Stochastic Optimal Prediction with Application to Averaged Euler Equations
Energy Technology Data Exchange (ETDEWEB)
Bell, John [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Chorin, Alexandre J. [Univ. of California, Berkeley, CA (United States); Crutchfield, William [Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
2017-04-24
Optimal prediction (OP) methods compensate for a lack of resolution in the numerical solution of complex problems through the use of an invariant measure as a prior measure in the Bayesian sense. In first-order OP, unresolved information is approximated by its conditional expectation with respect to the invariant measure. In higher-order OP, unresolved information is approximated by a stochastic estimator, leading to a system of random or stochastic differential equations. We explain the ideas through a simple example, and then apply them to the solution of Averaged Euler equations in two space dimensions.
Umut Caglar, Mehmet; Pal, Ranadip
2010-10-01
The central dogma of molecular biology states that ``information cannot be transferred back from protein to either protein or nucleic acid.'' However, this assumption is not exactly correct in most of the cases. There are a lot of feedback loops and interactions between different levels of systems. These types of interactions are hard to analyze due to the lack of data in the cellular level and probabilistic nature of interactions. Probabilistic models like Stochastic Master Equation (SME) or deterministic models like differential equations (DE) can be used to analyze these types of interactions. SME models based on chemical master equation (CME) can provide detailed representation of genetic regulatory system, but their use is restricted by the large data requirements and computational costs of calculations. The differential equations models on the other hand, have low calculation costs and much more adequate to generate control procedures on the system; but they are not adequate to investigate the probabilistic nature of interactions. In this work the success of the mapping between SME and DE is analyzed, and the success of a control policy generated by DE model with respect to SME model is examined. Index Terms--- Stochastic Master Equation models, Differential Equation Models, Control Policy Design, Systems biology
Stochastic Averaging and Stochastic Extremum Seeking
Liu, Shu-Jun
2012-01-01
Stochastic Averaging and Stochastic Extremum Seeking develops methods of mathematical analysis inspired by the interest in reverse engineering and analysis of bacterial convergence by chemotaxis and to apply similar stochastic optimization techniques in other environments. The first half of the text presents significant advances in stochastic averaging theory, necessitated by the fact that existing theorems are restricted to systems with linear growth, globally exponentially stable average models, vanishing stochastic perturbations, and prevent analysis over infinite time horizon. The second half of the text introduces stochastic extremum seeking algorithms for model-free optimization of systems in real time using stochastic perturbations for estimation of their gradients. Both gradient- and Newton-based algorithms are presented, offering the user the choice between the simplicity of implementation (gradient) and the ability to achieve a known, arbitrary convergence rate (Newton). The design of algorithms...
Singular stochastic differential equations
Cherny, Alexander S
2005-01-01
The authors introduce, in this research monograph on stochastic differential equations, a class of points termed isolated singular points. Stochastic differential equations possessing such points (called singular stochastic differential equations here) arise often in theory and in applications. However, known conditions for the existence and uniqueness of a solution typically fail for such equations. The book concentrates on the study of the existence, the uniqueness, and, what is most important, on the qualitative behaviour of solutions of singular stochastic differential equations. This is done by providing a qualitative classification of isolated singular points, into 48 possible types.
International Nuclear Information System (INIS)
Ichiguchi, Katsuji
1998-01-01
A new reduced set of resistive MHD equations is derived by averaging the full MHD equations on specified flux coordinates, which is consistent with 3D equilibria. It is confirmed that the total energy is conserved and the linearized equations for ideal modes are self-adjoint. (author)
Trajectory averaging for stochastic approximation MCMC algorithms
Liang, Faming
2010-01-01
to the stochastic approximation Monte Carlo algorithm [Liang, Liu and Carroll J. Amer. Statist. Assoc. 102 (2007) 305-320]. The application of the trajectory averaging estimator to other stochastic approximationMCMC algorithms, for example, a stochastic
Stochastic nonlinear beam equations
Czech Academy of Sciences Publication Activity Database
Brzezniak, Z.; Maslowski, Bohdan; Seidler, Jan
2005-01-01
Roč. 132, č. 1 (2005), s. 119-149 ISSN 0178-8051 R&D Projects: GA ČR(CZ) GA201/01/1197 Institutional research plan: CEZ:AV0Z10190503 Keywords : stochastic beam equation * stability Subject RIV: BA - General Mathematics Impact factor: 0.896, year: 2005
Stochastic porous media equations
Barbu, Viorel; Röckner, Michael
2016-01-01
Focusing on stochastic porous media equations, this book places an emphasis on existence theorems, asymptotic behavior and ergodic properties of the associated transition semigroup. Stochastic perturbations of the porous media equation have reviously been considered by physicists, but rigorous mathematical existence results have only recently been found. The porous media equation models a number of different physical phenomena, including the flow of an ideal gas and the diffusion of a compressible fluid through porous media, and also thermal propagation in plasma and plasma radiation. Another important application is to a model of the standard self-organized criticality process, called the "sand-pile model" or the "Bak-Tang-Wiesenfeld model". The book will be of interest to PhD students and researchers in mathematics, physics and biology.
Stochastic TDHF and the Boltzman-Langevin equation
International Nuclear Information System (INIS)
Suraud, E.; Reinhard, P.G.
1991-01-01
Outgoing from a time-dependent theory of correlations, we present a stochastic differential equation for the propagation of ensembles of Slater determinants, called Stochastic Time-Dependent Hartree-Fock (Stochastic TDHF). These ensembles are allowed to develop large fluctuations in the Hartree-Fock mean fields. An alternative stochastic differential equation, the Boltzmann-Langevin equation, can be derived from Stochastic TDHF by averaging over subensembles with small fluctuations
Stochastic partial differential equations
Lototsky, Sergey V
2017-01-01
Taking readers with a basic knowledge of probability and real analysis to the frontiers of a very active research discipline, this textbook provides all the necessary background from functional analysis and the theory of PDEs. It covers the main types of equations (elliptic, hyperbolic and parabolic) and discusses different types of random forcing. The objective is to give the reader the necessary tools to understand the proofs of existing theorems about SPDEs (from other sources) and perhaps even to formulate and prove a few new ones. Most of the material could be covered in about 40 hours of lectures, as long as not too much time is spent on the general discussion of stochastic analysis in infinite dimensions. As the subject of SPDEs is currently making the transition from the research level to that of a graduate or even undergraduate course, the book attempts to present enough exercise material to fill potential exams and homework assignments. Exercises appear throughout and are usually directly connected ...
Trajectory averaging for stochastic approximation MCMC algorithms
Liang, Faming
2010-10-01
The subject of stochastic approximation was founded by Robbins and Monro [Ann. Math. Statist. 22 (1951) 400-407]. After five decades of continual development, it has developed into an important area in systems control and optimization, and it has also served as a prototype for the development of adaptive algorithms for on-line estimation and control of stochastic systems. Recently, it has been used in statistics with Markov chain Monte Carlo for solving maximum likelihood estimation problems and for general simulation and optimizations. In this paper, we first show that the trajectory averaging estimator is asymptotically efficient for the stochastic approximation MCMC (SAMCMC) algorithm under mild conditions, and then apply this result to the stochastic approximation Monte Carlo algorithm [Liang, Liu and Carroll J. Amer. Statist. Assoc. 102 (2007) 305-320]. The application of the trajectory averaging estimator to other stochastic approximationMCMC algorithms, for example, a stochastic approximation MLE algorithm for missing data problems, is also considered in the paper. © Institute of Mathematical Statistics, 2010.
Maximal stochastic transport in the Lorenz equations
Energy Technology Data Exchange (ETDEWEB)
Agarwal, Sahil, E-mail: sahil.agarwal@yale.edu [Program in Applied Mathematics, Yale University, New Haven (United States); Wettlaufer, J.S., E-mail: john.wettlaufer@yale.edu [Program in Applied Mathematics, Yale University, New Haven (United States); Departments of Geology & Geophysics, Mathematics and Physics, Yale University, New Haven (United States); Mathematical Institute, University of Oxford, Oxford (United Kingdom); Nordita, Royal Institute of Technology and Stockholm University, Stockholm (Sweden)
2016-01-08
We calculate the stochastic upper bounds for the Lorenz equations using an extension of the background method. In analogy with Rayleigh–Bénard convection the upper bounds are for heat transport versus Rayleigh number. As might be expected, the stochastic upper bounds are larger than the deterministic counterpart of Souza and Doering [1], but their variation with noise amplitude exhibits interesting behavior. Below the transition to chaotic dynamics the upper bounds increase monotonically with noise amplitude. However, in the chaotic regime this monotonicity depends on the number of realizations in the ensemble; at a particular Rayleigh number the bound may increase or decrease with noise amplitude. The origin of this behavior is the coupling between the noise and unstable periodic orbits, the degree of which depends on the degree to which the ensemble represents the ergodic set. This is confirmed by examining the close returns plots of the full solutions to the stochastic equations and the numerical convergence of the noise correlations. The numerical convergence of both the ensemble and time averages of the noise correlations is sufficiently slow that it is the limiting aspect of the realization of these bounds. Finally, we note that the full solutions of the stochastic equations demonstrate that the effect of noise is equivalent to the effect of chaos.
Stochastic integration and differential equations
Protter, Philip E
2003-01-01
It has been 15 years since the first edition of Stochastic Integration and Differential Equations, A New Approach appeared, and in those years many other texts on the same subject have been published, often with connections to applications, especially mathematical finance. Yet in spite of the apparent simplicity of approach, none of these books has used the functional analytic method of presenting semimartingales and stochastic integration. Thus a 2nd edition seems worthwhile and timely, though it is no longer appropriate to call it "a new approach". The new edition has several significant changes, most prominently the addition of exercises for solution. These are intended to supplement the text, but lemmas needed in a proof are never relegated to the exercises. Many of the exercises have been tested by graduate students at Purdue and Cornell Universities. Chapter 3 has been completely redone, with a new, more intuitive and simultaneously elementary proof of the fundamental Doob-Meyer decomposition theorem, t...
Stochastic Levy Divergence and Maxwell's Equations
Directory of Open Access Journals (Sweden)
B. O. Volkov
2015-01-01
Full Text Available One of the main reasons for interest in the Levy Laplacian and its analogues such as Levy d'Alembertian is a connection of these operators with gauge fields. The theorem proved by Accardi, Gibillisco and Volovich stated that a connection in a bundle over a Euclidean space or over a Minkowski space is a solution of the Yang-Mills equations if and only if the corresponding parallel transport to the connection is a solution of the Laplace equation for the Levy Laplacian or of the d'Alembert equation for the Levy d'Alembertian respectively (see [5, 6]. There are two approaches to define Levy type operators, both of which date back to the original works of Levy [7]. The first is that the Levy Laplacian (or Levy d'Alembertian is defined as an integral functional generated by a special form of the second derivative. This approach is used in the works [5, 6], as well as in the paper [8] of Leandre and Volovich, where stochastic Levy-Laplacian is discussed. Another approach to the Levy Laplacian is defining it as the Cesaro mean of second order derivatives along the family of vectors, which is an orthonormal basis in the Hilbert space. This definition of the Levy Laplacian is used for the description of solutions of the Yang-Mills equations in the paper [10].The present work shows that the definitions of the Levy Laplacian and the Levy d'Alembertian based on Cesaro averaging of the second order directional derivatives can be transferred to the stochastic case. In the article the values of these operators on a stochastic parallel transport associated with a connection (vector potential are found. In this case, unlike the deterministic case and the stochastic case of Levy Laplacian from [8], these values are not equal to zero if the vector potential corresponding to the stochastic parallel transport is a solution of the Maxwell's equations. As a result, two approaches to definition of the Levy Laplacian in the stochastic case give different operators. This
Optimal Control for Stochastic Delay Evolution Equations
Energy Technology Data Exchange (ETDEWEB)
Meng, Qingxin, E-mail: mqx@hutc.zj.cn [Huzhou University, Department of Mathematical Sciences (China); Shen, Yang, E-mail: skyshen87@gmail.com [York University, Department of Mathematics and Statistics (Canada)
2016-08-15
In this paper, we investigate a class of infinite-dimensional optimal control problems, where the state equation is given by a stochastic delay evolution equation with random coefficients, and the corresponding adjoint equation is given by an anticipated backward stochastic evolution equation. We first prove the continuous dependence theorems for stochastic delay evolution equations and anticipated backward stochastic evolution equations, and show the existence and uniqueness of solutions to anticipated backward stochastic evolution equations. Then we establish necessary and sufficient conditions for optimality of the control problem in the form of Pontryagin’s maximum principles. To illustrate the theoretical results, we apply stochastic maximum principles to study two examples, an infinite-dimensional linear-quadratic control problem with delay and an optimal control of a Dirichlet problem for a stochastic partial differential equation with delay. Further applications of the two examples to a Cauchy problem for a controlled linear stochastic partial differential equation and an optimal harvesting problem are also considered.
Statistical Methods for Stochastic Differential Equations
Kessler, Mathieu; Sorensen, Michael
2012-01-01
The seventh volume in the SemStat series, Statistical Methods for Stochastic Differential Equations presents current research trends and recent developments in statistical methods for stochastic differential equations. Written to be accessible to both new students and seasoned researchers, each self-contained chapter starts with introductions to the topic at hand and builds gradually towards discussing recent research. The book covers Wiener-driven equations as well as stochastic differential equations with jumps, including continuous-time ARMA processes and COGARCH processes. It presents a sp
Stochastic differential equation model to Prendiville processes
Energy Technology Data Exchange (ETDEWEB)
Granita, E-mail: granitafc@gmail.com [Dept. of Mathematical Science, Universiti Teknologi Malaysia, 81310, Johor Malaysia (Malaysia); Bahar, Arifah [Dept. of Mathematical Science, Universiti Teknologi Malaysia, 81310, Johor Malaysia (Malaysia); UTM Center for Industrial & Applied Mathematics (UTM-CIAM) (Malaysia)
2015-10-22
The Prendiville process is another variation of the logistic model which assumes linearly decreasing population growth rate. It is a continuous time Markov chain (CTMC) taking integer values in the finite interval. The continuous time Markov chain can be approximated by stochastic differential equation (SDE). This paper discusses the stochastic differential equation of Prendiville process. The work started with the forward Kolmogorov equation in continuous time Markov chain of Prendiville process. Then it was formulated in the form of a central-difference approximation. The approximation was then used in Fokker-Planck equation in relation to the stochastic differential equation of the Prendiville process. The explicit solution of the Prendiville process was obtained from the stochastic differential equation. Therefore, the mean and variance function of the Prendiville process could be easily found from the explicit solution.
