previous one is used to estimate length and depth of the crack, lift off and two angles of the position of probes. Considering the calibration process, pseudorandom noise is considered in the model and statistical behavior is discussed.
Inverse problems with non-trivial priors: efficient solution through sequential Gibbs sampling
DEFF Research Database (Denmark)
Hansen, Thomas Mejer; Cordua, Knud Skou; Mosegaard, Klaus
2012-01-01
Markov chain Monte Carlo methods such as the Gibbs sampler and the Metropolis algorithm can be used to sample solutions to non-linear inverse problems. In principle, these methods allow incorporation of prior information of arbitrary complexity. If an analytical closed form description of the pri...... also reduce the computation time for the inversion dramatically. The method works for any statistical model for which sequential simulation can be used to generate realizations. This applies to most algorithms developed in the geostatistical community.......Markov chain Monte Carlo methods such as the Gibbs sampler and the Metropolis algorithm can be used to sample solutions to non-linear inverse problems. In principle, these methods allow incorporation of prior information of arbitrary complexity. If an analytical closed form description of the prior...... for applying the sequential Gibbs sampler and illustrate how it works. Through two case studies, we demonstrate the application of the method to a linear image restoration problem and to a non-linear cross-borehole inversion problem. We demonstrate how prior information can reduce the complexity of an inverse...
Anthropomorphic Coding of Speech and Audio: A Model Inversion Approach
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W. Bastiaan Kleijn
2005-06-01
Full Text Available Auditory modeling is a well-established methodology that provides insight into human perception and that facilitates the extraction of signal features that are most relevant to the listener. The aim of this paper is to provide a tutorial on perceptual speech and audio coding using an invertible auditory model. In this approach, the audio signal is converted into an auditory representation using an invertible auditory model. The auditory representation is quantized and coded. Upon decoding, it is then transformed back into the acoustic domain. This transformation converts a complex distortion criterion into a simple one, thus facilitating quantization with low complexity. We briefly review past work on auditory models and describe in more detail the components of our invertible model and its inversion procedure, that is, the method to reconstruct the signal from the output of the auditory model. We summarize attempts to use the auditory representation for low-bit-rate coding. Our approach also allows the exploitation of the inherent redundancy of the human auditory system for the purpose of multiple description (joint source-channel coding.
Chatterjee, Shre Kumar; Das, Saptarshi; Manzella, Veronica; Vitaletti, Andrea; Masi, Elisa; Santopolo, Luisa; Mancuso, Stefano; Maharatna, Koushik
2014-01-01
In this paper, system identification approach has been adopted to develop a novel dynamical model for describing the relationship between light as an environmental stimulus and the electrical response as the measured output for a bay leaf (Laurus nobilis) plant. More specifically, the target is to predict the characteristics of the input light stimulus (in terms of on-off timing, duration and intensity) from the measured electrical response - leading to an inverse problem. We explored two major classes of system estimators to develop dynamical models - linear and nonlinear - and their several variants for establishing a forward and also an inverse relationship between the light stimulus and plant electrical response. The best class of models are given by the Nonlinear Hammerstein-Wiener (NLHW) estimator showing good data fitting results over other linear and nonlinear estimators in a statistical sense. Consequently, a few set of models using different functional variants of NLHW has been developed and their a...
Stabilization of the Ball on the Beam System by Means of the Inverse Lyapunov Approach
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Carlos Aguilar-Ibañez
2012-01-01
Full Text Available A novel inverse Lyapunov approach in conjunction with the energy shaping technique is applied to derive a stabilizing controller for the ball on the beam system. The proposed strategy consists of shaping a candidate Lyapunov function as if it were an inverse stability problem. To this purpose, we fix a suitable dissipation function of the unknown energy function, with the property that the selected dissipation divides the corresponding time derivative of the candidate Lyapunov function. Afterwards, the stabilizing controller is directly obtained from the already shaped Lyapunov function. The stability analysis of the closed-loop system is carried out by using the invariance theorem of LaSalle. Simulation results to test the effectiveness of the obtained controller are presented.
Effects of geometric head model perturbations on the EEG forward and inverse problems.
von Ellenrieder, Nicolás; Muravchik, Carlos H; Nehorai, Arye
2006-03-01
We study the effect of geometric head model perturbations on the electroencephalography (EEG) forward and inverse problems. Small magnitude perturbations of the shape of the head could represent uncertainties in the head model due to errors on images or techniques used to construct the model. They could also represent small scale details of the shape of the surfaces not described in a deterministic model, such as the sulci and fissures of the cortical layer. We perform a first-order perturbation analysis, using a meshless method for computing the sensitivity of the solution of the forward problem to the geometry of the head model. The effect on the forward problem solution is treated as noise in the EEG measurements and the Cramér-Rao bound is computed to quantify the effect on the inverse problem performance. Our results show that, for a dipolar source, the effect of the perturbations on the inverse problem performance is under the level of the uncertainties due to the spontaneous brain activity. Thus, the results suggest that an extremely detailed model of the head may be unnecessary when solving the EEG inverse problem.
Maksimov, M. A.; Velímský, J.
2017-07-01
The deterministic approach to the inverse problem of the time-domain electromagnetic induction in a spherical Earth requires the calculation of the first derivative of a misfit function in every step of the minimization process. In addition, an a-posteriori error analysis can benefit from the knowledge of the Hessian, the matrix of the second derivatives. We present the derivation of the formulas for the fast calculation of the misfit gradient, the Hessian, and the Hessian-vector product, based on the solution of an adjoint problem. We validate our results on a synthetic model against a slow finite-difference scheme.
Institute of Scientific and Technical Information of China (English)
HUANG; Sixun; HAN; Wei; WU; Rongsheng
2004-01-01
In the present work, the data assimilation problem in meteorology and physical oceanography is re-examined using the variational optimal control approaches in combination with regularization techniques in inverse problem. Here the estimations of the initial condition,boundary condition and model parameters are performed simultaneously in the framework of variational data assimilation. To overcome the difficulty of ill-posedness, especially for the model parameters distributed in space and time, an additional term is added into the cost functional as a stabilized functional. Numerical experiments show that even with noisy observations the initial conditions and model parameters are recovered to an acceptable degree of accuracy.
Wang, Qian; Li, Xingwen; Song, Haoyong; Rong, Mingzhe
2010-04-01
Non-contact magnetic measurement method is an effective way to study the air arc behavior experimentally One of the crucial techniques is to solve an inverse problem for the electromagnetic field. This study is devoted to investigating different algorithms for this kind of inverse problem preliminarily, including the preconditioned conjugate gradient method, penalty function method and genetic algorithm. The feasibility of each algorithm is analyzed. It is shown that the preconditioned conjugate gradient method is valid only for few arc segments, the estimation accuracy of the penalty function method is dependent on the initial conditions, and the convergence of genetic algorithm should be studied further for more segments in an arc current.
Methane combustion kinetic rate constants determination: an ill-posed inverse problem analysis
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Bárbara D. L. Ferreira
2013-01-01
Full Text Available Methane combustion was studied by the Westbrook and Dryer model. This well-established simplified mechanism is very useful in combustion science, for computational effort can be notably reduced. In the inversion procedure to be studied, rate constants are obtained from [CO] concentration data. However, when inherent experimental errors in chemical concentrations are considered, an ill-conditioned inverse problem must be solved for which appropriate mathematical algorithms are needed. A recurrent neural network was chosen due to its numerical stability and robustness. The proposed methodology was compared against Simplex and Levenberg-Marquardt, the most used methods for optimization problems.
Solution of inverse heat conduction problem using the Tikhonov regularization method
Duda, Piotr
2017-02-01
It is hard to solve ill-posed problems, as calculated temperatures are very sensitive to errors made while calculating "measured" temperatures or performing real-time measurements. The errors can create temperature oscillation, which can be the cause of an unstable solution. In order to overcome such difficulties, a variety of techniques have been proposed in literature, including regularization, future time steps and smoothing digital filters. In this paper, the Tikhonov regularization is applied to stabilize the solution of the inverse heat conduction problem. The impact on the inverse solution stability and accuracy is demonstrated.
Entropy description of measured information in mathematical and physical inverse problems
Institute of Scientific and Technical Information of China (English)
2008-01-01
There are two types of inverse problems: Optimization designation and parameter identification. Before the parameter identification of mathematical and physical inverse problems, it is necessary to determine the number and position of measurement points in analysis and evaluation of a large number of measured data. In this paper, a mathematical methodology is proposed to describe the influence of the number and position of measurement points on the reconstruction precision. Information entropy and Bayesian theory are used in the description. Finally, a numerical experiment shows that the methodology is effective.
The Bayesian Formulation and Well-Posedness of Fractional Elliptic Inverse Problems
Trillos, Nicolas Garcia
2016-01-01
We study the inverse problem of recovering the order and the diffusion coefficient of an elliptic fractional partial differential equation from a finite number of noisy observations of the solution. We work in a Bayesian framework and show conditions under which the posterior distribution is given by a change of measure from the prior. Moreover, we show well-posedness of the inverse problem, in the sense that small perturbations of the observed solution lead to small Hellinger perturbations of the associated posterior measures. We thus provide a mathematical foundation to the Bayesian learning of the order ---and other inputs--- of fractional models.
Energy Technology Data Exchange (ETDEWEB)
Balci, Murat [Dept. of Mechanical Engineering, Bayburt University, Bayburt (Turkmenistan); Gundogdu, Omer [Dept. of Mechanical Engineering, Ataturk University, Erzurum (Turkmenistan)
2017-01-15
In this study, estimation of some physical properties of a laminated composite plate was conducted via the inverse vibration problem. Laminated composite plate was modelled and simulated to obtain vibration responses for different length-to-thickness ratio in ANSYS. Furthermore, a numerical finite element model was developed for the laminated composite utilizing the Kirchhoff plate theory and programmed in MATLAB for simulations. Optimizing the difference between these two vibration responses, inverse vibration problem was solved to obtain some of the physical properties of the laminated composite using genetic algorithms. The estimated parameters are compared with the theoretical results, and a very good correspondence was observed.
HT2DINV: A 2D forward and inverse code for steady-state and transient hydraulic tomography problems
Soueid Ahmed, A.; Jardani, A.; Revil, A.; Dupont, J. P.
2015-12-01
Hydraulic tomography is a technique used to characterize the spatial heterogeneities of storativity and transmissivity fields. The responses of an aquifer to a source of hydraulic stimulations are used to recover the features of the estimated fields using inverse techniques. We developed a 2D free source Matlab package for performing hydraulic tomography analysis in steady state and transient regimes. The package uses the finite elements method to solve the ground water flow equation for simple or complex geometries accounting for the anisotropy of the material properties. The inverse problem is based on implementing the geostatistical quasi-linear approach of Kitanidis combined with the adjoint-state method to compute the required sensitivity matrices. For undetermined inverse problems, the adjoint-state method provides a faster and more accurate approach for the evaluation of sensitivity matrices compared with the finite differences method. Our methodology is organized in a way that permits the end-user to activate parallel computing in order to reduce the computational burden. Three case studies are investigated demonstrating the robustness and efficiency of our approach for inverting hydraulic parameters.
Ebtehaj, Mohammad
approach, while taking into account the underlying sparsity in an appropriately chosen transform domain. This framework seeks solutions beyond the paradigm of the classic least squares by imposing a proper regularization. The results suggest that sparsity-promoting regularization can reduce uncertainty of estimation in hydro-meteorological inverse problems of downscaling, data fusion, and data assimilation. In continuation of the proposed methodologies, we also introduce a new data driven framework for multisensor spaceborne rainfall retrieval problem and present some preliminary and promising results.
Ramezanpour, A.
2016-06-01
We study the inverse problem of constructing an appropriate Hamiltonian from a physically reasonable set of orthogonal wave functions for a quantum spin system. Usually, we are given a local Hamiltonian and our goal is to characterize the relevant wave functions and energies (the spectrum) of the system. Here, we take the opposite approach; starting from a reasonable collection of orthogonal wave functions, we try to characterize the associated parent Hamiltonians, to see how the wave functions and the energy values affect the structure of the parent Hamiltonian. Specifically, we obtain (quasi) local Hamiltonians by a complete set of (multilayer) product states and a local mapping of the energy values to the wave functions. On the other hand, a complete set of tree wave functions (having a tree structure) results to nonlocal Hamiltonians and operators which flip simultaneously all the spins in a single branch of the tree graph. We observe that even for a given set of basis states, the energy spectrum can significantly change the nature of interactions in the Hamiltonian. These effects can be exploited in a quantum engineering problem optimizing an objective functional of the Hamiltonian.
Bürgel, Florian; Kazimierski, Kamil S.; Lechleiter, Armin
2017-06-01
We present a fast computational framework for the inverse medium problem in scattering, i.e. we look at discretization, reconstruction and numerical performance. The Helmholtz equation in two and three dimensions is used as a physical model of scattering including point sources and plane waves as incident fields as well as near and far field measurements. For the reconstruction of the medium, we set up a rapid variational regularization scheme and indicate favorable choices of the various parameters. The underlying paradigm is, roughly speaking, to minimize the discrepancy between the reconstruction and measured data while, at the same time, taking into account various structural a-priori information via suitable penalty terms. In particular, the involved penalty terms are designed to promote information expected in real-world environments. To this end, a combination of sparsity promoting terms, total variation, and physical bounds of the inhomogeneous medium, e.g. positivity constraints, is employed in the regularization penalty. A primal-dual algorithm is used to solve the minimization problem related to the variational regularization. The computational feasibility, performance and efficiency of the proposed approach is demonstrated for synthetic as well as experimentally measured data.
Karaoulis, M.; Revil, A.; Werkema, D.D.; Minsley, B.J.; Woodruff, W.F.; Kemna, A.
2011-01-01
Induced polarization (more precisely the magnitude and phase of impedance of the subsurface) is measured using a network of electrodes located at the ground surface or in boreholes. This method yields important information related to the distribution of permeability and contaminants in the shallow subsurface. We propose a new time-lapse 3-D modelling and inversion algorithm to image the evolution of complex conductivity over time. We discretize the subsurface using hexahedron cells. Each cell is assigned a complex resistivity or conductivity value. Using the finite-element approach, we model the in-phase and out-of-phase (quadrature) electrical potentials on the 3-D grid, which are then transformed into apparent complex resistivity. Inhomogeneous Dirichlet boundary conditions are used at the boundary of the domain. The calculation of the Jacobian matrix is based on the principles of reciprocity. The goal of time-lapse inversion is to determine the change in the complex resistivity of each cell of the spatial grid as a function of time. Each model along the time axis is called a 'reference space model'. This approach can be simplified into an inverse problem looking for the optimum of several reference space models using the approximation that the material properties vary linearly in time between two subsequent reference models. Regularizations in both space domain and time domain reduce inversion artefacts and improve the stability of the inversion problem. In addition, the use of the time-lapse equations allows the simultaneous inversion of data obtained at different times in just one inversion step (4-D inversion). The advantages of this new inversion algorithm are demonstrated on synthetic time-lapse data resulting from the simulation of a salt tracer test in a heterogeneous random material described by an anisotropic semi-variogram. ?? 2011 The Authors Geophysical Journal International ?? 2011 RAS.