Stochastic differential equation model to Prendiville processes
International Nuclear Information System (INIS)
Granita; Bahar, Arifah
2015-01-01
The Prendiville process is another variation of the logistic model which assumes linearly decreasing population growth rate. It is a continuous time Markov chain (CTMC) taking integer values in the finite interval. The continuous time Markov chain can be approximated by stochastic differential equation (SDE). This paper discusses the stochastic differential equation of Prendiville process. The work started with the forward Kolmogorov equation in continuous time Markov chain of Prendiville process. Then it was formulated in the form of a central-difference approximation. The approximation was then used in Fokker-Planck equation in relation to the stochastic differential equation of the Prendiville process. The explicit solution of the Prendiville process was obtained from the stochastic differential equation. Therefore, the mean and variance function of the Prendiville process could be easily found from the explicit solution
Perturbation theory for continuous stochastic equations
International Nuclear Information System (INIS)
Chechetkin, V.R.; Lutovinov, V.S.
1987-01-01
The various general perturbational schemes for continuous stochastic equations are considered. These schemes have many analogous features with the iterational solution of Schwinger equation for S-matrix. The following problems are discussed: continuous stochastic evolution equations for probability distribution functionals, evolution equations for equal time correlators, perturbation theory for Gaussian and Poissonian additive noise, perturbation theory for birth and death processes, stochastic properties of systems with multiplicative noise. The general results are illustrated by diffusion-controlled reactions, fluctuations in closed systems with chemical processes, propagation of waves in random media in parabolic equation approximation, and non-equilibrium phase transitions in systems with Poissonian breeding centers. The rate of irreversible reaction X + X → A (Smoluchowski process) is calculated with the use of general theory based on continuous stochastic equations for birth and death processes. The threshold criterion and range of fluctuational region for synergetic phase transition in system with Poissonian breeding centers are also considered. (author)
A microscopic derivation of stochastic differential equations
International Nuclear Information System (INIS)
Arimitsu, Toshihico
1996-01-01
With the help of the formulation of Non-Equilibrium Thermo Field Dynamics, a unified canonical operator formalism is constructed for the quantum stochastic differential equations. In the course of its construction, it is found that there are at least two formulations, i.e. one is non-hermitian and the other is hermitian. Having settled which framework should be satisfied by the quantum stochastic differential equations, a microscopic derivation is performed for these stochastic differential equations by extending the projector methods. This investigation may open a new field for quantum systems in order to understand the deeper meaning of dissipation
Ambit processes and stochastic partial differential equations
DEFF Research Database (Denmark)
Barndorff-Nielsen, Ole; Benth, Fred Espen; Veraart, Almut
Ambit processes are general stochastic processes based on stochastic integrals with respect to Lévy bases. Due to their flexible structure, they have great potential for providing realistic models for various applications such as in turbulence and finance. This papers studies the connection betwe...... ambit processes and solutions to stochastic partial differential equations. We investigate this relationship from two angles: from the Walsh theory of martingale measures and from the viewpoint of the Lévy noise analysis....
Numerical Solution of Heun Equation Via Linear Stochastic Differential Equation
Directory of Open Access Journals (Sweden)
Hamidreza Rezazadeh
2014-05-01
Full Text Available In this paper, we intend to solve special kind of ordinary differential equations which is called Heun equations, by converting to a corresponding stochastic differential equation(S.D.E.. So, we construct a stochastic linear equation system from this equation which its solution is based on computing fundamental matrix of this system and then, this S.D.E. is solved by numerically methods. Moreover, its asymptotic stability and statistical concepts like expectation and variance of solutions are discussed. Finally, the attained solutions of these S.D.E.s compared with exact solution of corresponding differential equations.
Stochastic effects on the nonlinear Schroedinger equation
International Nuclear Information System (INIS)
Flessas, G P; Leach, P G L; Yannacopoulos, A N
2004-01-01
The aim of this article is to provide a brief review of recent advances in the field of stochastic effects on the nonlinear Schroedinger equation. The article reviews rigorous and perturbative results. (review article)
Asymptotic problems for stochastic partial differential equations
Salins, Michael
Stochastic partial differential equations (SPDEs) can be used to model systems in a wide variety of fields including physics, chemistry, and engineering. The main SPDEs of interest in this dissertation are the semilinear stochastic wave equations which model the movement of a material with constant mass density that is exposed to both determinstic and random forcing. Cerrai and Freidlin have shown that on fixed time intervals, as the mass density of the material approaches zero, the solutions of the stochastic wave equation converge uniformly to the solutions of a stochastic heat equation, in probability. This is called the Smoluchowski-Kramers approximation. In Chapter 2, we investigate some of the multi-scale behaviors that these wave equations exhibit. In particular, we show that the Freidlin-Wentzell exit place and exit time asymptotics for the stochastic wave equation in the small noise regime can be approximated by the exit place and exit time asymptotics for the stochastic heat equation. We prove that the exit time and exit place asymptotics are characterized by quantities called quasipotentials and we prove that the quasipotentials converge. We then investigate the special case where the equation has a gradient structure and show that we can explicitly solve for the quasipotentials, and that the quasipotentials for the heat equation and wave equation are equal. In Chapter 3, we study the Smoluchowski-Kramers approximation in the case where the material is electrically charged and exposed to a magnetic field. Interestingly, if the system is frictionless, then the Smoluchowski-Kramers approximation does not hold. We prove that the Smoluchowski-Kramers approximation is valid for systems exposed to both a magnetic field and friction. Notably, we prove that the solutions to the second-order equations converge to the solutions of the first-order equation in an Lp sense. This strengthens previous results where convergence was proved in probability.
Asymptotic analysis for functional stochastic differential equations
Bao, Jianhai; Yuan, Chenggui
2016-01-01
This brief treats dynamical systems that involve delays and random disturbances. The study is motivated by a wide variety of systems in real life in which random noise has to be taken into consideration and the effect of delays cannot be ignored. Concentrating on such systems that are described by functional stochastic differential equations, this work focuses on the study of large time behavior, in particular, ergodicity. This brief is written for probabilists, applied mathematicians, engineers, and scientists who need to use delay systems and functional stochastic differential equations in their work. Selected topics from the brief can also be used in a graduate level topics course in probability and stochastic processes.
Exactly averaged equations for flow and transport in random media
International Nuclear Information System (INIS)
Shvidler, Mark; Karasaki, Kenzi
2001-01-01
It is well known that exact averaging of the equations of flow and transport in random porous media can be realized only for a small number of special, occasionally exotic, fields. On the other hand, the properties of approximate averaging methods are not yet fully understood. For example, the convergence behavior and the accuracy of truncated perturbation series. Furthermore, the calculation of the high-order perturbations is very complicated. These problems for a long time have stimulated attempts to find the answer for the question: Are there in existence some exact general and sufficiently universal forms of averaged equations? If the answer is positive, there arises the problem of the construction of these equations and analyzing them. There exist many publications related to these problems and oriented on different applications: hydrodynamics, flow and transport in porous media, theory of elasticity, acoustic and electromagnetic waves in random fields, etc. We present a method of finding the general form of exactly averaged equations for flow and transport in random fields by using (1) an assumption of the existence of Green's functions for appropriate stochastic problems, (2) some general properties of the Green's functions, and (3) the some basic information about the random fields of the conductivity, porosity and flow velocity. We present a general form of the exactly averaged non-local equations for the following cases. 1. Steady-state flow with sources in porous media with random conductivity. 2. Transient flow with sources in compressible media with random conductivity and porosity. 3. Non-reactive solute transport in random porous media. We discuss the problem of uniqueness and the properties of the non-local averaged equations, for the cases with some types of symmetry (isotropic, transversal isotropic, orthotropic) and we analyze the hypothesis of the structure non-local equations in general case of stochastically homogeneous fields. (author)
Multiphase averaging of periodic soliton equations
International Nuclear Information System (INIS)
Forest, M.G.
1979-01-01
The multiphase averaging of periodic soliton equations is considered. Particular attention is given to the periodic sine-Gordon and Korteweg-deVries (KdV) equations. The periodic sine-Gordon equation and its associated inverse spectral theory are analyzed, including a discussion of the spectral representations of exact, N-phase sine-Gordon solutions. The emphasis is on physical characteristics of the periodic waves, with a motivation from the well-known whole-line solitons. A canonical Hamiltonian approach for the modulational theory of N-phase waves is prescribed. A concrete illustration of this averaging method is provided with the periodic sine-Gordon equation; explicit averaging results are given only for the N = 1 case, laying a foundation for a more thorough treatment of the general N-phase problem. For the KdV equation, very general results are given for multiphase averaging of the N-phase waves. The single-phase results of Whitham are extended to general N phases, and more importantly, an invariant representation in terms of Abelian differentials on a Riemann surface is provided. Several consequences of this invariant representation are deduced, including strong evidence for the Hamiltonian structure of N-phase modulational equations
Modeling animal movements using stochastic differential equations
Haiganoush K. Preisler; Alan A. Ager; Bruce K. Johnson; John G. Kie
2004-01-01
We describe the use of bivariate stochastic differential equations (SDE) for modeling movements of 216 radiocollared female Rocky Mountain elk at the Starkey Experimental Forest and Range in northeastern Oregon. Spatially and temporally explicit vector fields were estimated using approximating difference equations and nonparametric regression techniques. Estimated...
Energy Technology Data Exchange (ETDEWEB)
Paul, Wolfgang; Koeppe, Jeanette [Institut fuer Physik, Martin Luther Universitaet, 06099 Halle (Germany); Grecksch, Wilfried [Institut fuer Mathematik, Martin Luther Universitaet, 06099 Halle (Germany)
2016-07-01
The standard approach to solve a non-relativistic quantum problem is through analytical or numerical solution of the Schroedinger equation. We show a way to go around it. This way is based on the derivation of the Schroedinger equation from conservative diffusion processes and the establishment of (several) stochastic variational principles leading to the Schroedinger equation under the assumption of a kinematics described by Nelson's diffusion processes. Mathematically, the variational principle can be considered as a stochastic optimal control problem linked to the forward-backward stochastic differential equations of Nelson's stochastic mechanics. The Hamilton-Jacobi-Bellmann equation of this control problem is the Schroedinger equation. We present the mathematical background and how to turn it into a numerical scheme for analyzing a quantum system without using the Schroedinger equation and exemplify the approach for a simple 1d problem.
Stochastic differential equations and diffusion processes
Ikeda, N
1989-01-01
Being a systematic treatment of the modern theory of stochastic integrals and stochastic differential equations, the theory is developed within the martingale framework, which was developed by J.L. Doob and which plays an indispensable role in the modern theory of stochastic analysis.A considerable number of corrections and improvements have been made for the second edition of this classic work. In particular, major and substantial changes are in Chapter III and Chapter V where the sections treating excursions of Brownian Motion and the Malliavin Calculus have been expanded and refined. Sectio
Stochastic partial differential equations an introduction
Liu, Wei
2015-01-01
This book provides an introduction to the theory of stochastic partial differential equations (SPDEs) of evolutionary type. SPDEs are one of the main research directions in probability theory with several wide ranging applications. Many types of dynamics with stochastic influence in nature or man-made complex systems can be modelled by such equations. The theory of SPDEs is based both on the theory of deterministic partial differential equations, as well as on modern stochastic analysis. Whilst this volume mainly follows the ‘variational approach’, it also contains a short account on the ‘semigroup (or mild solution) approach’. In particular, the volume contains a complete presentation of the main existence and uniqueness results in the case of locally monotone coefficients. Various types of generalized coercivity conditions are shown to guarantee non-explosion, but also a systematic approach to treat SPDEs with explosion in finite time is developed. It is, so far, the only book where the latter and t...
An introduction to stochastic differential equations
Evans, Lawrence C
2014-01-01
These notes provide a concise introduction to stochastic differential equations and their application to the study of financial markets and as a basis for modeling diverse physical phenomena. They are accessible to non-specialists and make a valuable addition to the collection of texts on the topic. -Srinivasa Varadhan, New York University This is a handy and very useful text for studying stochastic differential equations. There is enough mathematical detail so that the reader can benefit from this introduction with only a basic background in mathematical analysis and probability. -George Papa
Stochastic 2-D Navier-Stokes Equation
International Nuclear Information System (INIS)
Menaldi, J.L.; Sritharan, S.S.
2002-01-01
In this paper we prove the existence and uniqueness of strong solutions for the stochastic Navier-Stokes equation in bounded and unbounded domains. These solutions are stochastic analogs of the classical Lions-Prodi solutions to the deterministic Navier-Stokes equation. Local monotonicity of the nonlinearity is exploited to obtain the solutions in a given probability space and this significantly improves the earlier techniques for obtaining strong solutions, which depended on pathwise solutions to the Navier-Stokes martingale problem where the probability space is also obtained as a part of the solution
Stochastic Differential Equations and Kondratiev Spaces
Energy Technology Data Exchange (ETDEWEB)
Vaage, G.
1995-05-01
The purpose of this mathematical thesis was to improve the understanding of physical processes such as fluid flow in porous media. An example is oil flowing in a reservoir. In the first of five included papers, Hilbert space methods for elliptic boundary value problems are used to prove the existence and uniqueness of a large family of elliptic differential equations with additive noise without using the Hermite transform. The ideas are then extended to the multidimensional case and used to prove existence and uniqueness of solution of the Stokes equations with additive noise. The second paper uses functional analytic methods for partial differential equations and presents a general framework for proving existence and uniqueness of solutions to stochastic partial differential equations with multiplicative noise, for a large family of noises. The methods are applied to equations of elliptic, parabolic as well as hyperbolic type. The framework presented can be extended to the multidimensional case. The third paper shows how the ideas from the second paper can be extended to study the moving boundary value problem associated with the stochastic pressure equation. The fourth paper discusses a set of stochastic differential equations. The fifth paper studies the relationship between the two families of Kondratiev spaces used in the thesis. 102 refs.
Efficient Estimating Functions for Stochastic Differential Equations
DEFF Research Database (Denmark)
Jakobsen, Nina Munkholt
The overall topic of this thesis is approximate martingale estimating function-based estimationfor solutions of stochastic differential equations, sampled at high frequency. Focuslies on the asymptotic properties of the estimators. The first part of the thesis deals with diffusions observed over...
Stochastic differential equations used to model conjugation
DEFF Research Database (Denmark)
Philipsen, Kirsten Riber; Christiansen, Lasse Engbo
Stochastic differential equations (SDEs) are used to model horizontal transfer of antibiotic resis- tance by conjugation. The model describes the concentration of donor, recipient, transconjugants and substrate. The strength of the SDE model over the traditional ODE models is that the noise can...