An Augmented Lagrangian Approach for Scheduling Problems
Nishi, Tatsushi; Konishi, Masami
The paper describes an augmented Lagrangian decomposition and coordination approach for solving single machine scheduling problems to minimize the total weighted tardiness. The problem belongs to the class of NP-hard combinatorial optimization problem. We propose an augmented Lagrangian decomposition and coordination approach, which is commonly used for continuous optimization problems, for solving scheduling problems despite the fact that the problem is nonconvex and non-differentiable. The proposed method shows a good convergence to a feasible solution without heuristically constructing a feasible solution. The performance of the proposed method is compared with that of an ordinary Lagrangian relaxation.
An automatization of Barnsley's algorithm for the inverse problem of iterated function systems.
Wadströmer, Niclas
2003-01-01
We present an automatization of Barnsley's manual algorithm for the solution of the inverse problem of iterated function systems (IFSs). The problem is to retrieve the number of mappings and the parameters of an IFS from a digital binary image approximating the attractor induced by the IFS. M.F. Barnsley et al. described a way to solve manually the inverse problem by identifying the fragments of which the collage is composed, and then computing the parameters of the mappings (Barnsley et al., Proc. Nat. Acad. Sci. USA, vol.83, p.1975-7, 1986; Barnsley, "Fractals Everywhere", Academic, 1988; Barnsley and Hurd, L., "Fractal Image Compression", A.K. Peters, 1992). The automatic algorithm searches through a finite set of points in the parameter space determining a set of affine mappings. The algorithm uses the collage theorem and the Hausdorff metric. The inverse problem of IFSs is related to the image coding of binary images. If the number of mappings and the parameters of an IFS, with not too many mappings, could be obtained from a binary image, then this would give an efficient representation of the image. It is shown that the inverse problem solved by the automatic algorithm has a solution and some experiments show that the automatic algorithm is able to retrieve an IFS, including the number of mappings, from a digital binary image approximating the attractor induced by the IFS.
Solution of the nonlinear inverse scattering problem by T -matrix completion. II. Simulations
Levinson, Howard W.; Markel, Vadim A.
2016-10-01
This is Part II of the paper series on data-compatible T -matrix completion (DCTMC), which is a method for solving nonlinear inverse problems. Part I of the series [H. W. Levinson and V. A. Markel, Phys. Rev. E 94, 043317 (2016), 10.1103/PhysRevE.94.043317] contains theory and here we present simulations for inverse scattering of scalar waves. The underlying mathematical model is the scalar wave equation and the object function that is reconstructed is the medium susceptibility. The simulations are relevant to ultrasound tomographic imaging and seismic tomography. It is shown that DCTMC is a viable method for solving strongly nonlinear inverse problems with large data sets. It provides not only the overall shape of the object, but the quantitative contrast, which can correspond, for instance, to the variable speed of sound in the imaged medium.
The application of finite element method to forward and inverse seismic problems in frequency domain
Energy Technology Data Exchange (ETDEWEB)
Shaoyou, J.; Xiangheng, J.; Shizhe, X.
1987-04-01
Unstable result is obtained when the boundary problem of wave equation is solved using finite element method in time domain. However, when the boundary problem of wave equation is solved by finite element method in frequency domain, not only the unstablity can be avoided but also computation is speeded up because of using FFT. The procedure for solving the boundary problem using finite element method in frequency domain is as follows: 1. the wave equation is transformed into Helmholtz equation by making one-dimensional Fourier transform with respect to time; 2. Helmholtz equation is solved using finite element method in frequency domain; 3. the obtained result is returned to time domain by making inverse Fourier transform. Both forward and inverse seismic problems can be solved by this method.
Moment theory and some inverse problems in potential theory and heat conduction
Ang, Dang Dinh; Le, Vy Khoi; Trong, Dang Duc
2002-01-01
Moment Theory is not a new subject; however, in classical treatments, the ill-posedness of the problem is not taken into account - hence this monograph. Assuming a "true" solution to be uniquely determined by a sequence of moments (given as integrals) of which only finitely many are inaccurately given, the authors describe and analyze several regularization methods and derive stability estimates. Mathematically, the task often consists in the reconstruction of an analytic or harmonic function, as is natural from concrete applications discussed (e.g. inverse heat conduction problems, Cauchy's problem for the Laplace equation, gravimetry). The book can be used in a graduate or upper undergraduate course in Inverse Problems, or as supplementary reading for a course on Applied Partial Differential Equations.
Numerical study of a parametric parabolic equation and a related inverse boundary value problem
Mustonen, Lauri
2016-10-01
We consider a time-dependent linear diffusion equation together with a related inverse boundary value problem. The aim of the inverse problem is to determine, based on observations on the boundary, the nonhomogeneous diffusion coefficient in the interior of an object. The method in this paper relies on solving the forward problem for a whole family of diffusivities by using a spectral Galerkin method in the high-dimensional parameter domain. The evaluation of the parametric solution and its derivatives is then completely independent of spatial and temporal discretizations. In the case of a quadratic approximation for the parameter dependence and a direct solver for linear least squares problems, we show that the evaluation of the parametric solution does not increase the complexity of any linearized subproblem arising from a Gauss-Newtonian method that is used to minimize a Tikhonov functional. The feasibility of the proposed algorithm is demonstrated by diffusivity reconstructions in two and three spatial dimensions.
A compressive sensing approach to the calculation of the inverse data space
Khan, Babar Hasan
2012-01-01
Seismic processing in the Inverse Data Space (IDS) has its advantages like the task of removing the multiples simply becomes muting the zero offset and zero time data in the inverse domain. Calculation of the Inverse Data Space by sparse inversion techniques has seen mitigation of some artifacts. We reformulate the problem by taking advantage of some of the developments from the field of Compressive Sensing. The seismic data is compressed at the sensor level by recording projections of the traces. We then process this compressed data directly to estimate the inverse data space. Due to the smaller number of data set we also gain in terms of computational complexity.
Solution of a multiple-scattering inverse problem: electron diffraction from surfaces.
Saldin, D K; Seubert, A; Heinz, K
2002-03-18
We present a solution to the multiple-scattering inverse problem for low-energy electron diffraction that enables the determination of the three-dimensional atomic structure of an entire surface unit cell directly from measured data. The solution requires a knowledge of the structure of the underlying bulk crystal and is implemented by a maximum entropy algorithm.
Large scale inverse problems computational methods and applications in the earth sciences
Scheichl, Robert; Freitag, Melina A; Kindermann, Stefan
2013-01-01
This book is thesecond volume of three volume series recording the ""Radon Special Semester 2011 on Multiscale Simulation & Analysis in Energy and the Environment"" taking place in Linz, Austria, October 3-7, 2011. The volume addresses the common ground in the mathematical and computational procedures required for large-scale inverse problems and data assimilation in forefront applications.
An inverse problem for a semilinear parabolic equation arising from cardiac electrophysiology
Beretta, Elena; Cavaterra, Cecilia; Cerutti, M. Cristina; Manzoni, Andrea; Ratti, Luca
2017-10-01
In this paper we develop theoretical analysis and numerical reconstruction techniques for the solution of an inverse boundary value problem dealing with the nonlinear, time-dependent monodomain equation, which models the evolution of the electric potential in the myocardial tissue. The goal is the detection of an inhomogeneity \
Ryzhikov, I. S.; Semenkin, E. S.
2017-02-01
This study is focused on solving an inverse mathematical modelling problem for dynamical systems based on observation data and control inputs. The mathematical model is being searched in the form of a linear differential equation, which determines the system with multiple inputs and a single output, and a vector of the initial point coordinates. The described problem is complex and multimodal and for this reason the proposed evolutionary-based optimization technique, which is oriented on a dynamical system identification problem, was applied. To improve its performance an algorithm restart operator was implemented.
Open forward and inverse problems in theoretical modeling of bone tissue adaptation.
Zadpoor, Amir Abbas
2013-11-01
Theoretical modeling of bone tissue adaptation started several decades ago. Many important problems have been addressed in this area of research during the last decades. However, many important questions remain unanswered. In this paper, an overview of open problems in theoretical modeling of bone tissue adaptation is presented. First, the principal elements of bone tissue adaptation models are defined and briefly reviewed. Based on these principal elements, four categories of open problems are identified. Two of these categories primarily include forward problems, while two others include inverse problems. In every one of the identified categories, important open problems are highlighted and their importance is discussed. It is shown that most of previous studies on the theoretical modeling of bone tissue adaptation have been focused on the problems of the first category and not much is done in three other categories. The paper tries to highlight these potentially important problems that have been so far largely overlooked and to inspire new avenues of research.
Oblique projections and standard-form transformations for discrete inverse problems
DEFF Research Database (Denmark)
Hansen, Per Christian
2013-01-01
This tutorial paper considers a specific computational tool for the numerical solution of discrete inverse problems, known as the standard-form transformation, by which we can treat general Tikhonov regularization problems efficiently. In the tradition of B. N. Datta's expositions of numerical li...... linear algebra, we use the close relationship between oblique projections, pseudoinverses, and matrix computations to derive a simple geometric motivation and algebraic formulation of the standard-form transformation....
Solving the Axisymmetric Inverse Heat Conduction Problem by a Wavelet Dual Least Squares Method
Directory of Open Access Journals (Sweden)
Fu Chu-Li
2009-01-01
Full Text Available We consider an axisymmetric inverse heat conduction problem of determining the surface temperature from a fixed location inside a cylinder. This problem is ill-posed; the solution (if it exists does not depend continuously on the data. A special project method—dual least squares method generated by the family of Shannon wavelet is applied to formulate regularized solution. Meanwhile, an order optimal error estimate between the approximate solution and exact solution is proved.
An inverse source problem of the Poisson equation with Cauchy data
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Ji-Chuan Liu
2017-05-01
Full Text Available In this article, we study an inverse source problem of the Poisson equation with Cauchy data. We want to find iterative algorithms to detect the hidden source within a body from measurements on the boundary. Our goal is to reconstruct the location, the size and the shape of the hidden source. This problem is ill-posed, regularization techniques should be employed to obtain the regularized solution. Numerical examples show that our proposed algorithms are valid and effective.
A VARIATIONAL EXPECTATION-MAXIMIZATION METHOD FOR THE INVERSE BLACK BODY RADIATION PROBLEM
Institute of Scientific and Technical Information of China (English)
Jiantao Cheng; Tie Zhou
2008-01-01
The inverse black body radiation problem, which is to reconstruct the area tempera-ture distribution from the measurement of power spectrum distribution, is a well-known ill-posed problem. In this paper, a variational expectation-maximization (EM) method is developed and its convergence is studied. Numerical experiments demonstrate that the variational EM method is more efficient and accurate than the traditional methods, in-cluding the Tikhonov regularization method, the Landweber method and the conjugate gradient method.
Cui, Tiangang; Marzouk, Youssef; Willcox, Karen
2016-06-01
Two major bottlenecks to the solution of large-scale Bayesian inverse problems are the scaling of posterior sampling algorithms to high-dimensional parameter spaces and the computational cost of forward model evaluations. Yet incomplete or noisy data, the state variation and parameter dependence of the forward model, and correlations in the prior collectively provide useful structure that can be exploited for dimension reduction in this setting-both in the parameter space of the inverse problem and in the state space of the forward model. To this end, we show how to jointly construct low-dimensional subspaces of the parameter space and the state space in order to accelerate the Bayesian solution of the inverse problem. As a byproduct of state dimension reduction, we also show how to identify low-dimensional subspaces of the data in problems with high-dimensional observations. These subspaces enable approximation of the posterior as a product of two factors: (i) a projection of the posterior onto a low-dimensional parameter subspace, wherein the original likelihood is replaced by an approximation involving a reduced model; and (ii) the marginal prior distribution on the high-dimensional complement of the parameter subspace. We present and compare several strategies for constructing these subspaces using only a limited number of forward and adjoint model simulations. The resulting posterior approximations can rapidly be characterized using standard sampling techniques, e.g., Markov chain Monte Carlo. Two numerical examples demonstrate the accuracy and efficiency of our approach: inversion of an integral equation in atmospheric remote sensing, where the data dimension is very high; and the inference of a heterogeneous transmissivity field in a groundwater system, which involves a partial differential equation forward model with high dimensional state and parameters.
FOREWORD: 2nd International Workshop on New Computational Methods for Inverse Problems (NCMIP 2012)
Blanc-Féraud, Laure; Joubert, Pierre-Yves
2012-09-01
Conference logo This volume of Journal of Physics: Conference Series is dedicated to the scientific contributions presented during the 2nd International Workshop on New Computational Methods for Inverse Problems, (NCMIP 2012). This workshop took place at Ecole Normale Supérieure de Cachan, in Cachan, France, on 15 May 2012, at the initiative of Institut Farman. The first edition of NCMIP also took place in Cachan, France, within the scope of the ValueTools Conference, in May 2011 (http://www.ncmip.org/2011/). The NCMIP Workshop focused on recent advances in the resolution of inverse problems. Indeed inverse problems appear in numerous scientific areas such as geophysics, biological and medical imaging, material and structure characterization, electrical, mechanical and civil engineering, and finance. The resolution of inverse problems consists of estimating the parameters of the observed system or structure from data collected by an instrumental sensing or imaging device. Its success firstly requires the collection of relevant observation data. It also requires accurate models describing the physical interactions between the instrumental device and the observed system, as well as the intrinsic properties of the solution itself. Finally, it requires the design of robust, accurate and efficient inversion algorithms. Advanced sensor arrays and imaging devices provide high rate and high volume data; in this context, the efficient resolution of the inverse problem requires the joint development of new models and inversion methods, taking computational and implementation aspects into account. During this one-day workshop, researchers had the opportunity to bring to light and share new techniques and results in the field of inverse problems. The topics of the workshop were: algorithms and computational aspects of inversion, Bayesian estimation, kernel methods, learning methods, convex optimization, free discontinuity problems, metamodels, proper orthogonal decomposition
FOREWORD: 3rd International Workshop on New Computational Methods for Inverse Problems (NCMIP 2013)
Blanc-Féraud, Laure; Joubert, Pierre-Yves
2013-10-01
Conference logo This volume of Journal of Physics: Conference Series is dedicated to the scientific contributions presented during the 3rd International Workshop on New Computational Methods for Inverse Problems, NCMIP 2013 (http://www.farman.ens-cachan.fr/NCMIP_2013.html). This workshop took place at Ecole Normale Supérieure de Cachan, in Cachan, France, on 22 May 2013, at the initiative of Institut Farman. The prior editions of NCMIP also took place in Cachan, France, firstly within the scope of the ValueTools Conference, in May 2011 (http://www.ncmip.org/2011/), and secondly at the initiative of Institut Farman, in May 2012 (http://www.farman.ens-cachan.fr/NCMIP_2012.html). The NCMIP Workshop focused on recent advances in the resolution of inverse problems. Indeed inverse problems appear in numerous scientific areas such as geophysics, biological and medical imaging, material and structure characterization, electrical, mechanical and civil engineering, and finances. The resolution of inverse problems consists of estimating the parameters of the observed system or structure from data collected by an instrumental sensing or imaging device. Its success firstly requires the collection of relevant observation data. It also requires accurate models describing the physical interactions between the instrumental device and the observed system, as well as the intrinsic properties of the solution itself. Finally, it requires the design of robust, accurate and efficient inversion algorithms. Advanced sensor arrays and imaging devices provide high rate and high volume data; in this context, the efficient resolution of the inverse problem requires the joint development of new models and inversion methods, taking computational and implementation aspects into account. During this one-day workshop, researchers had the opportunity to bring to light and share new techniques and results in the field of inverse problems. The topics of the workshop were: algorithms and computational
Risk evaluation of uranium mining: A geochemical inverse modelling approach
Rillard, J.; Zuddas, P.; Scislewski, A.