Degenerate parabolic stochastic partial differential equations
Czech Academy of Sciences Publication Activity Database
span class="emphasis">Hofmanová, Martinaspan>
2013-01-01
Roč. 123, č. 12 (2013), s. 4294-4336 ISSN 0304-4149 R&D Projects: GA ČR GAP201/10/0752 Institutional support: RVO:67985556 Keywords : kinetic solutions * degenerate stochastic parabolic equations Subject RIV: BA - General Mathematics Impact factor: 1.046, year: 2013 http://library.utia.cas.cz/separaty/2013/SI/hofmanova-0397241.pdf
Potential in stochastic differential equations: novel construction
International Nuclear Information System (INIS)
Ao, P
2004-01-01
There is a whole range of emergent phenomena in a complex network such as robustness, adaptiveness, multiple-equilibrium, hysteresis, oscillation and feedback. Those non-equilibrium behaviours can often be described by a set of stochastic differential equations. One persistent important question is the existence of a potential function. Here we demonstrate that a dynamical structure built into stochastic differential equation allows us to construct such a global optimization potential function. We present an explicit construction procedure to obtain the potential and relevant quantities. In the procedure no reference to the Fokker-Planck equation is needed. The availability of the potential suggests that powerful statistical mechanics tools can be used in nonequilibrium situations. (letter to the editor)
Stochastic differential equations and a biological system
DEFF Research Database (Denmark)
Wang, Chunyan
1994-01-01
The purpose of this Ph.D. study is to explore the property of a growth process. The study includes solving and simulating of the growth process which is described in terms of stochastic differential equations. The identification of the growth and variability parameters of the process based...... on experimental data is considered. As an example, the growth of bacteria Pseudomonas fluorescens is taken. Due to the specific features of stochastic differential equations, namely that their solutions do not exist in the general sense, two new integrals - the Ito integral and the Stratonovich integral - have...... description. In order to identify the parameters, a Maximum likelihood estimation method is used together with a simplified truncated second order filter. Because of the continuity feature of the predictor equation, two numerical integration methods, called the Odeint and the Discretization method...
On stochastic differential equations with random delay
International Nuclear Information System (INIS)
Krapivsky, P L; Luck, J M; Mallick, K
2011-01-01
We consider stochastic dynamical systems defined by differential equations with a uniform random time delay. The latter equations are shown to be equivalent to deterministic higher-order differential equations: for an nth-order equation with random delay, the corresponding deterministic equation has order n + 1. We analyze various examples of dynamical systems of this kind, and find a number of unusual behaviors. For instance, for the harmonic oscillator with random delay, the energy grows as exp((3/2) t 2/3 ) in reduced units. We then investigate the effect of introducing a discrete time step ε. At variance with the continuous situation, the discrete random recursion relations thus obtained have intrinsic fluctuations. The crossover between the fluctuating discrete problem and the deterministic continuous one as ε goes to zero is studied in detail on the example of a first-order linear differential equation
A stochastic solution of the advective transport equation with uncertainty
International Nuclear Information System (INIS)
Williams, M.M.R.
1991-01-01
A model has been developed for calculating the transport of water-borne radionuclides through layers of porous materials, such as rock or clay. The model is based upon a purely advective transport equation, in which the fluid velocity is a random variable, thereby simulating dispersion in a more realistic manner than the ad hoc introduction of a dispersivity. In addition to a random velocity field, which is an observable physical phenomenon, allowance is made for uncertainty in our knowledge of the parameters which enter the equation, e.g. the retardation coefficient. This too, is assumed to be a random variable and contributes to the stochasticity of the resulting partial differential equation of transport. The stochastic differential equation can be solved analytically and then ensemble averages taken over the associated probability distribution of velocity and retardation coefficient. A method based upon a novel form of the central limit theorem of statistics is employed to obtain tractable solutions of a system consisting of many serial legs of varying properties. One interesting conclusion is that the total flux out of a medium is significantly underestimated by using the deterministic solution with an average transit time compared with that from the stochastically averaged solution. The theory is illustrated numerically for a number of physically relevant cases. (author) 8 figs., 4 tabs., 7 refs
Parameter estimation in stochastic differential equations
Bishwal, Jaya P N
2008-01-01
Parameter estimation in stochastic differential equations and stochastic partial differential equations is the science, art and technology of modelling complex phenomena and making beautiful decisions. The subject has attracted researchers from several areas of mathematics and other related fields like economics and finance. This volume presents the estimation of the unknown parameters in the corresponding continuous models based on continuous and discrete observations and examines extensively maximum likelihood, minimum contrast and Bayesian methods. Useful because of the current availability of high frequency data is the study of refined asymptotic properties of several estimators when the observation time length is large and the observation time interval is small. Also space time white noise driven models, useful for spatial data, and more sophisticated non-Markovian and non-semimartingale models like fractional diffusions that model the long memory phenomena are examined in this volume.
Stochastic Evolution Equations Driven by Fractional Noises
2016-11-28
paper is to establish the weak convergence, in the topology of the Skorohod space, of the ν-symmetric Riemann sums for functionals of the fractional...stochastic heat equation with fractional-colored noise: existence of the solution. ALEA Lat. Am. J. Probab. Math . Stat. 4 (2008), 57–87. [8] P. Carmona, Y...Hu: Strong disorder implies strong localization for directed polymers in a random environment. ALEA Lat. Am. J. Probab. Math . Stat. 2 (2006), 217
Modelling Evolutionary Algorithms with Stochastic Differential Equations.
Heredia, Jorge Pérez
2017-11-20
There has been renewed interest in modelling the behaviour of evolutionary algorithms (EAs) by more traditional mathematical objects, such as ordinary differential equations or Markov chains. The advantage is that the analysis becomes greatly facilitated due to the existence of well established methods. However, this typically comes at the cost of disregarding information about the process. Here, we introduce the use of stochastic differential equations (SDEs) for the study of EAs. SDEs can produce simple analytical results for the dynamics of stochastic processes, unlike Markov chains which can produce rigorous but unwieldy expressions about the dynamics. On the other hand, unlike ordinary differential equations (ODEs), they do not discard information about the stochasticity of the process. We show that these are especially suitable for the analysis of fixed budget scenarios and present analogues of the additive and multiplicative drift theorems from runtime analysis. In addition, we derive a new more general multiplicative drift theorem that also covers non-elitist EAs. This theorem simultaneously allows for positive and negative results, providing information on the algorithm's progress even when the problem cannot be optimised efficiently. Finally, we provide results for some well-known heuristics namely Random Walk (RW), Random Local Search (RLS), the (1+1) EA, the Metropolis Algorithm (MA), and the Strong Selection Weak Mutation (SSWM) algorithm.
Modeling and Prediction Using Stochastic Differential Equations
DEFF Research Database (Denmark)
Juhl, Rune; Møller, Jan Kloppenborg; Jørgensen, John Bagterp
2016-01-01
Pharmacokinetic/pharmakodynamic (PK/PD) modeling for a single subject is most often performed using nonlinear models based on deterministic ordinary differential equations (ODEs), and the variation between subjects in a population of subjects is described using a population (mixed effects) setup...... deterministic and can predict the future perfectly. A more realistic approach would be to allow for randomness in the model due to e.g., the model be too simple or errors in input. We describe a modeling and prediction setup which better reflects reality and suggests stochastic differential equations (SDEs...
Causal interpretation of stochastic differential equations
DEFF Research Database (Denmark)
Sokol, Alexander; Hansen, Niels Richard
2014-01-01
We give a causal interpretation of stochastic differential equations (SDEs) by defining the postintervention SDE resulting from an intervention in an SDE. We show that under Lipschitz conditions, the solution to the postintervention SDE is equal to a uniform limit in probability of postintervention...... structural equation models based on the Euler scheme of the original SDE, thus relating our definition to mainstream causal concepts. We prove that when the driving noise in the SDE is a Lévy process, the postintervention distribution is identifiable from the generator of the SDE....
Modelling conjugation with stochastic differential equations
DEFF Research Database (Denmark)
Philipsen, Kirsten Riber; Christiansen, Lasse Engbo; Hasman, Henrik
2010-01-01
Enterococcus faecium strains in a rich exhaustible media. The model contains a new expression for a substrate dependent conjugation rate. A maximum likelihood based method is used to estimate the model parameters. Different models including different noise structure for the system and observations are compared......Conjugation is an important mechanism involved in the transfer of resistance between bacteria. In this article a stochastic differential equation based model consisting of a continuous time state equation and a discrete time measurement equation is introduced to model growth and conjugation of two...... using a likelihood-ratio test and Akaike's information criterion. Experiments indicating conjugation on the agar plates selecting for transconjugants motivates the introduction of an extended model, for which conjugation on the agar plate is described in the measurement equation. This model is compared...
Sparse learning of stochastic dynamical equations
Boninsegna, Lorenzo; Nüske, Feliks; Clementi, Cecilia
2018-06-01
With the rapid increase of available data for complex systems, there is great interest in the extraction of physically relevant information from massive datasets. Recently, a framework called Sparse Identification of Nonlinear Dynamics (SINDy) has been introduced to identify the governing equations of dynamical systems from simulation data. In this study, we extend SINDy to stochastic dynamical systems which are frequently used to model biophysical processes. We prove the asymptotic correctness of stochastic SINDy in the infinite data limit, both in the original and projected variables. We discuss algorithms to solve the sparse regression problem arising from the practical implementation of SINDy and show that cross validation is an essential tool to determine the right level of sparsity. We demonstrate the proposed methodology on two test systems, namely, the diffusion in a one-dimensional potential and the projected dynamics of a two-dimensional diffusion process.
PC analysis of stochastic differential equations driven by Wiener noise
Le Maitre, Olivier; Knio, Omar
2015-01-01
A polynomial chaos (PC) analysis with stochastic expansion coefficients is proposed for stochastic differential equations driven by additive or multiplicative Wiener noise. It is shown that for this setting, a Galerkin formalism naturally leads
Multivalued stochastic delay differential equations and related ...
African Journals Online (AJOL)
We study the existence and uniqueness of a solution for the multivalued stochastic differential equation with delay (the multivalued term is of subdifferential type):. dX(t) + aφ (X(t))dt ∍ b(t,X(t), Y(t), Z(t)) dt. ⎨ +σ (t, X (t), Y (t), Z (t)) dW (t), t ∈ (s, T). X(t) = ξ (t - s), t ∈ [s - δ, s]. Specify that in this case the coefficients at time t ...
Polynomial chaos functions and stochastic differential equations
International Nuclear Information System (INIS)
Williams, M.M.R.
2006-01-01
The Karhunen-Loeve procedure and the associated polynomial chaos expansion have been employed to solve a simple first order stochastic differential equation which is typical of transport problems. Because the equation has an analytical solution, it provides a useful test of the efficacy of polynomial chaos. We find that the convergence is very rapid in some cases but that the increased complexity associated with many random variables can lead to very long computational times. The work is illustrated by exact and approximate solutions for the mean, variance and the probability distribution itself. The usefulness of a white noise approximation is also assessed. Extensive numerical results are given which highlight the weaknesses and strengths of polynomial chaos. The general conclusion is that the method is promising but requires further detailed study by application to a practical problem in transport theory
Energy Technology Data Exchange (ETDEWEB)
Silva, Milena Wollmann da; Vilhena, Marco Tullio M.B.; Bodmann, Bardo Ernst J.; Vasques, Richard, E-mail: milena.wollmann@ufrgs.br, E-mail: vilhena@mat.ufrgs.br, E-mail: bardobodmann@ufrgs.br, E-mail: richard.vasques@fulbrightmail.org [Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, RS (Brazil). Programa de Pos-Graduacao em Engenharia Mecanica
2015-07-01
The neutron point kinetics equation, which models the time-dependent behavior of nuclear reactors, is often used to understand the dynamics of nuclear reactor operations. It consists of a system of coupled differential equations that models the interaction between (i) the neutron population; and (II) the concentration of the delayed neutron precursors, which are radioactive isotopes formed in the fission process that decay through neutron emission. These equations are deterministic in nature, and therefore can provide only average values of the modeled populations. However, the actual dynamical process is stochastic: the neutron density and the delayed neutron precursor concentrations vary randomly with time. To address this stochastic behavior, Hayes and Allen have generalized the standard deterministic point kinetics equation. They derived a system of stochastic differential equations that can accurately model the random behavior of the neutron density and the precursor concentrations in a point reactor. Due to the stiffness of these equations, this system was numerically implemented using a stochastic piecewise constant approximation method (Stochastic PCA). Here, we present a study of the influence of stochastic fluctuations on the results of the neutron point kinetics equation. We reproduce the stochastic formulation introduced by Hayes and Allen and compute Monte Carlo numerical results for examples with constant and time-dependent reactivity, comparing these results with stochastic and deterministic methods found in the literature. Moreover, we introduce a modified version of the stochastic method to obtain a non-stiff solution, analogue to a previously derived deterministic approach. (author)
International Nuclear Information System (INIS)
Silva, Milena Wollmann da; Vilhena, Marco Tullio M.B.; Bodmann, Bardo Ernst J.; Vasques, Richard
2015-01-01
The neutron point kinetics equation, which models the time-dependent behavior of nuclear reactors, is often used to understand the dynamics of nuclear reactor operations. It consists of a system of coupled differential equations that models the interaction between (i) the neutron population; and (II) the concentration of the delayed neutron precursors, which are radioactive isotopes formed in the fission process that decay through neutron emission. These equations are deterministic in nature, and therefore can provide only average values of the modeled populations. However, the actual dynamical process is stochastic: the neutron density and the delayed neutron precursor concentrations vary randomly with time. To address this stochastic behavior, Hayes and Allen have generalized the standard deterministic point kinetics equation. They derived a system of stochastic differential equations that can accurately model the random behavior of the neutron density and the precursor concentrations in a point reactor. Due to the stiffness of these equations, this system was numerically implemented using a stochastic piecewise constant approximation method (Stochastic PCA). Here, we present a study of the influence of stochastic fluctuations on the results of the neutron point kinetics equation. We reproduce the stochastic formulation introduced by Hayes and Allen and compute Monte Carlo numerical results for examples with constant and time-dependent reactivity, comparing these results with stochastic and deterministic methods found in the literature. Moreover, we introduce a modified version of the stochastic method to obtain a non-stiff solution, analogue to a previously derived deterministic approach. (author)
Semilinear Kolmogorov Equations and Applications to Stochastic Optimal Control
International Nuclear Information System (INIS)
Masiero, Federica
2005-01-01
Semilinear parabolic differential equations are solved in a mild sense in an infinite-dimensional Hilbert space. Applications to stochastic optimal control problems are studied by solving the associated Hamilton-Jacobi-Bellman equation. These results are applied to some controlled stochastic partial differential equations
Stochastic reliability analysis using Fokker Planck equations
International Nuclear Information System (INIS)
Hari Prasad, M.; Rami Reddy, G.; Srividya, A.; Verma, A.K.
2011-01-01
The Fokker-Planck equation describes the time evolution of the probability density function of the velocity of a particle, and can be generalized to other observables as well. It is also known as the Kolmogorov forward equation (diffusion). Hence, for any process, which evolves with time, the probability density function as a function of time can be represented with Fokker-Planck equation. In stochastic reliability analysis one is more interested in finding out the reliability or failure probability of the components or structures as a function of time rather than instantaneous failure probabilities. In this analysis the variables are represented with random processes instead of random variables. A random processes can be either stationary or non stationary. If the random process is stationary then the failure probability doesn't change with time where as in the case of non stationary processes the failure probability changes with time. In the present paper Fokker Planck equations have been used to find out the probability density function of the non stationary random processes. In this paper a flow chart has been provided which describes step by step process for carrying out stochastic reliability analysis using Fokker-Planck equations. As a first step one has to identify the failure function as a function of random processes. Then one has to solve the Fokker-Planck equation for each random process. In this paper the Fokker-Planck equation has been solved by using Finite difference method. As a result one gets the probability density values of the random process in the sample space as well as time space. Later at each time step appropriate probability distribution has to be identified based on the available probability density values. For checking the better fitness of the data Kolmogorov-Smirnov Goodness of fit test has been performed. In this way one can find out the distribution of the random process at each time step. Once one has the probability distribution
Invariant measures for stochastic nonlinear beam and wave equations
Czech Academy of Sciences Publication Activity Database
Brzezniak, Z.; Ondreját, Martin; Seidler, Jan
2016-01-01
Roč. 260, č. 5 (2016), s. 4157-4179 ISSN 0022-0396 R&D Projects: GA ČR GAP201/10/0752 Institutional support: RVO:67985556 Keywords : stochastic partial differential equation * stochastic beam equation * stochastic wave equation * invariant measure Subject RIV: BA - General Mathematics Impact factor: 1.988, year: 2016 http://library.utia.cas.cz/separaty/2016/SI/ondrejat-0453412.pdf
Bipartite Fuzzy Stochastic Differential Equations with Global Lipschitz Condition
Directory of Open Access Journals (Sweden)
Marek T. Malinowski
2016-01-01
Full Text Available We introduce and analyze a new type of fuzzy stochastic differential equations. We consider equations with drift and diffusion terms occurring at both sides of equations. Therefore we call them the bipartite fuzzy stochastic differential equations. Under the Lipschitz and boundedness conditions imposed on drifts and diffusions coefficients we prove existence of a unique solution. Then, insensitivity of the solution under small changes of data of equation is examined. Finally, we mention that all results can be repeated for solutions to bipartite set-valued stochastic differential equations.
Lyapunov functionals and stability of stochastic functional differential equations
Shaikhet, Leonid
2013-01-01
Stability conditions for functional differential equations can be obtained using Lyapunov functionals. Lyapunov Functionals and Stability of Stochastic Functional Differential Equations describes the general method of construction of Lyapunov functionals to investigate the stability of differential equations with delays. This work continues and complements the author’s previous book Lyapunov Functionals and Stability of Stochastic Difference Equations, where this method is described for discrete- and continuous-time difference equations. The text begins with a description of the peculiarities of deterministic and stochastic functional differential equations. There follow basic definitions for stability theory of stochastic hereditary systems, and a formal procedure of Lyapunov functionals construction is presented. Stability investigation is conducted for stochastic linear and nonlinear differential equations with constant and distributed delays. The proposed method is used for stability investigation of di...
Efficient Estimating Functions for Stochastic Differential Equations
DEFF Research Database (Denmark)
Jakobsen, Nina Munkholt
The overall topic of this thesis is approximate martingale estimating function-based estimationfor solutions of stochastic differential equations, sampled at high frequency. Focuslies on the asymptotic properties of the estimators. The first part of the thesis deals with diffusions observed over...... a fixed time interval. Rate optimal and effcient estimators areobtained for a one-dimensional diffusion parameter. Stable convergence in distribution isused to achieve a practically applicable Gaussian limit distribution for suitably normalisedestimators. In a simulation example, the limit distributions...... multidimensional parameter. Conditions for rate optimality and effciency of estimatorsof drift-jump and diffusion parameters are given in some special cases. Theseconditions are found to extend the pre-existing conditions applicable to continuous diffusions,and impose much stronger requirements on the estimating...
Optimal control of stochastic difference Volterra equations an introduction
Shaikhet, Leonid
2015-01-01
This book showcases a subclass of hereditary systems, that is, systems with behaviour depending not only on their current state but also on their past history; it is an introduction to the mathematical theory of optimal control for stochastic difference Volterra equations of neutral type. As such, it will be of much interest to researchers interested in modelling processes in physics, mechanics, automatic regulation, economics and finance, biology, sociology and medicine for all of which such equations are very popular tools. The text deals with problems of optimal control such as meeting given performance criteria, and stabilization, extending them to neutral stochastic difference Volterra equations. In particular, it contrasts the difference analogues of solutions to optimal control and optimal estimation problems for stochastic integral Volterra equations with optimal solutions for corresponding problems in stochastic difference Volterra equations. Optimal Control of Stochastic Difference Volterra Equation...
Numerical analysis of systems of ordinary and stochastic differential equations
Artemiev, S S
1997-01-01
This text deals with numerical analysis of systems of both ordinary and stochastic differential equations. It covers numerical solution problems of the Cauchy problem for stiff ordinary differential equations (ODE) systems by Rosenbrock-type methods (RTMs).
Perturbative approach to non-Markovian stochastic Schroedinger equations
International Nuclear Information System (INIS)
Gambetta, Jay; Wiseman, H.M.
2002-01-01
In this paper we present a perturbative procedure that allows one to numerically solve diffusive non-Markovian stochastic Schroedinger equations, for a wide range of memory functions. To illustrate this procedure numerical results are presented for a classically driven two-level atom immersed in an environment with a simple memory function. It is observed that as the order of the perturbation is increased the numerical results for the ensemble average state ρ red (t) approach the exact reduced state found via Imamog-barlu ' s enlarged system method [Phys. Rev. A 50, 3650 (1994)
Numerical methods for stochastic partial differential equations with white noise
Zhang, Zhongqiang
2017-01-01
This book covers numerical methods for stochastic partial differential equations with white noise using the framework of Wong-Zakai approximation. The book begins with some motivational and background material in the introductory chapters and is divided into three parts. Part I covers numerical stochastic ordinary differential equations. Here the authors start with numerical methods for SDEs with delay using the Wong-Zakai approximation and finite difference in time. Part II covers temporal white noise. Here the authors consider SPDEs as PDEs driven by white noise, where discretization of white noise (Brownian motion) leads to PDEs with smooth noise, which can then be treated by numerical methods for PDEs. In this part, recursive algorithms based on Wiener chaos expansion and stochastic collocation methods are presented for linear stochastic advection-diffusion-reaction equations. In addition, stochastic Euler equations are exploited as an application of stochastic collocation methods, where a numerical compa...
On Volatility Induced Stationarity for Stochastic Differential Equations
DEFF Research Database (Denmark)
Albin, J.M.P.; Astrup Jensen, Bjarne; Muszta, Anders
2006-01-01
This article deals with stochastic differential equations with volatility induced stationarity. We study of theoretical properties of such equations, as well as numerical aspects, together with a detailed study of three examples.......This article deals with stochastic differential equations with volatility induced stationarity. We study of theoretical properties of such equations, as well as numerical aspects, together with a detailed study of three examples....
Symmetries of stochastic differential equations: A geometric approach
Energy Technology Data Exchange (ETDEWEB)
De Vecchi, Francesco C., E-mail: francesco.devecchi@unimi.it; Ugolini, Stefania, E-mail: stefania.ugolini@unimi.it [Dipartimento di Matematica, Università degli Studi di Milano, via Saldini 50, Milano (Italy); Morando, Paola, E-mail: paola.morando@unimi.it [DISAA, Università degli Studi di Milano, via Celoria 2, Milano (Italy)
2016-06-15
A new notion of stochastic transformation is proposed and applied to the study of both weak and strong symmetries of stochastic differential equations (SDEs). The correspondence between an algebra of weak symmetries for a given SDE and an algebra of strong symmetries for a modified SDE is proved under suitable regularity assumptions. This general approach is applied to a stochastic version of a two dimensional symmetric ordinary differential equation and to the case of two dimensional Brownian motion.
Numerical Analysis for Stochastic Partial Differential Delay Equations with Jumps
Li, Yan; Hu, Junhao
2013-01-01
We investigate the convergence rate of Euler-Maruyama method for a class of stochastic partial differential delay equations driven by both Brownian motion and Poisson point processes. We discretize in space by a Galerkin method and in time by using a stochastic exponential integrator. We generalize some results of Bao et al. (2011) and Jacob et al. (2009) in finite dimensions to a class of stochastic partial differential delay equations with jumps in infinite dimensions.
On Some Fractional Stochastic Integrodifferential Equations in Hilbert Space
Directory of Open Access Journals (Sweden)
Hamdy M. Ahmed
2009-01-01
Full Text Available We study a class of fractional stochastic integrodifferential equations considered in a real Hilbert space. The existence and uniqueness of the Mild solutions of the considered problem is also studied. We also give an application for stochastic integropartial differential equations of fractional order.
Stochastic equations for complex systems theoretical and computational topics
Bessaih, Hakima
2015-01-01
Mathematical analyses and computational predictions of the behavior of complex systems are needed to effectively deal with weather and climate predictions, for example, and the optimal design of technical processes. Given the random nature of such systems and the recognized relevance of randomness, the equations used to describe such systems usually need to involve stochastics. The basic goal of this book is to introduce the mathematics and application of stochastic equations used for the modeling of complex systems. A first focus is on the introduction to different topics in mathematical analysis. A second focus is on the application of mathematical tools to the analysis of stochastic equations. A third focus is on the development and application of stochastic methods to simulate turbulent flows as seen in reality. This book is primarily oriented towards mathematics and engineering PhD students, young and experienced researchers, and professionals working in the area of stochastic differential equations ...
On the Langevin equation for stochastic quantization of gravity
International Nuclear Information System (INIS)
Nakazawa, Naohito.
1989-10-01
We study the Langevin equation for stochastic quantization of gravity. By introducing two independent variables with a second-class constraint for the gravitational field, we formulate a pair of the Langevin equations for gravity which couples with white noises. After eliminating the multiplier field for the second-class constraint, we show that the equations leads to stochastic quantization of gravity including an unique superspace metric. (author)
Symmetries of th-Order Approximate Stochastic Ordinary Differential Equations
Fredericks, E.; Mahomed, F. M.
2012-01-01
Symmetries of $n$ th-order approximate stochastic ordinary differential equations (SODEs) are studied. The determining equations of these SODEs are derived in an Itô calculus context. These determining equations are not stochastic in nature. SODEs are normally used to model nature (e.g., earthquakes) or for testing the safety and reliability of models in construction engineering when looking at the impact of random perturbations.
A general comparison theorem for backward stochastic differential equations
Cohen, Samuel N.; Elliott, Robert J.; Pearce, Charles E. M.
2010-01-01
A useful result when dealing with backward stochastic differential equations is the comparison theorem of Peng (1992). When the equations are not based on Brownian motion, the comparison theorem no longer holds in general. In this paper we present a condition for a comparison theorem to hold for backward stochastic differential equations based on arbitrary martingales. This theorem applies to both vector and scalar situations. Applications to the theory of nonlinear expectat...
Neutral Backward Stochastic Functional Differential Equations and Their Application
Wei, Wenning
2013-01-01
In this paper we are concerned with a new type of backward equations with anticipation which we call neutral backward stochastic functional differential equations. We obtain the existence and uniqueness and prove a comparison theorem. As an application, we discuss the optimal control of neutral stochastic functional differential equations, establish a Pontryagin maximum principle, and give an explicit optimal value for the linear optimal control.
Backward stochastic differential equations from linear to fully nonlinear theory
Zhang, Jianfeng
2017-01-01
This book provides a systematic and accessible approach to stochastic differential equations, backward stochastic differential equations, and their connection with partial differential equations, as well as the recent development of the fully nonlinear theory, including nonlinear expectation, second order backward stochastic differential equations, and path dependent partial differential equations. Their main applications and numerical algorithms, as well as many exercises, are included. The book focuses on ideas and clarity, with most results having been solved from scratch and most theories being motivated from applications. It can be considered a starting point for junior researchers in the field, and can serve as a textbook for a two-semester graduate course in probability theory and stochastic analysis. It is also accessible for graduate students majoring in financial engineering.
PC analysis of stochastic differential equations driven by Wiener noise
Le Maitre, Olivier
2015-03-01
A polynomial chaos (PC) analysis with stochastic expansion coefficients is proposed for stochastic differential equations driven by additive or multiplicative Wiener noise. It is shown that for this setting, a Galerkin formalism naturally leads to the definition of a hierarchy of stochastic differential equations governing the evolution of the PC modes. Under the mild assumption that the Wiener and uncertain parameters can be treated as independent random variables, it is also shown that the Galerkin formalism naturally separates parametric uncertainty and stochastic forcing dependences. This enables us to perform an orthogonal decomposition of the process variance, and consequently identify contributions arising from the uncertainty in parameters, the stochastic forcing, and a coupled term. Insight gained from this decomposition is illustrated in light of implementation to simplified linear and non-linear problems; the case of a stochastic bifurcation is also considered.
Analysis of stability for stochastic delay integro-differential equations.
Zhang, Yu; Li, Longsuo
2018-01-01
In this paper, we concern stability of numerical methods applied to stochastic delay integro-differential equations. For linear stochastic delay integro-differential equations, it is shown that the mean-square stability is derived by the split-step backward Euler method without any restriction on step-size, while the Euler-Maruyama method could reproduce the mean-square stability under a step-size constraint. We also confirm the mean-square stability of the split-step backward Euler method for nonlinear stochastic delay integro-differential equations. The numerical experiments further verify the theoretical results.