2011-12-01
It is well known that uranium extraction operations can increase risks linked to radiation exposure. The toxicity of uranium and associated heavy metals is the main environmental concern regarding exploitation and processing of U-ore. In areas where U mining is planned, a careful assessment of toxic and radioactive element concentrations is recommended before the start of mining activities. A background evaluation of harmful elements is important in order to prevent and/or quantify future water contamination resulting from possible migration of toxic metals coming from ore and waste water interaction. Controlled leaching experiments were carried out to investigate processes of ore and waste (leached ore) degradation, using samples from the uranium exploitation site located in Caetité-Bahia, Brazil. In experiments in which the reaction of waste with water was tested, we found that the water had low pH and high levels of sulphates and aluminium. On the other hand, in experiments in which ore was tested, the water had a chemical composition comparable to natural water found in the region of Caetité. On the basis of our experiments, we suggest that waste resulting from sulphuric acid treatment can induce acidification and salinization of surface and ground water. For this reason proper storage of waste is imperative. As a tool to evaluate the risks, a geochemical inverse modelling approach was developed to estimate the water-mineral interaction involving the presence of toxic elements. We used a method earlier described by Scislewski and Zuddas 2010 (Geochim. Cosmochim. Acta 74, 6996-7007) in which the reactive surface area of mineral dissolution can be estimated. We found that the reactive surface area of rock parent minerals is not constant during time but varies according to several orders of magnitude in only two months of interaction. We propose that parent mineral heterogeneity and particularly, neogenic phase formation may explain the observed variation of the
Analysis of forward and inverse problems in chemical dynamics and spectroscopy
Energy Technology Data Exchange (ETDEWEB)
Rabitz, H. [Princeton Univ., NJ (United States)
1993-12-01
The overall scope of this research concerns the development and application of forward and inverse analysis tools for problems in chemical dynamics and chemical kinetics. The chemical dynamics work is specifically associated with relating features in potential surfaces and resultant dynamical behavior. The analogous inverse research aims to provide stable algorithms for extracting potential surfaces from laboratory data. In the case of chemical kinetics, the focus is on the development of systematic means to reduce the complexity of chemical kinetic models. Recent progress in these directions is summarized below.
Orazov, Isabek; Makhatova, Anar
2014-08-01
In this paper, we consider one family of problems simulating the determination of target components and density of sources from given values of the initial and final states. The mathematical statement of these problems leads to the inverse problem for the diffusion equation, where it is required to find not only a solution of the problem, but also its right-hand side that depends only on a spatial variable. A specific feature of the considered problems is that the system of eigenfunctions of the multiple differentiation operator subject to boundary conditions of the initial problem does not have the basis property. We prove the unique existence of a generalized solution of the problem.
Inverse Problem of Air Filtration of Nanoparticles: Optimal Quality Factors of Fibrous Filters
Directory of Open Access Journals (Sweden)
Dahua Shou
2015-01-01
Full Text Available Application of nanofibers has become an emerging approach to enhance filtration efficiency, but questions arise about the decrease in Quality factor (QF for certain particles due to the rapidly increasing pressure drop. In this paper, we theoretically investigate the QF of dual-layer filters for filtration of monodisperse and polydisperse nanoparticles. The inverse problem of air filtration, as defined in this work, consists in determining the optimal construction of the two-layer fibrous filter with the maximum QF. In comparison to a single-layer substrate, improved QF values for dual-layer filters are found when a second layer with proper structural parameters is added. The influences of solidity, fiber diameter, filter thickness, face velocity, and particle size on the optimization of QF are studied. The maximum QF values for realistic polydisperse particles with a lognormal size distribution are also found. Furthermore, we propose a modified QF (MQF accounting for the effects of energy cost and flow velocity, which are significant in certain operations. The optimal MQF of the dual-layer filter is found to be over twice that of the first layer. This work provides a quick tool for designing and optimizing fibrous structures with better performance for the air filtration of specific nanoparticles.
From capture to simulation: connecting forward and inverse problems in fluids
Gregson, James
2014-07-27
We explore the connection between fluid capture, simulation and proximal methods, a class of algorithms commonly used for inverse problems in image processing and computer vision. Our key finding is that the proximal operator constraining fluid velocities to be divergence-free is directly equivalent to the pressure-projection methods commonly used in incompressible flow solvers. This observation lets us treat the inverse problem of fluid tracking as a constrained flow problem all while working in an efficient, modular framework. In addition it lets us tightly couple fluid simulation into flow tracking, providing a global prior that significantly increases tracking accuracy and temporal coherence as compared to previous techniques. We demonstrate how we can use these improved results for a variety of applications, such as re-simulation, detail enhancement, and domain modification. We furthermore give an outlook of the applications beyond fluid tracking that our proximal operator framework could enable by exploring the connection of deblurring and fluid guiding.
Kirsch, Andreas; Rieder, Andreas
2016-08-01
It is common knowledge—mainly based on experience—that parameter identification problems in partial differential equations are ill-posed. Yet, a mathematical sound argumentation is missing, except for some special cases. We present a general theory for inverse problems related to abstract evolution equations which explains not only their local ill-posedness but also provides the Fréchet derivative and its adjoint of the corresponding parameter-to-solution map which are needed, e.g., in Newton-like solvers. Our abstract results are applied to inverse problems related to the following first order hyperbolic systems: Maxwell’s equation (electromagnetic scattering in conducting media) and elastic wave equation (seismic imaging).
Energy Technology Data Exchange (ETDEWEB)
Barbone, Paul E; Rivas, Carlos E [College of Engineering, Boston University, Boston, MA (United States); Harari, Isaac; Albocher, Uri [Faculty of Engineering, Tel Aviv University, 69978 Ramat Aviv (Israel); Oberai, Assad A; Goenzen, Sevan [Mechanical, Aerospace, and Nuclear Engineering, Rensselaer Polytechnic Inst., Troy, NY (United States)], E-mail: barbone@bu.edu, E-mail: harari@eng.tau.ac.il, E-mail: oberaa@rpi.edu
2008-11-01
We describe a novel variational formulation of the inverse elasticity problem given interior data. The strong form of this problem is governed by equations of pure advective transport. To address this problem, we generalize the adjoint-weighted variational equation (AWE) formulation, originally developed for flow of a passive scalar. Here, we shall study the properties of the AWE formulation in the context of inverse plane stress elasticity imaging. We show that the solution of the AWE formulation is equivalent to that of the strong form when both are well posed. We prove that the Galerkin discretization of the AWE formulation leads to a stable, convergent numerical method, and prove optimal rates of convergence.