Jacobian elliptic function expansion solutions of nonlinear stochastic equations
International Nuclear Information System (INIS)
Wei Caimin; Xia Zunquan; Tian Naishuo
2005-01-01
Jacobian elliptic function expansion method is extended and applied to construct the exact solutions of the nonlinear Wick-type stochastic partial differential equations (SPDEs) and some new exact solutions are obtained via this method and Hermite transformation
Almost Periodic Solutions for Impulsive Fractional Stochastic Evolution Equations
Directory of Open Access Journals (Sweden)
Toufik Guendouzi
2014-08-01
Full Text Available In this paper, we consider the existence of square-mean piecewise almost periodic solutions for impulsive fractional stochastic evolution equations involving Caputo fractional derivative. The main results are obtained by means of the theory of operators semi-group, fractional calculus, fixed point technique and stochastic analysis theory and methods adopted directly from deterministic fractional equations. Some known results are improved and generalized.
Introduction to stochastic analysis integrals and differential equations
Mackevicius, Vigirdas
2013-01-01
This is an introduction to stochastic integration and stochastic differential equations written in an understandable way for a wide audience, from students of mathematics to practitioners in biology, chemistry, physics, and finances. The presentation is based on the naïve stochastic integration, rather than on abstract theories of measure and stochastic processes. The proofs are rather simple for practitioners and, at the same time, rather rigorous for mathematicians. Detailed application examples in natural sciences and finance are presented. Much attention is paid to simulation diffusion pro
Renormalization group equations in the stochastic quantization scheme
International Nuclear Information System (INIS)
Pugnetti, S.
1987-01-01
We show that there exists a remarkable link between the stochastic quantization and the theory of critical phenomena and dynamical statistical systems. In the stochastic quantization of a field theory, the stochastic Green functions coverge to the quantum ones when the frictious time goes to infinity. We therefore use the typical techniques of the Renormalization Group equations developed in the framework of critical phenomena to discuss some features of the convergence of the stochastic theory. We are also able, in this way, to compute some dynamical critical exponents and give new numerical valuations for them. (orig.)
Numerical studies of the stochastic Korteweg-de Vries equation
International Nuclear Information System (INIS)
Lin Guang; Grinberg, Leopold; Karniadakis, George Em
2006-01-01
We present numerical solutions of the stochastic Korteweg-de Vries equation for three cases corresponding to additive time-dependent noise, multiplicative space-dependent noise and a combination of the two. We employ polynomial chaos for discretization in random space, and discontinuous Galerkin and finite difference for discretization in physical space. The accuracy of the stochastic solutions is investigated by comparing the first two moments against analytical and Monte Carlo simulation results. Of particular interest is the interplay of spatial discretization error with the stochastic approximation error, which is examined for different orders of spatial and stochastic approximation
Theory of time-averaged neutral dynamics with environmental stochasticity
Danino, Matan; Shnerb, Nadav M.
2018-04-01
Competition is the main driver of population dynamics, which shapes the genetic composition of populations and the assembly of ecological communities. Neutral models assume that all the individuals are equivalent and that the dynamics is governed by demographic (shot) noise, with a steady state species abundance distribution (SAD) that reflects a mutation-extinction equilibrium. Recently, many empirical and theoretical studies emphasized the importance of environmental variations that affect coherently the relative fitness of entire populations. Here we consider two generic time-averaged neutral models; in both the relative fitness of each species fluctuates independently in time but its mean is zero. The first (model A) describes a system with local competition and linear fitness dependence of the birth-death rates, while in the second (model B) the competition is global and the fitness dependence is nonlinear. Due to this nonlinearity, model B admits a noise-induced stabilization mechanism that facilitates the invasion of new mutants. A self-consistent mean-field approach is used to reduce the multispecies problem to two-species dynamics, and the large-N asymptotics of the emerging set of Fokker-Planck equations is presented and solved. Our analytic expressions are shown to fit the SADs obtained from extensive Monte Carlo simulations and from numerical solutions of the corresponding master equations.
Effective action for stochastic partial differential equations
International Nuclear Information System (INIS)
Hochberg, David; Molina-Paris, Carmen; Perez-Mercader, Juan; Visser, Matt
1999-01-01
Stochastic partial differential equations (SPDEs) are the basic tool for modeling systems where noise is important. SPDEs are used for models of turbulence, pattern formation, and the structural development of the universe itself. It is reasonably well known that certain SPDEs can be manipulated to be equivalent to (nonquantum) field theories that nevertheless exhibit deep and important relationships with quantum field theory. In this paper we systematically extend these ideas: We set up a functional integral formalism and demonstrate how to extract all the one-loop physics for an arbitrary SPDE subject to arbitrary Gaussian noise. It is extremely important to realize that Gaussian noise does not imply that the field variables undergo Gaussian fluctuations, and that these nonquantum field theories are fully interacting. The limitation to one loop is not as serious as might be supposed: Experience with quantum field theories (QFTs) has taught us that one-loop physics is often quite adequate to give a good description of the salient issues. The limitation to one loop does, however, offer marked technical advantages: Because at one loop almost any field theory can be rendered finite using zeta function technology, we can sidestep the complications inherent in the Martin-Siggia-Rose formalism (the SPDE analog of the Becchi-Rouet-Stora-Tyutin formalism used in QFT) and instead focus attention on a minimalist approach that uses only the physical fields (this ''direct approach'' is the SPDE analog of canonical quantization using physical fields). After setting up the general formalism for the characteristic functional (partition function), we show how to define the effective action to all loops, and then focus on the one-loop effective action and its specialization to constant fields: the effective potential. The physical interpretation of the effective action and effective potential for SPDEs is addressed and we show that key features carry over from QFT to the case of
Effective action for stochastic partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Hochberg, David [Laboratorio de Astrofisica Espacial y Fisica Fundamental, Apartado 50727, 28080 Madrid, (Spain); Centro de Astrobiologia, INTA, Carratera Ajalvir, Km. 4, 28850 Torrejon, Madrid, (Spain); Molina-Paris, Carmen [Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (United States); Perez-Mercader, Juan [Laboratorio de Astrofisica Espacial y Fisica Fundamental, Apartado 50727, 28080 Madrid, (Spain); Visser, Matt [Physics Department, Washington University, Saint Louis, Missouri 63130-4899 (United States)
1999-12-01
Stochastic partial differential equations (SPDEs) are the basic tool for modeling systems where noise is important. SPDEs are used for models of turbulence, pattern formation, and the structural development of the universe itself. It is reasonably well known that certain SPDEs can be manipulated to be equivalent to (nonquantum) field theories that nevertheless exhibit deep and important relationships with quantum field theory. In this paper we systematically extend these ideas: We set up a functional integral formalism and demonstrate how to extract all the one-loop physics for an arbitrary SPDE subject to arbitrary Gaussian noise. It is extremely important to realize that Gaussian noise does not imply that the field variables undergo Gaussian fluctuations, and that these nonquantum field theories are fully interacting. The limitation to one loop is not as serious as might be supposed: Experience with quantum field theories (QFTs) has taught us that one-loop physics is often quite adequate to give a good description of the salient issues. The limitation to one loop does, however, offer marked technical advantages: Because at one loop almost any field theory can be rendered finite using zeta function technology, we can sidestep the complications inherent in the Martin-Siggia-Rose formalism (the SPDE analog of the Becchi-Rouet-Stora-Tyutin formalism used in QFT) and instead focus attention on a minimalist approach that uses only the physical fields (this ''direct approach'' is the SPDE analog of canonical quantization using physical fields). After setting up the general formalism for the characteristic functional (partition function), we show how to define the effective action to all loops, and then focus on the one-loop effective action and its specialization to constant fields: the effective potential. The physical interpretation of the effective action and effective potential for SPDEs is addressed and we show that key features carry over from
Derivation of exact master equation with stochastic description: dissipative harmonic oscillator.
Li, Haifeng; Shao, Jiushu; Wang, Shikuan
2011-11-01
A systematic procedure for deriving the master equation of a dissipative system is reported in the framework of stochastic description. For the Caldeira-Leggett model of the harmonic-oscillator bath, a detailed and elementary derivation of the bath-induced stochastic field is presented. The dynamics of the system is thereby fully described by a stochastic differential equation, and the desired master equation would be acquired with statistical averaging. It is shown that the existence of a closed-form master equation depends on the specificity of the system as well as the feature of the dissipation characterized by the spectral density function. For a dissipative harmonic oscillator it is observed that the correlation between the stochastic field due to the bath and the system can be decoupled, and the master equation naturally results. Such an equation possesses the Lindblad form in which time-dependent coefficients are determined by a set of integral equations. It is proved that the obtained master equation is equivalent to the well-known Hu-Paz-Zhang equation based on the path-integral technique. The procedure is also used to obtain the master equation of a dissipative harmonic oscillator in time-dependent fields.
Feynman path integral related to stochastic schroedinger equation
International Nuclear Information System (INIS)
Belavkin, V.P.; Smolyanov, O.G.
1998-01-01
The derivation of the Schroedinger equation describing the continuous measurement process is presented. The representation of the solution of the stochastic Schroedinger equation for continuous measurements is obtained by means of the Feynman path integral. The connection with the heuristic approach to the description of continuous measurements is considered. The connection with the Senon paradox is established [ru
STABILITY OF SOME KIND OF STOCHASTIC DIFFERENTIAL EQUATION
Institute of Scientific and Technical Information of China (English)
无
2011-01-01
In this paper,a kind of stochastic differential equation is investigated and the almost sure exponential stability of the equation is obtained using Gronwall's inequality.Further,we also give other noise intensity function to keep the stability of the system.
COMPARISON THEOREM OF BACKWARD DOUBLY STOCHASTIC DIFFERENTIAL EQUATIONS
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
This paper is devoted to deriving a comparison theorem of solutions to backward doubly stochastic differential equations driven by Brownian motion and backward It-Kunita integral. By the application of this theorem, we give an existence result of the solutions to these equations with continuous coefficients.
Stochastic analysis of complex reaction networks using binomial moment equations.
Barzel, Baruch; Biham, Ofer
2012-09-01
The stochastic analysis of complex reaction networks is a difficult problem because the number of microscopic states in such systems increases exponentially with the number of reactive species. Direct integration of the master equation is thus infeasible and is most often replaced by Monte Carlo simulations. While Monte Carlo simulations are a highly effective tool, equation-based formulations are more amenable to analytical treatment and may provide deeper insight into the dynamics of the network. Here, we present a highly efficient equation-based method for the analysis of stochastic reaction networks. The method is based on the recently introduced binomial moment equations [Barzel and Biham, Phys. Rev. Lett. 106, 150602 (2011)]. The binomial moments are linear combinations of the ordinary moments of the probability distribution function of the population sizes of the interacting species. They capture the essential combinatorics of the reaction processes reflecting their stoichiometric structure. This leads to a simple and transparent form of the equations, and allows a highly efficient and surprisingly simple truncation scheme. Unlike ordinary moment equations, in which the inclusion of high order moments is prohibitively complicated, the binomial moment equations can be easily constructed up to any desired order. The result is a set of equations that enables the stochastic analysis of complex reaction networks under a broad range of conditions. The number of equations is dramatically reduced from the exponential proliferation of the master equation to a polynomial (and often quadratic) dependence on the number of reactive species in the binomial moment equations. The aim of this paper is twofold: to present a complete derivation of the binomial moment equations; to demonstrate the applicability of the moment equations for a representative set of example networks, in which stochastic effects play an important role.
A concise course on stochastic partial differential equations
Prévôt, Claudia
2007-01-01
These lectures concentrate on (nonlinear) stochastic partial differential equations (SPDE) of evolutionary type. All kinds of dynamics with stochastic influence in nature or man-made complex systems can be modelled by such equations. To keep the technicalities minimal we confine ourselves to the case where the noise term is given by a stochastic integral w.r.t. a cylindrical Wiener process.But all results can be easily generalized to SPDE with more general noises such as, for instance, stochastic integral w.r.t. a continuous local martingale. There are basically three approaches to analyze SPDE: the "martingale measure approach", the "mild solution approach" and the "variational approach". The purpose of these notes is to give a concise and as self-contained as possible an introduction to the "variational approach". A large part of necessary background material, such as definitions and results from the theory of Hilbert spaces, are included in appendices.
On the Stochastic Wave Equation with Nonlinear Damping
International Nuclear Information System (INIS)
Kim, Jong Uhn
2008-01-01
We discuss an initial boundary value problem for the stochastic wave equation with nonlinear damping. We establish the existence and uniqueness of a solution. Our method for the existence of pathwise solutions consists of regularization of the equation and data, the Galerkin approximation and an elementary measure-theoretic argument. We also prove the existence of an invariant measure when the equation has pure nonlinear damping
New analytic solutions of stochastic coupled KdV equations
International Nuclear Information System (INIS)
Dai Chaoqing; Chen Junlang
2009-01-01
In this paper, firstly, we use the exp-function method to seek new exact solutions of the Riccati equation. Then, with the help of Hermit transformation, we employ the Riccati equation and its new exact solutions to find new analytic solutions of the stochastic coupled KdV equation in the white noise environment. As some special examples, some analytic solutions can degenerate into these solutions reported in open literatures.
International Nuclear Information System (INIS)
Ge, Gen; Li, ZePeng
2016-01-01
A modified stochastic averaging method on single-degree-of-freedom (SDOF) oscillators under white noise excitations with strongly nonlinearity was proposed. Considering the existing approach dealing with strongly nonlinear SDOFs derived by Zhu and Huang [14, 15] is quite time consuming in calculating the drift coefficient and diffusion coefficients and the expressions of them are considerable long, the so-called He's energy balance method was applied to overcome the minor defect of the Zhu and Huang's method. The modified method can offer more concise approximate expressions of the drift and diffusion coefficients without weakening the accuracy of predicting the responses of the systems too much by giving an averaged frequency beforehand. Three examples, a cubic and quadratic nonlinearity coexisting oscillator, a quadratic nonlinear oscillator under external white noise excitations and an externally excited Duffing–Rayleigh oscillator, were given to illustrate the approach we proposed. The three examples were excited by the Gaussian white noise and the Gaussian colored noise separately. The stationary responses of probability density of amplitudes and energy, together with joint probability density of displacement and velocity are studied to verify the presented approach. The reliability of the systems were also investigated to offer further support. Digital simulations were carried out and the output of that are coincide with the theoretical approximations well.
Gao, Peng
2018-04-01
This work concerns the problem associated with averaging principle for a higher order nonlinear Schrödinger equation perturbed by a oscillating term arising as the solution of a stochastic reaction-diffusion equation evolving with respect to the fast time. This model can be translated into a multiscale stochastic partial differential equations. Stochastic averaging principle is a powerful tool for studying qualitative analysis of stochastic dynamical systems with different time-scales. To be more precise, under suitable conditions, we prove that there is a limit process in which the fast varying process is averaged out and the limit process which takes the form of the higher order nonlinear Schrödinger equation is an average with respect to the stationary measure of the fast varying process. Finally, by using the Khasminskii technique we can obtain the rate of strong convergence for the slow component towards the solution of the averaged equation, and as a consequence, the system can be reduced to a single higher order nonlinear Schrödinger equation with a modified coefficient.
Gao, Peng
2018-06-01
This work concerns the problem associated with averaging principle for a higher order nonlinear Schrödinger equation perturbed by a oscillating term arising as the solution of a stochastic reaction-diffusion equation evolving with respect to the fast time. This model can be translated into a multiscale stochastic partial differential equations. Stochastic averaging principle is a powerful tool for studying qualitative analysis of stochastic dynamical systems with different time-scales. To be more precise, under suitable conditions, we prove that there is a limit process in which the fast varying process is averaged out and the limit process which takes the form of the higher order nonlinear Schrödinger equation is an average with respect to the stationary measure of the fast varying process. Finally, by using the Khasminskii technique we can obtain the rate of strong convergence for the slow component towards the solution of the averaged equation, and as a consequence, the system can be reduced to a single higher order nonlinear Schrödinger equation with a modified coefficient.
Multiple-scale stochastic processes: Decimation, averaging and beyond
Energy Technology Data Exchange (ETDEWEB)
Bo, Stefano, E-mail: stefano.bo@nordita.org [Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-106 91 Stockholm (Sweden); Celani, Antonio [Quantitative Life Sciences, The Abdus Salam International Centre for Theoretical Physics (ICTP), Strada Costiera 11, I-34151 - Trieste (Italy)
2017-02-07
The recent experimental progresses in handling microscopic systems have allowed to probe them at levels where fluctuations are prominent, calling for stochastic modeling in a large number of physical, chemical and biological phenomena. This has provided fruitful applications for established stochastic methods and motivated further developments. These systems often involve processes taking place on widely separated time scales. For an efficient modeling one usually focuses on the slower degrees of freedom and it is of great importance to accurately eliminate the fast variables in a controlled fashion, carefully accounting for their net effect on the slower dynamics. This procedure in general requires to perform two different operations: decimation and coarse-graining. We introduce the asymptotic methods that form the basis of this procedure and discuss their application to a series of physical, biological and chemical examples. We then turn our attention to functionals of the stochastic trajectories such as residence times, counting statistics, fluxes, entropy production, etc. which have been increasingly studied in recent years. For such functionals, the elimination of the fast degrees of freedom can present additional difficulties and naive procedures can lead to blatantly inconsistent results. Homogenization techniques for functionals are less covered in the literature and we will pedagogically present them here, as natural extensions of the ones employed for the trajectories. We will also discuss recent applications of these techniques to the thermodynamics of small systems and their interpretation in terms of information-theoretic concepts.
Mélykúti, Bence; Burrage, Kevin; Zygalakis, Konstantinos C.
2010-01-01
The Chemical Langevin Equation (CLE), which is a stochastic differential equation driven by a multidimensional Wiener process, acts as a bridge between the discrete stochastic simulation algorithm and the deterministic reaction rate equation when
Equations for the stochastic cumulative multiplying chain
Energy Technology Data Exchange (ETDEWEB)
Lewins, J D [Cambridge Univ. (UK). Dept. of Engineering
1980-01-01
The forward and backward equations for the conditional probability of the neutron multiplying chain are derived in a new generalization accounting for the chain length and admitting time dependent properties. These Kolmogorov equations form the basis of a variational and hence complete description of the 'lumped' multiplying system. The equations reduce to the marginal distribution, summed over all chain lengths, and to the simpler equations previously derived for that problem. The method of derivation, direct and in the probability space with the minimum of mathematical manipulations, is perhaps the chief attraction: the equations are also displayed in conventional generating function form. As such, they appear to apply to number of problems in areas of social anthropology, polymer chemistry, genetics and cell biology as well as neutron reactor theory and radiation damage.
Equations for the stochastic cumulative multiplying chain
International Nuclear Information System (INIS)
Lewins, J.D.
1980-01-01
The forward and backward equations for the conditional probability of the neutron multiplying chain are derived in a new generalization accounting for the chain length and admitting time dependent properties. These Kolmogorov equations form the basis of a variational and hence complete description of the 'lumped' multiplying system. The equations reduce to the marginal distribution, summed over all chain lengths, and to the simpler equations previously derived for that problem. The method of derivation, direct and in the probability space with the minimum of mathematical manipulations, is perhaps the chief attraction: the equations are also displayed in conventional generating function form. As such, they appear to apply to number of problems in areas of social anthropology, polymer chemistry, genetics and cell biology as well as neutron reactor theory and radiation damage. (author)
Probabilistic Forecasts of Solar Irradiance by Stochastic Differential Equations
DEFF Research Database (Denmark)
Iversen, Jan Emil Banning; Morales González, Juan Miguel; Møller, Jan Kloppenborg
2014-01-01
approach allows for characterizing both the interdependence structure of prediction errors of short-term solar irradiance and their predictive distribution. Three different stochastic differential equation models are first fitted to a training data set and subsequently evaluated on a one-year test set...... included in probabilistic forecasts may be paramount for decision makers to efficiently make use of this uncertain and variable generation. In this paper, a stochastic differential equation framework for modeling the uncertainty associated with the solar irradiance point forecast is proposed. This modeling...
2–stage stochastic Runge–Kutta for stochastic delay differential equations
Energy Technology Data Exchange (ETDEWEB)
Rosli, Norhayati; Jusoh Awang, Rahimah [Faculty of Industrial Science and Technology, Universiti Malaysia Pahang, Lebuhraya Tun Razak, 26300, Gambang, Pahang (Malaysia); Bahar, Arifah; Yeak, S. H. [Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 Johor Bahru, Johor (Malaysia)
2015-05-15
This paper proposes a newly developed one-step derivative-free method, that is 2-stage stochastic Runge-Kutta (SRK2) to approximate the solution of stochastic delay differential equations (SDDEs) with a constant time lag, r > 0. General formulation of stochastic Runge-Kutta for SDDEs is introduced and Stratonovich Taylor series expansion for numerical solution of SRK2 is presented. Local truncation error of SRK2 is measured by comparing the Stratonovich Taylor expansion of the exact solution with the computed solution. Numerical experiment is performed to assure the validity of the method in simulating the strong solution of SDDEs.
A Simple Stochastic Differential Equation with Discontinuous Drift
DEFF Research Database (Denmark)
Simonsen, Maria; Leth, John-Josef; Schiøler, Henrik
2013-01-01
In this paper we study solutions to stochastic differential equations (SDEs) with discontinuous drift. We apply two approaches: The Euler-Maruyama method and the Fokker-Planck equation and show that a candidate density function based on the Euler-Maruyama method approximates a candidate density...... function based on the stationary Fokker-Planck equation. Furthermore, we introduce a smooth function which approximates the discontinuous drift and apply the Euler-Maruyama method and the Fokker-Planck equation with this input. The point of departure for this work is a particular SDE with discontinuous...
Stochastic fractional differential equations: Modeling, method and analysis
International Nuclear Information System (INIS)
Pedjeu, Jean-C.; Ladde, Gangaram S.
2012-01-01
By introducing a concept of dynamic process operating under multi-time scales in sciences and engineering, a mathematical model described by a system of multi-time scale stochastic differential equations is formulated. The classical Picard–Lindelöf successive approximations scheme is applied to the model validation problem, namely, existence and uniqueness of solution process. Naturally, this leads to the problem of finding closed form solutions of both linear and nonlinear multi-time scale stochastic differential equations of Itô–Doob type. Finally, to illustrate the scope of ideas and presented results, multi-time scale stochastic models for ecological and epidemiological processes in population dynamic are outlined.
Master equations and the theory of stochastic path integrals
Weber, Markus F.; Frey, Erwin
2017-04-01
This review provides a pedagogic and self-contained introduction to master equations and to their representation by path integrals. Since the 1930s, master equations have served as a fundamental tool to understand the role of fluctuations in complex biological, chemical, and physical systems. Despite their simple appearance, analyses of master equations most often rely on low-noise approximations such as the Kramers-Moyal or the system size expansion, or require ad-hoc closure schemes for the derivation of low-order moment equations. We focus on numerical and analytical methods going beyond the low-noise limit and provide a unified framework for the study of master equations. After deriving the forward and backward master equations from the Chapman-Kolmogorov equation, we show how the two master equations can be cast into either of four linear partial differential equations (PDEs). Three of these PDEs are discussed in detail. The first PDE governs the time evolution of a generalized probability generating function whose basis depends on the stochastic process under consideration. Spectral methods, WKB approximations, and a variational approach have been proposed for the analysis of the PDE. The second PDE is novel and is obeyed by a distribution that is marginalized over an initial state. It proves useful for the computation of mean extinction times. The third PDE describes the time evolution of a ‘generating functional’, which generalizes the so-called Poisson representation. Subsequently, the solutions of the PDEs are expressed in terms of two path integrals: a ‘forward’ and a ‘backward’ path integral. Combined with inverse transformations, one obtains two distinct path integral representations of the conditional probability distribution solving the master equations. We exemplify both path integrals in analysing elementary chemical reactions. Moreover, we show how a well-known path integral representation of averaged observables can be recovered from
Master equations and the theory of stochastic path integrals.
Weber, Markus F; Frey, Erwin
2017-04-01
This review provides a pedagogic and self-contained introduction to master equations and to their representation by path integrals. Since the 1930s, master equations have served as a fundamental tool to understand the role of fluctuations in complex biological, chemical, and physical systems. Despite their simple appearance, analyses of master equations most often rely on low-noise approximations such as the Kramers-Moyal or the system size expansion, or require ad-hoc closure schemes for the derivation of low-order moment equations. We focus on numerical and analytical methods going beyond the low-noise limit and provide a unified framework for the study of master equations. After deriving the forward and backward master equations from the Chapman-Kolmogorov equation, we show how the two master equations can be cast into either of four linear partial differential equations (PDEs). Three of these PDEs are discussed in detail. The first PDE governs the time evolution of a generalized probability generating function whose basis depends on the stochastic process under consideration. Spectral methods, WKB approximations, and a variational approach have been proposed for the analysis of the PDE. The second PDE is novel and is obeyed by a distribution that is marginalized over an initial state. It proves useful for the computation of mean extinction times. The third PDE describes the time evolution of a 'generating functional', which generalizes the so-called Poisson representation. Subsequently, the solutions of the PDEs are expressed in terms of two path integrals: a 'forward' and a 'backward' path integral. Combined with inverse transformations, one obtains two distinct path integral representations of the conditional probability distribution solving the master equations. We exemplify both path integrals in analysing elementary chemical reactions. Moreover, we show how a well-known path integral representation of averaged observables can be recovered from them. Upon
New derivation of quantum equations from classical stochastic arguments
Bergeron, H.
2003-01-01
In a previous article [H. Bergeron, J. Math. Phys. 42, 3983 (2001)], we presented a method to obtain a continuous transition from classical to quantum mechanics starting from the usual phase space formulation of classical mechanics. This procedure was based on a Koopman-von Neumann approach where classical equations are reformulated into a quantumlike form. In this article, we develop a different derivation of quantum equations, based on purely classical stochastic arguments, taking some elem...
Derivation of Stochastic Equations for Computational Uncertainties ...
African Journals Online (AJOL)
ADOWIE PERE
Investment, Harvard Business Review, 42, No.1, p. 95-106. Freedman, R., Ausburn, B.E. (1985). “The Waxman-. Smits Equation for Shaly Sands: Simple Methods of Solution, Error Analysis'': The Log Analyst,. March-April, pp11-24. Hook, J. R, (1983). “The Precision of Core Analysis. Data and Some Implication for Reservoir.
Derivation of the Schroedinger equation from stochastic mechanics
International Nuclear Information System (INIS)
Wallstrom, T.C.
1988-01-01
The thesis is divided into four largely independent chapters. The first three chapters treat mathematical problems in the theory of stochastic mechanics. The fourth chapter deals with stochastic mechanisms as a physical theory and shows that the Schroedinger equation cannot be derived from existing formulations of stochastic mechanics, as had previously been believed. Since the drift coefficients of stochastic mechanical diffusions are undefined on the nodes, or zeros of the density, an important problem has been to show that the sample paths stay away from the nodes. In Chapter 1, it is shown that for a smooth wavefunction, the closest approach to the nodes can be bounded solely in terms of the time-integrated energy. The ergodic properties of stochastic mechanical diffusions are greatly complicated by the tendency of the particles to avoid the nodes. In Chapter 2, it is shown that a sufficient condition for a stationary process to be ergodic is that there exist positive t and c such that for all x and y, p t (x,y) > cp(y), and this result is applied to show that the set of spin-1/2 diffusions is uniformly ergodic. Nelson has conjectured that in the limit as the particle's moment of inertia I goes to zero, the projections of the Bopp-Haag-Dankel diffusions onto IR 3 converge to a Markovian limit process. This conjecture is proved for the spin-1/2 case in Chapter 3, and the limit process identified as the diffusion naturally associated with the solution to the regular Pauli equation. In Chapter 4 it is shown that the general solution of the stochastic Newton equation does not correspond to a solution of the Schroedinger equation
A stochastic differential equation framework for the turbulent velocity field
DEFF Research Database (Denmark)
Barndorff-Nielsen, Ole Eiler; Schmiegel, Jürgen
We discuss a stochastic differential equation, as a modelling framework for the turbulent velocity field, that is capable of capturing basic stylized facts of the statistics of velocity increments. In particular, we focus on the evolution of the probability density of velocity increments...
Mild Solutions of Neutral Stochastic Partial Functional Differential Equations
Directory of Open Access Journals (Sweden)
T. E. Govindan
2011-01-01
Full Text Available This paper studies the existence and uniqueness of a mild solution for a neutral stochastic partial functional differential equation using a local Lipschitz condition. When the neutral term is zero and even in the deterministic special case, the result obtained here appears to be new. An example is included to illustrate the theory.
A class of degenerate stochastic differential equations with non ...