Least-Squares Solution of Inverse Problem for Hermitian Anti-reflexive Matrices and Its Appoximation
Institute of Scientific and Technical Information of China (English)
Zhen Yun PENG; Yuan Bei DENG; Jin Wang LIU
2006-01-01
In this paper, we first consider the least-squares solution of the matrix inverse problem as follows: Find a hermitian anti-reflexive matrix corresponding to a given generalized reflection matrix J such that for given matrices X, B we have minA‖AX - B‖. The existence theorems are obtained, and a general representation of such a matrix is presented. We denote the set of such matrices by SE. Then the matrix nearness problem for the matrix inverse problem is discussed. That is: Given an arbitrary A*, find a matrix A ∈ SE which is nearest to A* in Frobenius norm. We show that the nearest matrix is unique and provide an expression for this nearest matrix.
On computational experiments in some inverse problems of heat and mass transfer
Bilchenko, G. G.; Bilchenko, N. G.
2016-11-01
The results of mathematical modeling of effective heat and mass transfer on hypersonic aircraft permeable surfaces are considered. The physic-chemical processes (the dissociation and the ionization) in laminar boundary layer of compressible gas are appreciated. Some algorithms of control restoration are suggested for the interpolation and approximation statements of heat and mass transfer inverse problems. The differences between the methods applied for the problem solutions search for these statements are discussed. Both the algorithms are realized as programs. Many computational experiments were accomplished with the use of these programs. The parameters of boundary layer obtained by means of the A.A.Dorodnicyn's generalized integral relations method from solving the direct problems have been used to obtain the inverse problems solutions. Two types of blowing laws restoration for the inverse problem in interpolation statement are presented as the examples. The influence of the temperature factor on the blowing restoration is investigated. The different character of sensitivity of controllable parameters (the local heat flow and local tangent friction) respectively to step (discrete) changing of control (the blowing) and the switching point position is studied.
Menke, William
2017-02-01
We prove that the problem of inverting Rayleigh wave phase velocity functions c( k ) , where k is wavenumber, for density ρ ( z ) , rigidity μ ( z ) and Lamé parameter λ ( z ) , where z is depth, is fully non-unique, at least in the highly-idealized case where the base Earth model is an isotropic half space. The model functions completely trade off. This is one special case of a common inversion scenario in which one seeks to determine several model functions from a single data function. We explore the circumstances under which this broad class of problems is unique, starting with very simple scenarios, building up to the somewhat more complicated (and common) case where data and model functions are related by convolutions, and then finally, to scale-independent problems (which include the Rayleigh wave problem). The idealized cases that we examine analytically provide insight into the kinds of nonuniqueness that are inherent in the much more complicated problems encountered in modern geophysical imaging (though they do not necessarily provide methods for solving those problems). We also define what is meant by a Backus and Gilbert resolution kernel in this kind of inversion and show under what circumstances a unique localized average of a single model function can be constructed.
Menke, William
2017-04-01
We prove that the problem of inverting Rayleigh wave phase velocity functions c( k ), where k is wavenumber, for density ρ ( z ), rigidity μ ( z ) and Lamé parameter λ ( z ), where z is depth, is fully non-unique, at least in the highly-idealized case where the base Earth model is an isotropic half space. The model functions completely trade off. This is one special case of a common inversion scenario in which one seeks to determine several model functions from a single data function. We explore the circumstances under which this broad class of problems is unique, starting with very simple scenarios, building up to the somewhat more complicated (and common) case where data and model functions are related by convolutions, and then finally, to scale-independent problems (which include the Rayleigh wave problem). The idealized cases that we examine analytically provide insight into the kinds of nonuniqueness that are inherent in the much more complicated problems encountered in modern geophysical imaging (though they do not necessarily provide methods for solving those problems). We also define what is meant by a Backus and Gilbert resolution kernel in this kind of inversion and show under what circumstances a unique localized average of a single model function can be constructed.
Schumacher, F.; Friederich, W.; Lamara, S.
2016-02-01
We present a new conceptual approach to scattering-integral-based seismic full waveform inversion (FWI) that allows a flexible, extendable, modular and both computationally and storage-efficient numerical implementation. To achieve maximum modularity and extendability, interactions between the three fundamental steps carried out sequentially in each iteration of the inversion procedure, namely, solving the forward problem, computing waveform sensitivity kernels and deriving a model update, are kept at an absolute minimum and are implemented by dedicated interfaces. To realize storage efficiency and maximum flexibility, the spatial discretization of the inverted earth model is allowed to be completely independent of the spatial discretization employed by the forward solver. For computational efficiency reasons, the inversion is done in the frequency domain. The benefits of our approach are as follows: (1) Each of the three stages of an iteration is realized by a stand-alone software program. In this way, we avoid the monolithic, unflexible and hard-to-modify codes that have often been written for solving inverse problems. (2) The solution of the forward problem, required for kernel computation, can be obtained by any wave propagation modelling code giving users maximum flexibility in choosing the forward modelling method. Both time-domain and frequency-domain approaches can be used. (3) Forward solvers typically demand spatial discretizations that are significantly denser than actually desired for the inverted model. Exploiting this fact by pre-integrating the kernels allows a dramatic reduction of disk space and makes kernel storage feasible. No assumptions are made on the spatial discretization scheme employed by the forward solver. (4) In addition, working in the frequency domain effectively reduces the amount of data, the number of kernels to be computed and the number of equations to be solved. (5) Updating the model by solving a large equation system can be
Inverse Scattering Transform of the Coupled Sasa-Satsuma Equation by Riemann-Hilbert Approach
Wu, Jian-Ping; Geng, Xian-Guo
2017-05-01
The inverse scattering transform of a coupled Sasa-Satsuma equation is studied via Riemann-Hilbert approach. Firstly, the spectral analysis is performed for the coupled Sasa-Satsuma equation, from which a Riemann-Hilbert problem is formulated. Then the Riemann-Hilbert problem corresponding to the reflection-less case is solved. As applications, multi-soliton solutions are obtained for the coupled Sasa-Satsuma equation. Moreover, some figures are given to describe the soliton behaviors, including breather types, single-hump solitons, double-hump solitons, and two-bell solitons. Supported by the National Natural Science Foundation of China under Project Nos. 11331008 and 11171312 and the Collaborative Innovation Center for Aviation Economy Development of Henan Province
Inverse Kinematics of a Humanoid Robot with Non-Spherical Hip: A Hybrid Algorithm Approach
Directory of Open Access Journals (Sweden)
Rafael Cisneros Limón
2013-04-01
Full Text Available This paper describes an approach to solve the inverse kinematics problem of humanoid robots whose construction shows a small but non negligible offset at the hip which prevents any purely analytical solution to be developed. Knowing that a purely numerical solution is not feasible due to variable efficiency problems, the proposed one first neglects the offset presence in order to obtain an approximate “solution” by means of an analytical algorithm based on screw theory, and then uses it as the initial condition of a numerical refining procedure based on the Levenberg‐Marquardt algorithm. In this way, few iterations are needed for any specified attitude, making it possible to implement the algorithm for real‐time applications. As a way to show the algorithm’s implementation, one case of study is considered throughout the paper, represented by the SILO2 humanoid robot.
Analysis of forward and inverse problems in chemical dynamics and spectroscopy
Energy Technology Data Exchange (ETDEWEB)
Rabitz, H.