Indian Academy of Sciences (India)
Introduction. In this article we consider (possibly degenerate) stochastic differential equations (SDEs) with non-Lipschitz coefficients. If the coefficients are Lipschitz, we can prove the existence of a unique strong solution (see [9]). But uniqueness fails in the case of non-Lipschitz coefficients. The literature on this topic is not ...
STABILITY OF STOCHASTIC DIFFERENTIAL EQUATIONS WITH UNBOUNDED DELAY
Institute of Scientific and Technical Information of China (English)
无
2009-01-01
In this paper,we obtain suffcient conditions for the stability in p-th moment of the analytical solutions and the mean square stability of a stochastic differential equation with unbounded delay proposed in [6,10] using the explicit Euler method.
Efficient Numerical Methods for Stochastic Differential Equations in Computational Finance
Happola, Juho
2017-09-19
Stochastic Differential Equations (SDE) offer a rich framework to model the probabilistic evolution of the state of a system. Numerical approximation methods are typically needed in evaluating relevant Quantities of Interest arising from such models. In this dissertation, we present novel effective methods for evaluating Quantities of Interest relevant to computational finance when the state of the system is described by an SDE.
Reflected backward stochastic differential equations in an orthant
Indian Academy of Sciences (India)
R. Narasimhan (Krishtel eMaging) 1461 1996 Oct 15 13:05:22
Since backward stochastic differential equations were introduced about a decade back there has been a lot .... is given in ([10] pp. 76–80) in connection with financial networks; these authors call ..... Applying Theorem 3.2 to (̂Y. (k) i. , ̂Z. (k).
Efficient Numerical Methods for Stochastic Differential Equations in Computational Finance
Happola, Juho
2017-01-01
Stochastic Differential Equations (SDE) offer a rich framework to model the probabilistic evolution of the state of a system. Numerical approximation methods are typically needed in evaluating relevant Quantities of Interest arising from such models. In this dissertation, we present novel effective methods for evaluating Quantities of Interest relevant to computational finance when the state of the system is described by an SDE.
High Weak Order Methods for Stochastic Differential Equations Based on Modified Equations
Abdulle, Assyr
2012-01-01
© 2012 Society for Industrial and Applied Mathematics. Inspired by recent advances in the theory of modified differential equations, we propose a new methodology for constructing numerical integrators with high weak order for the time integration of stochastic differential equations. This approach is illustrated with the constructions of new methods of weak order two, in particular, semi-implicit integrators well suited for stiff (meansquare stable) stochastic problems, and implicit integrators that exactly conserve all quadratic first integrals of a stochastic dynamical system. Numerical examples confirm the theoretical results and show the versatility of our methodology.
Approximate Controllability for Linear Stochastic Differential Equations in Infinite Dimensions
International Nuclear Information System (INIS)
Goreac, D.
2009-01-01
The objective of the paper is to investigate the approximate controllability property of a linear stochastic control system with values in a separable real Hilbert space. In a first step we prove the existence and uniqueness for the solution of the dual linear backward stochastic differential equation. This equation has the particularity that in addition to an unbounded operator acting on the Y-component of the solution there is still another one acting on the Z-component. With the help of this dual equation we then deduce the duality between approximate controllability and observability. Finally, under the assumption that the unbounded operator acting on the state process of the forward equation is an infinitesimal generator of an exponentially stable semigroup, we show that the generalized Hautus test provides a necessary condition for the approximate controllability. The paper generalizes former results by Buckdahn, Quincampoix and Tessitore (Stochastic Partial Differential Equations and Applications, Series of Lecture Notes in Pure and Appl. Math., vol. 245, pp. 253-260, Chapman and Hall, London, 2006) and Goreac (Applied Analysis and Differential Equations, pp. 153-164, World Scientific, Singapore, 2007) from the finite dimensional to the infinite dimensional case
Stochastic Averaging Principle for Spatial Birth-and-Death Evolutions in the Continuum
Friesen, Martin; Kondratiev, Yuri
2018-06-01
We study a spatial birth-and-death process on the phase space of locally finite configurations Γ^+ × Γ^- over R}^d. Dynamics is described by an non-equilibrium evolution of states obtained from the Fokker-Planck equation and associated with the Markov operator L^+(γ ^-) + 1/ɛ L^-, ɛ > 0. Here L^- describes the environment process on Γ^- and L^+(γ ^-) describes the system process on Γ^+, where γ ^- indicates that the corresponding birth-and-death rates depend on another locally finite configuration γ ^- \\in Γ^-. We prove that, for a certain class of birth-and-death rates, the corresponding Fokker-Planck equation is well-posed, i.e. there exists a unique evolution of states μ _t^{ɛ } on Γ^+ × Γ^-. Moreover, we give a sufficient condition such that the environment is ergodic with exponential rate. Let μ _{inv} be the invariant measure for the environment process on Γ^-. In the main part of this work we establish the stochastic averaging principle, i.e. we prove that the marginal of μ _t^{ɛ } onto Γ^+ converges weakly to an evolution of states on {Γ}^+ associated with the averaged Markov birth-and-death operator {\\overline{L}} = \\int _{Γ}^- L^+(γ ^-)d μ _{inv}(γ ^-).
Stochastic Averaging Principle for Spatial Birth-and-Death Evolutions in the Continuum
Friesen, Martin; Kondratiev, Yuri
2018-04-01
We study a spatial birth-and-death process on the phase space of locally finite configurations Γ^+ × Γ^- over R^d . Dynamics is described by an non-equilibrium evolution of states obtained from the Fokker-Planck equation and associated with the Markov operator L^+(γ ^-) + 1/ɛ L^- , ɛ > 0 . Here L^- describes the environment process on Γ^- and L^+(γ ^-) describes the system process on Γ^+ , where γ ^- indicates that the corresponding birth-and-death rates depend on another locally finite configuration γ ^- \\in Γ^- . We prove that, for a certain class of birth-and-death rates, the corresponding Fokker-Planck equation is well-posed, i.e. there exists a unique evolution of states μ _t^{ɛ } on Γ^+ × Γ^- . Moreover, we give a sufficient condition such that the environment is ergodic with exponential rate. Let μ _{inv} be the invariant measure for the environment process on Γ^- . In the main part of this work we establish the stochastic averaging principle, i.e. we prove that the marginal of μ _t^{ɛ } onto Γ^+ converges weakly to an evolution of states on Γ^+ associated with the averaged Markov birth-and-death operator \\overline{L} = \\int _{Γ}^-}L^+(γ ^-)d μ _{inv}(γ ^-).
Chekroun, Mickaël D; Wang, Shouhong
2015-01-01
In this second volume, a general approach is developed to provide approximate parameterizations of the "small" scales by the "large" ones for a broad class of stochastic partial differential equations (SPDEs). This is accomplished via the concept of parameterizing manifolds (PMs), which are stochastic manifolds that improve, for a given realization of the noise, in mean square error the partial knowledge of the full SPDE solution when compared to its projection onto some resolved modes. Backward-forward systems are designed to give access to such PMs in practice. The key idea consists of representing the modes with high wave numbers as a pullback limit depending on the time-history of the modes with low wave numbers. Non-Markovian stochastic reduced systems are then derived based on such a PM approach. The reduced systems take the form of stochastic differential equations involving random coefficients that convey memory effects. The theory is illustrated on a stochastic Burgers-type equation.
Improved stochastic approximation methods for discretized parabolic partial differential equations
Guiaş, Flavius
2016-12-01
We present improvements of the stochastic direct simulation method, a known numerical scheme based on Markov jump processes which is used for approximating solutions of ordinary differential equations. This scheme is suited especially for spatial discretizations of evolution partial differential equations (PDEs). By exploiting the full path simulation of the stochastic method, we use this first approximation as a predictor and construct improved approximations by Picard iterations, Runge-Kutta steps, or a combination. This has as consequence an increased order of convergence. We illustrate the features of the improved method at a standard benchmark problem, a reaction-diffusion equation modeling a combustion process in one space dimension (1D) and two space dimensions (2D).
Application of Stochastic Partial Differential Equations to Reservoir Property Modelling
Potsepaev, R.
2010-09-06
Existing algorithms of geostatistics for stochastic modelling of reservoir parameters require a mapping (the \\'uvt-transform\\') into the parametric space and reconstruction of a stratigraphic co-ordinate system. The parametric space can be considered to represent a pre-deformed and pre-faulted depositional environment. Existing approximations of this mapping in many cases cause significant distortions to the correlation distances. In this work we propose a coordinate free approach for modelling stochastic textures through the application of stochastic partial differential equations. By avoiding the construction of a uvt-transform and stratigraphic coordinates, one can generate realizations directly in the physical space in the presence of deformations and faults. In particular the solution of the modified Helmholtz equation driven by Gaussian white noise is a zero mean Gaussian stationary random field with exponential correlation function (in 3-D). This equation can be used to generate realizations in parametric space. In order to sample in physical space we introduce a stochastic elliptic PDE with tensor coefficients, where the tensor is related to correlation anisotropy and its variation is physical space.
Stochastic Calculus and Differential Equations for Physics and Finance
McCauley, Joseph L.
2013-02-01
1. Random variables and probability distributions; 2. Martingales, Markov, and nonstationarity; 3. Stochastic calculus; 4. Ito processes and Fokker-Planck equations; 5. Selfsimilar Ito processes; 6. Fractional Brownian motion; 7. Kolmogorov's PDEs and Chapman-Kolmogorov; 8. Non Markov Ito processes; 9. Black-Scholes, martingales, and Feynman-Katz; 10. Stochastic calculus with martingales; 11. Statistical physics and finance, a brief history of both; 12. Introduction to new financial economics; 13. Statistical ensembles and time series analysis; 14. Econometrics; 15. Semimartingales; References; Index.
Malliavin Calculus With Applications to Stochastic Partial Differential Equations
Sanz-Solé, Marta
2005-01-01
Developed in the 1970s to study the existence and smoothness of density for the probability laws of random vectors, Malliavin calculus--a stochastic calculus of variation on the Wiener space--has proven fruitful in many problems in probability theory, particularly in probabilistic numerical methods in financial mathematics.This book presents applications of Malliavin calculus to the analysis of probability laws of solutions to stochastic partial differential equations driven by Gaussian noises that are white in time and coloured in space. The first five chapters introduce the calculus itself
Linear stochastic differential equations with anticipating initial conditions
DEFF Research Database (Denmark)
Khalifa, Narjess; Kuo, Hui-Hsiung; Ouerdiane, Habib
In this paper we use the new stochastic integral introduced by Ayed and Kuo (2008) and the results obtained by Kuo et al. (2012b) to find a solution to a drift-free linear stochastic differential equation with anticipating initial condition. Our solution is based on well-known results from...... classical Itô theory and anticipative Itô formula results from Kue et al. (2012b). We also show that the solution obtained by our method is consistent with the solution obtained by the methods of Malliavin calculus, e.g. Buckdahn and Nualart (1994)....
Indirect Inference for Stochastic Differential Equations Based on Moment Expansions
Ballesio, Marco
2016-01-06
We provide an indirect inference method to estimate the parameters of timehomogeneous scalar diffusion and jump diffusion processes. We obtain a system of ODEs that approximate the time evolution of the first two moments of the process by the approximation of the stochastic model applying a second order Taylor expansion of the SDE s infinitesimal generator in the Dynkin s formula. This method allows a simple and efficient procedure to infer the parameters of such stochastic processes given the data by the maximization of the likelihood of an approximating Gaussian process described by the two moments equations. Finally, we perform numerical experiments for two datasets arising from organic and inorganic fouling deposition phenomena.
Reflected stochastic differential equation models for constrained animal movement
Hanks, Ephraim M.; Johnson, Devin S.; Hooten, Mevin B.
2017-01-01
Movement for many animal species is constrained in space by barriers such as rivers, shorelines, or impassable cliffs. We develop an approach for modeling animal movement constrained in space by considering a class of constrained stochastic processes, reflected stochastic differential equations. Our approach generalizes existing methods for modeling unconstrained animal movement. We present methods for simulation and inference based on augmenting the constrained movement path with a latent unconstrained path and illustrate this augmentation with a simulation example and an analysis of telemetry data from a Steller sea lion (Eumatopias jubatus) in southeast Alaska.
Study of the stochastic point reactor kinetic equation
International Nuclear Information System (INIS)
Gotoh, Yorio
1980-01-01
Diagrammatic technique is used to solve the stochastic point reactor kinetic equation. The method gives exact results which are derived from Fokker-Plank theory. A Green's function dressed with the clouds of noise is defined, which is a transfer function of point reactor with fluctuating reactivity. An integral equation for the correlation function of neutron power is derived using the following assumptions: 1) Green's funntion should be dressed with noise, 2) The ladder type diagrams only contributes to the correlation function. For a white noise and the one delayed neutron group approximation, the norm of the integral equation and the variance to mean-squared ratio are analytically obtained. (author)
Ren, Jiagang; Wu, Jing; Zhang, Hua
2015-01-01
In this paper, we prove a large deviation principle of Freidlin-Wentzell's type for the multivalued stochastic differential equations. As an application, we derive a functional iterated logarithm law for the solutions of multivalued stochastic differential equations.
On the Existence and the Applications of Modified Equations for Stochastic Differential Equations
Zygalakis, K. C.
2011-01-01
In this paper we describe a general framework for deriving modified equations for stochastic differential equations (SDEs) with respect to weak convergence. Modified equations are derived for a variety of numerical methods, such as the Euler or the Milstein method. Existence of higher order modified equations is also discussed. In the case of linear SDEs, using the Gaussianity of the underlying solutions, we derive an SDE which the numerical method solves exactly in the weak sense. Applications of modified equations in the numerical study of Langevin equations is also discussed. © 2011 Society for Industrial and Applied Mathematics.
EXISTENCE AND UNIQUENESS OF SOLUTIONS TO STOCHASTIC DIFFERENTIAL EQUATION WITH RANDOM COEFFICIENTS
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
This paper mainly deals with a stochastic differential equation (SDE) with random coefficients. Sufficient conditions which guarantee the existence and uniqueness of solutions to the equation are given.