1991-01-01
This research is concerned with the development and application of advanced analysis tools for studying dynamics, kinetics, and spectroscopic phenomena from a forward and inverse perspective. In particular, the forward problem is concerned with understanding how detailed interatomic potential information maps onto a hierarchy of chemical dynamic and kinetic observables. The inverse aspects of the research are concerned with exactly the reverse of this process, whereby we desire to understand how particular measurements project back to yield information regarding the potential surface. Thus, in the latter domain, our research is concerned with the development of theoretically based tools ultimately aimed at applications to the inversion of quality laboratory data for the extraction of microscopic potential information.
Fault estimation - A standard problem approach
DEFF Research Database (Denmark)
Stoustrup, J.; Niemann, Hans Henrik
2002-01-01
This paper presents a range of optimization based approaches to fault diagnosis. A variety of fault diagnosis problems are reformulated in the so-called standard problem set-up introduced in the literature on robust control. Once the standard problem formulations are given, the fault diagnosis pr...... problems can be solved by standard optimization techniques. The proposed methods include (1) fault diagnosis (fault estimation, (FE)) for systems with model uncertainties; FE for systems with parametric faults, and FE for a class of nonlinear systems. Copyright......This paper presents a range of optimization based approaches to fault diagnosis. A variety of fault diagnosis problems are reformulated in the so-called standard problem set-up introduced in the literature on robust control. Once the standard problem formulations are given, the fault diagnosis...
Honarvar, Mohammad; Sahebjavaher, Ramin S; Rohling, Robert; Salcudean, Septimiu E
2017-08-01
In quantitative elastography, maps of the mechanical properties of soft tissue, or elastograms, are calculated from the measured displacement data by solving an inverse problem. The model assumptions have a significant effect on elastograms. Motivated by the high sensitivity of imaging results to the model assumptions for in vivo magnetic resonance elastography of the prostate, we compared elastograms obtained with four different methods. Two finite-element method (FEM)-based methods developed by our group were compared with two other commonly used methods, local frequency estimator (LFE) and curl-based direct inversion (c-DI). All the methods assume a linear isotropic elastic model, but the methods vary in their assumptions, such as local homogeneity or incompressibility, and in the specific approach used. We report results using simulations, phantom, and ex vivo and in vivo data. The simulation and phantom studies show, for regions with an inclusion, that the contrast to noise ratio (CNR) for the FEM methods is about three to five times higher than the CNR for the LFE and c-DI and the rms error is about half. The LFE method produces very smooth results (i.e., low CNR) and is fast. c-DI is faster than the FEM methods but it is only accurate in areas where elasticity variations are small. The artifacts resulting from the homogeneity assumption in c-DI is detrimental in regions with large variations. The ex vivo and in vivo results also show similar trends as the simulation and phantom studies. The c-FEM method is more sensitive to noise compared with the mixed-FEM due to higher orders derivatives. This is especially evident at lower frequencies, where the wave curvature is smaller and it is more prone to such error, causing a discrepancy in the absolute values between the mixed-FEM and c-FEM in our in vivo results. In general, the proposed FEMs use fewer simplifying assumptions and outperform the other methods but they are computationally more expensive.
Well-posedness of inverse problems for systems with time dependent parameters
DEFF Research Database (Denmark)
Banks, H. T.; Pedersen, Michael
2009-01-01
H is identified with its dual and denotes the associated duality product. We show under reasonable assumptions on the time-dependent sesquilinear forms a (t;.,.) : V x V -> C and d (t;.,.) : V-D x V-D -> C that this model allows a unique solution and that the solution depends continuously...... on the data of the problem. We also consider well-posedness as well as finite element type approximations in associated inverse problems. The problem above is a weak formulation that includes models in abstract differential operator form that include plate, beam and shell equations with several important...
On an Inverse Eigenvalue Problem for a Semilinear Sturm-Liouville Operator
Zhidkov, P E
2005-01-01
The following problem is considered: $-u''+f(u)=\\lambda u, x\\in (0,1), u=u(x), u(0)=1, u'(0)=u(1)=0,$ where $\\lambda $ is a spectral parameter. We study the inverse problem: for a given part of the spectrum $\\lambda _n\\to +\\infty $ we seek odd $f$. We obtain a description of the whole class of solutions of this problem. In addition, we show that there exists at most one function $f$ such that an auxiliary function is nondecreasing.
Regularization and Bayesian methods for inverse problems in signal and image processing
Giovannelli , Jean-François
2015-01-01
The focus of this book is on "ill-posed inverse problems". These problems cannot be solved only on the basis of observed data. The building of solutions involves the recognition of other pieces of a priori information. These solutions are then specific to the pieces of information taken into account. Clarifying and taking these pieces of information into account is necessary for grasping the domain of validity and the field of application for the solutions built. For too long, the interest in these problems has remained very limited in the signal-image community. However, the community has si
Towards clinical prostate ultrasound elastography using full inversion approach.
Mousavi, Seyed Reza; Sadeghi-Naini, Ali; Czarnota, Gregory J; Samani, Abbas
2014-03-01
Various types of cancers including prostate cancer are known to be associated with biological changes that lead to tissue stiffening. Digital rectal examination is based on manually palpating the prostate tissue via the rectum. This test lacks sufficient accuracy required for early diagnosis which is necessary for effective management of prostate cancer. To develop an effective prostate cancer diagnostic technique, the authors propose an imaging technique that maps the distribution of the relative prostate tissue's elasticity modulus. Unlike digital rectal examination, this technique is quantitative, capable of accurately detecting small prostate lesions that cannot be sensed by manual palpation, and its accuracy is independent of the physician's experience. The proposed technique is a quasistatic elastography technique which uses ultrasound imaging to acquire tissue displacements resulting from transrectal ultrasound mechanical stimulation. The system involves a standard ultrasound imaging unit with accessibility to its radiofrequency data. The displacements are used as data for the tissue elasticity reconstruction. This reconstruction does not require tissue segmentation and is based on physics governing tissue mechanics. It is formulated using an inverse problem framework where elastic tissue deformation equations are fully inverted using an iterative scheme where each iteration involves stress calculation followed by elastic modulus updating until convergence is achieved.In silico and tissue mimicking phantom studies were conducted to validate the proposed technique, followed by a clinical pilot study involving two prostate cancer patients with whole-mount histopathology analysis on prostatectomy specimens to confirm a cancer location. The phantom studies demonstrated robustness and reasonably high accuracy of the proposed method. Obtained Young's modulus ratios indicated reconstruction errors of less than 12%. Reconstructed elastic modulus images of the two
Directory of Open Access Journals (Sweden)
M. Giudici
2014-10-01
Full Text Available Numerical modelling of the dynamic evolution of ice sheets and glaciers requires the solution of discrete equations which are based on physical principles (e.g. conservation of mass, linear momentum and energy and phenomenological constitutive laws (e.g. Glen's and Fourier's laws. These equations must be accompanied by information on the forcing term and by initial and boundary conditions (IBCs on ice velocity, stress and temperature; on the other hand the constitutive laws involve many physical parameters, some of which depend on the ice thermodynamical state. The proper forecast of the dynamics of ice sheets and glaciers requires a precise knowledge of several quantities which appear in the IBCs, in the forcing terms and in the phenomenological laws. As these quantities cannot be easily measured at the study scale in the field, they are often obtained through model calibration by solving an inverse problem (IP. The objective of this paper is to provide a thorough and rigorous conceptual framework for IPs in cryospheric studies and in particular: to clarify the role of experimental and monitoring data to determine the calibration targets and the values of the parameters that can be considered to be fixed; to define and characterise identifiability, a property related to the solution to the forward problem; to study well-posedness in a correct way, without confusing instability with ill-conditioning or with the properties of the method applied to compute a solution; to cast sensitivity analysis in a general framework and to differentiate between the computation of local sensitivity indicators with a one-at-a-time approach and first-order sensitivity indicators that consider the whole possible variability of the model parameters. The conceptual framework and the relevant properties are illustrated by means of a simple numerical example of isothermal ice flow, based on the shallow-ice approximation.
Arnold, Alexander; Bruhns, Otto T.; Mosler, Jörn
2011-07-01
A novel finite element formulation suitable for computing efficiently the stiffness distribution in soft biological tissue is presented in this paper. For that purpose, the inverse problem of finite strain hyperelasticity is considered and solved iteratively. In line with Arnold et al (2010 Phys. Med. Biol. 55 2035), the computing time is effectively reduced by using adaptive finite element methods. In sharp contrast to previous approaches, the novel mesh adaption relies on an r-adaption (re-allocation of the nodes within the finite element triangulation). This method allows the detection of material interfaces between healthy and diseased tissue in a very effective manner. The evolution of the nodal positions is canonically driven by the same minimization principle characterizing the inverse problem of hyperelasticity. Consequently, the proposed mesh adaption is variationally consistent. Furthermore, it guarantees that the quality of the numerical solution is improved. Since the proposed r-adaption requires only a relatively coarse triangulation for detecting material interfaces, the underlying finite element spaces are usually not rich enough for predicting the deformation field sufficiently accurately (the forward problem). For this reason, the novel variational r-refinement is combined with the variational h-adaption (Arnold et al 2010) to obtain a variational hr-refinement algorithm. The resulting approach captures material interfaces well (by using r-adaption) and predicts a deformation field in good agreement with that observed experimentally (by using h-adaption).
Opal shell structures: direct assembly versus inversion approach.
Deng, Tian-Song; Sharifi, Parvin; Marlow, Frank
2013-09-16
Opal shell structures can be fabricated in two ways: By direct assembly from hollow spheres (hs-opal) or by infiltration of precursors into opal templates and inversion. The resulting lattice disturbances were characterized by scanning electron microscopy (SEM), optical microscopy, and transmission spectra. The hs-opal system shows much lower disturbances, for example, a lower number of cracks and lattice deformations. The strong suppression of crack formation in one of these inverse opal structures can be considered as promising candidates for the fabrication of more perfect photonic crystals.
A Recursive Born Approach to Nonlinear Inverse Scattering
Kamilov, Ulugbek S; Mansour, Hassan; Boufounos, Petros T
2016-01-01
The Iterative Born Approximation (IBA) is a well-known method for describing waves scattered by semi-transparent objects. In this paper, we present a novel nonlinear inverse scattering method that combines IBA with an edge-preserving total variation (TV) regularizer. The proposed method is obtained by relating iterations of IBA to layers of a feedforward neural network and developing a corresponding error backpropagation algorithm for efficiently estimating the permittivity of the object. Simulations illustrate that, by accounting for multiple scattering, the method successfully recovers the permittivity distribution where the traditional linear inverse scattering fails.
Inverse problems with Poisson data: statistical regularization theory, applications and algorithms
Hohage, Thorsten; Werner, Frank
2016-09-01
Inverse problems with Poisson data arise in many photonic imaging modalities in medicine, engineering and astronomy. The design of regularization methods and estimators for such problems has been studied intensively over the last two decades. In this review we give an overview of statistical regularization theory for such problems, the most important applications, and the most widely used algorithms. The focus is on variational regularization methods in the form of penalized maximum likelihood estimators, which can be analyzed in a general setup. Complementing a number of recent convergence rate results we will establish consistency results. Moreover, we discuss estimators based on a wavelet-vaguelette decomposition of the (necessarily linear) forward operator. As most prominent applications we briefly introduce Positron emission tomography, inverse problems in fluorescence microscopy, and phase retrieval problems. The computation of a penalized maximum likelihood estimator involves the solution of a (typically convex) minimization problem. We also review several efficient algorithms which have been proposed for such problems over the last five years.
Constrained neural approaches to quadratic assignment problems.
Ishii, S; Sato, M
1998-08-01
In this paper, we discuss analog neural approaches to the quadratic assignment problem (QAP). These approaches employ a hard constraints scheme to restrict the domain space, and are able to obtain much improved solutions over conventional neural approaches. Since only a few strong heuristics for QAP have been known to date, our approaches are good alternatives, capable of obtaining fairly good solutions in a short period of time. Some of them can also be applied to large-scale problems, say of size N>/=300.
Nielsen, Bjørn Fredrik; Lysaker, Marius; Tveito, Aslak
2007-01-01
The electrical activity in the heart is modeled by a complex, nonlinear, fully coupled system of differential equations. Several scientists have studied how this model, referred to as the bidomain model, can be modified to incorporate the effect of heart infarctions on simulated ECG (electrocardiogram) recordings. We are concerned with the associated inverse problem; how can we use ECG recordings and mathematical models to identify the position, size and shape of heart infarctions? Due to the extreme CPU efforts needed to solve the bidomain equations, this model, in its full complexity, is not well-suited for this kind of problems. In this paper we show how biological knowledge about the resting potential in the heart and level set techniques can be combined to derive a suitable stationary model, expressed in terms of an elliptic PDE, for such applications. This approach leads to a nonlinear ill-posed minimization problem, which we propose to regularize and solve with a simple iterative scheme. Finally, our theoretical findings are illuminated through a series of computer simulations for an experimental setup involving a realistic heart in torso geometry. More specifically, experiments with synthetic ECG recordings, produced by solving the bidomain model, indicate that our method manages to identify the physical characteristics of the ischemic region(s) in the heart. Furthermore, the ill-posed nature of this inverse problem is explored, i.e. several quantitative issues of our scheme are explored.
Methods and Algorithms for Solving Inverse Problems for Fractional Advection-Dispersion Equations
Aldoghaither, Abeer
2015-11-12
Fractional calculus has been introduced as an e cient tool for modeling physical phenomena, thanks to its memory and hereditary properties. For example, fractional models have been successfully used to describe anomalous di↵usion processes such as contaminant transport in soil, oil flow in porous media, and groundwater flow. These models capture important features of particle transport such as particles with velocity variations and long-rest periods. Mathematical modeling of physical phenomena requires the identification of pa- rameters and variables from available measurements. This is referred to as an inverse problem. In this work, we are interested in studying theoretically and numerically inverse problems for space Fractional Advection-Dispersion Equation (FADE), which is used to model solute transport in porous media. Identifying parameters for such an equa- tion is important to understand how chemical or biological contaminants are trans- ported throughout surface aquifer systems. For instance, an estimate of the di↵eren- tiation order in groundwater contaminant transport model can provide information about soil properties, such as the heterogeneity of the medium. Our main contribution is to propose a novel e cient algorithm based on modulat-ing functions to estimate the coe cients and the di↵erentiation order for space FADE, which can be extended to general fractional Partial Di↵erential Equation (PDE). We also show how the method can be applied to the source inverse problem. This work is divided into two parts: In part I, the proposed method is described and studied through an extensive numerical analysis. The local convergence of the proposed two-stage algorithm is proven for 1D space FADE. The properties of this method are studied along with its limitations. Then, the algorithm is generalized to the 2D FADE. In part II, we analyze direct and inverse source problems for a space FADE. The problem consists of recovering the source term using final