Stochastic solution of population balance equations for reactor networks
International Nuclear Information System (INIS)
Menz, William J.; Akroyd, Jethro; Kraft, Markus
2014-01-01
This work presents a sequential modular approach to solve a generic network of reactors with a population balance model using a stochastic numerical method. Full-coupling to the gas-phase is achieved through operator-splitting. The convergence of the stochastic particle algorithm in test networks is evaluated as a function of network size, recycle fraction and numerical parameters. These test cases are used to identify methods through which systematic and statistical error may be reduced, including by use of stochastic weighted algorithms. The optimal algorithm was subsequently used to solve a one-dimensional example of silicon nanoparticle synthesis using a multivariate particle model. This example demonstrated the power of stochastic methods in resolving particle structure by investigating the transient and spatial evolution of primary polydispersity, degree of sintering and TEM-style images. Highlights: •An algorithm is presented to solve reactor networks with a population balance model. •A stochastic method is used to solve the population balance equations. •The convergence and efficiency of the reported algorithms are evaluated. •The algorithm is applied to simulate silicon nanoparticle synthesis in a 1D reactor. •Particle structure is reported as a function of reactor length and time
Front propagation and clustering in the stochastic nonlocal Fisher equation
Ganan, Yehuda A.; Kessler, David A.
2018-04-01
In this work, we study the problem of front propagation and pattern formation in the stochastic nonlocal Fisher equation. We find a crossover between two regimes: a steadily propagating regime for not too large interaction range and a stochastic punctuated spreading regime for larger ranges. We show that the former regime is well described by the heuristic approximation of the system by a deterministic system where the linear growth term is cut off below some critical density. This deterministic system is seen not only to give the right front velocity, but also predicts the onset of clustering for interaction kernels which give rise to stable uniform states, such as the Gaussian kernel, for sufficiently large cutoff. Above the critical cutoff, distinct clusters emerge behind the front. These same features are present in the stochastic model for sufficiently small carrying capacity. In the latter, punctuated spreading, regime, the population is concentrated on clusters, as in the infinite range case, which divide and separate as a result of the stochastic noise. Due to the finite interaction range, if a fragment at the edge of the population separates sufficiently far, it stabilizes as a new cluster, and the processes begins anew. The deterministic cutoff model does not have this spreading for large interaction ranges, attesting to its purely stochastic origins. We show that this mode of spreading has an exponentially small mean spreading velocity, decaying with the range of the interaction kernel.
Numerical Solution of Stochastic Nonlinear Fractional Differential Equations
El-Beltagy, Mohamed A.; Al-Juhani, Amnah
2015-01-01
Using Wiener-Hermite expansion (WHE) technique in the solution of the stochastic partial differential equations (SPDEs) has the advantage of converting the problem to a system of deterministic equations that can be solved efficiently using the standard deterministic numerical methods [1]. WHE is the only known expansion that handles the white/colored noise exactly. This work introduces a numerical estimation of the stochastic response of the Duffing oscillator with fractional or variable order damping and driven by white noise. The WHE technique is integrated with the Grunwald-Letnikov approximation in case of fractional order and with Coimbra approximation in case of variable-order damping. The numerical solver was tested with the analytic solution and with Monte-Carlo simulations. The developed mixed technique was shown to be efficient in simulating SPDEs.
Numerical Solution of Stochastic Nonlinear Fractional Differential Equations
El-Beltagy, Mohamed A.
2015-01-07
Using Wiener-Hermite expansion (WHE) technique in the solution of the stochastic partial differential equations (SPDEs) has the advantage of converting the problem to a system of deterministic equations that can be solved efficiently using the standard deterministic numerical methods [1]. WHE is the only known expansion that handles the white/colored noise exactly. This work introduces a numerical estimation of the stochastic response of the Duffing oscillator with fractional or variable order damping and driven by white noise. The WHE technique is integrated with the Grunwald-Letnikov approximation in case of fractional order and with Coimbra approximation in case of variable-order damping. The numerical solver was tested with the analytic solution and with Monte-Carlo simulations. The developed mixed technique was shown to be efficient in simulating SPDEs.
Different seeds to solve the equations of stochastic point kinetics using the Euler-Maruyama method
International Nuclear Information System (INIS)
Suescun D, D.; Oviedo T, M.
2017-09-01
In this paper, a numerical study of stochastic differential equations that describe the kinetics in a nuclear reactor is presented. These equations, known as the stochastic equations of punctual kinetics they model temporal variations in neutron population density and concentrations of deferred neutron precursors. Because these equations are probabilistic in nature (since random oscillations in the neutrons and population of precursors were considered to be approximately normally distributed, and these equations also possess strong coupling and stiffness properties) the proposed method for the numerical simulations is the Euler-Maruyama scheme that provides very good approximations for calculating the neutron population and concentrations of deferred neutron precursors. The method proposed for this work was computationally tested for different seeds, initial conditions, experimental data and forms of reactivity for a group of precursors and then for six groups of deferred neutron precursors at each time step with 5000 Brownian movements per seed. In a paper reported in the literature, the Euler-Maruyama method was proposed, but there are many doubts about the reported values, in addition to not reporting the seed used, so in this work is expected to rectify the reported values. After taking the average of the different seeds used to generate the pseudo-random numbers the results provided by the Euler-Maruyama scheme will be compared in mean and standard deviation with other methods reported in the literature and results of the deterministic model of the equations of the punctual kinetics. This comparison confirms in particular that the Euler-Maruyama scheme is an efficient method to solve the equations of stochastic point kinetics but different from the values found and reported by another author. The Euler-Maruyama method is simple and easy to implement, provides acceptable results for neutron population density and concentration of deferred neutron precursors and
A stochastic Galerkin method for the Euler equations with Roe variable transformation
Pettersson, Per; Iaccarino, Gianluca; Nordströ m, Jan
2014-01-01
The Euler equations subject to uncertainty in the initial and boundary conditions are investigated via the stochastic Galerkin approach. We present a new fully intrusive method based on a variable transformation of the continuous equations. Roe variables are employed to get quadratic dependence in the flux function and a well-defined Roe average matrix that can be determined without matrix inversion.In previous formulations based on generalized polynomial chaos expansion of the physical variables, the need to introduce stochastic expansions of inverse quantities, or square roots of stochastic quantities of interest, adds to the number of possible different ways to approximate the original stochastic problem. We present a method where the square roots occur in the choice of variables, resulting in an unambiguous problem formulation.The Roe formulation saves computational cost compared to the formulation based on expansion of conservative variables. Moreover, the Roe formulation is more robust and can handle cases of supersonic flow, for which the conservative variable formulation fails to produce a bounded solution. For certain stochastic basis functions, the proposed method can be made more effective and well-conditioned. This leads to increased robustness for both choices of variables. We use a multi-wavelet basis that can be chosen to include a large number of resolution levels to handle more extreme cases (e.g. strong discontinuities) in a robust way. For smooth cases, the order of the polynomial representation can be increased for increased accuracy. © 2013 Elsevier Inc.
Homogenization of the stochastic Navier–Stokes equation with a stochastic slip boundary condition
Bessaih, Hakima
2015-11-02
The two-dimensional Navier–Stokes equation in a perforated domain with a dynamical slip boundary condition is considered. We assume that the dynamic is driven by a stochastic perturbation on the interior of the domain and another stochastic perturbation on the boundaries of the holes. We consider a scaling (ᵋ for the viscosity and 1 for the density) that will lead to a time-dependent limit problem. However, the noncritical scaling (ᵋ, β > 1) is considered in front of the nonlinear term. The homogenized system in the limit is obtained as a Darcy’s law with memory with two permeabilities and an extra term that is due to the stochastic perturbation on the boundary of the holes. The nonhomogeneity on the boundary contains a stochastic part that yields in the limit an additional term in the Darcy’s law. We use the two-scale convergence method after extending the solution with 0 inside the holes to pass to the limit. By Itô stochastic calculus, we get uniform estimates on the solution in appropriate spaces. Due to the stochastic integral, the pressure that appears in the variational formulation does not have enough regularity in time. This fact made us rely only on the variational formulation for the passage to the limit on the solution. We obtain a variational formulation for the limit that is solution of a Stokes system with two pressures. This two-scale limit gives rise to three cell problems, two of them give the permeabilities while the third one gives an extra term in the Darcy’s law due to the stochastic perturbation on the boundary of the holes.
International Nuclear Information System (INIS)
Wu, Fuke; Tian, Tianhai; Rawlings, James B.; Yin, George
2016-01-01
The frequently used reduction technique is based on the chemical master equation for stochastic chemical kinetics with two-time scales, which yields the modified stochastic simulation algorithm (SSA). For the chemical reaction processes involving a large number of molecular species and reactions, the collection of slow reactions may still include a large number of molecular species and reactions. Consequently, the SSA is still computationally expensive. Because the chemical Langevin equations (CLEs) can effectively work for a large number of molecular species and reactions, this paper develops a reduction method based on the CLE by the stochastic averaging principle developed in the work of Khasminskii and Yin [SIAM J. Appl. Math. 56, 1766–1793 (1996); ibid. 56, 1794–1819 (1996)] to average out the fast-reacting variables. This reduction method leads to a limit averaging system, which is an approximation of the slow reactions. Because in the stochastic chemical kinetics, the CLE is seen as the approximation of the SSA, the limit averaging system can be treated as the approximation of the slow reactions. As an application, we examine the reduction of computation complexity for the gene regulatory networks with two-time scales driven by intrinsic noise. For linear and nonlinear protein production functions, the simulations show that the sample average (expectation) of the limit averaging system is close to that of the slow-reaction process based on the SSA. It demonstrates that the limit averaging system is an efficient approximation of the slow-reaction process in the sense of the weak convergence.
Modelling biochemical reaction systems by stochastic differential equations with reflection.
Niu, Yuanling; Burrage, Kevin; Chen, Luonan
2016-05-07
In this paper, we gave a new framework for modelling and simulating biochemical reaction systems by stochastic differential equations with reflection not in a heuristic way but in a mathematical way. The model is computationally efficient compared with the discrete-state Markov chain approach, and it ensures that both analytic and numerical solutions remain in a biologically plausible region. Specifically, our model mathematically ensures that species numbers lie in the domain D, which is a physical constraint for biochemical reactions, in contrast to the previous models. The domain D is actually obtained according to the structure of the corresponding chemical Langevin equations, i.e., the boundary is inherent in the biochemical reaction system. A variant of projection method was employed to solve the reflected stochastic differential equation model, and it includes three simple steps, i.e., Euler-Maruyama method was applied to the equations first, and then check whether or not the point lies within the domain D, and if not perform an orthogonal projection. It is found that the projection onto the closure D¯ is the solution to a convex quadratic programming problem. Thus, existing methods for the convex quadratic programming problem can be employed for the orthogonal projection map. Numerical tests on several important problems in biological systems confirmed the efficiency and accuracy of this approach. Copyright © 2016 Elsevier Ltd. All rights reserved.
Stochastic Differential Equation-Based Flexible Software Reliability Growth Model
Directory of Open Access Journals (Sweden)
P. K. Kapur
2009-01-01
Full Text Available Several software reliability growth models (SRGMs have been developed by software developers in tracking and measuring the growth of reliability. As the size of software system is large and the number of faults detected during the testing phase becomes large, so the change of the number of faults that are detected and removed through each debugging becomes sufficiently small compared with the initial fault content at the beginning of the testing phase. In such a situation, we can model the software fault detection process as a stochastic process with continuous state space. In this paper, we propose a new software reliability growth model based on Itô type of stochastic differential equation. We consider an SDE-based generalized Erlang model with logistic error detection function. The model is estimated and validated on real-life data sets cited in literature to show its flexibility. The proposed model integrated with the concept of stochastic differential equation performs comparatively better than the existing NHPP-based models.
Stochastic partial differential equations a modeling, white noise functional approach
Holden, Helge; Ubøe, Jan; Zhang, Tusheng
1996-01-01
This book is based on research that, to a large extent, started around 1990, when a research project on fluid flow in stochastic reservoirs was initiated by a group including some of us with the support of VISTA, a research coopera tion between the Norwegian Academy of Science and Letters and Den norske stats oljeselskap A.S. (Statoil). The purpose of the project was to use stochastic partial differential equations (SPDEs) to describe the flow of fluid in a medium where some of the parameters, e.g., the permeability, were stochastic or "noisy". We soon realized that the theory of SPDEs at the time was insufficient to handle such equations. Therefore it became our aim to develop a new mathematically rigorous theory that satisfied the following conditions. 1) The theory should be physically meaningful and realistic, and the corre sponding solutions should make sense physically and should be useful in applications. 2) The theory should be general enough to handle many of the interesting SPDEs that occur in r...
DEFF Research Database (Denmark)
Chon, K H; Hoyer, D; Armoundas, A A
1999-01-01
In this study, we introduce a new approach for estimating linear and nonlinear stochastic autoregressive moving average (ARMA) model parameters, given a corrupt signal, using artificial recurrent neural networks. This new approach is a two-step approach in which the parameters of the deterministic...... part of the stochastic ARMA model are first estimated via a three-layer artificial neural network (deterministic estimation step) and then reestimated using the prediction error as one of the inputs to the artificial neural networks in an iterative algorithm (stochastic estimation step). The prediction...... error is obtained by subtracting the corrupt signal of the estimated ARMA model obtained via the deterministic estimation step from the system output response. We present computer simulation examples to show the efficacy of the proposed stochastic recurrent neural network approach in obtaining accurate...
New Exact Solutions for the Wick-Type Stochastic Kudryashov–Sinelshchikov Equation
International Nuclear Information System (INIS)
Ray, S. Saha; Singh, S.
2017-01-01
In this article, exact solutions of Wick-type stochastic Kudryashov–Sinelshchikov equation have been obtained by using improved Sub-equation method. We have used Hermite transform for transforming the Wick-type stochastic Kudryashov–Sinelshchikov equation to deterministic partial differential equation. Also we have applied inverse Hermite transform for obtaining a set of stochastic solutions in the white noise space. (paper)
Stochastic wave-function unravelling of the generalized Lindblad equation using correlated states
International Nuclear Information System (INIS)
Moodley, Mervlyn; Nsio Nzundu, T; Paul, S
2012-01-01
We perform a stochastic wave-function unravelling of the generalized Lindblad master equation using correlated states, a combination of the system state vectors and the environment population. The time-convolutionless projection operator method using correlated projection superoperators is applied to a two-state system, a qubit, that is coupled to an environment consisting of two energy bands which are both populated. These results are compared to the data obtained from Monte Carlo wave-function simulations based on the unravelling of the master equation. We also show a typical quantum trajectory and the average time evolution of the state vector on the Bloch sphere. (paper)
Stochastic Landau equation with time-dependent drift
International Nuclear Information System (INIS)
Swift, J.B.; Hohenberg, P.C.; Ahlers, G.
1991-01-01
The stochastic differential equation τ 0 ∂ tA =ε(t)A-g 3 A 3 +bar f(t), where bar f(t) is Gaussian white noise, is studied for arbitrary time dependence of ε(t). In particular, cases are considered where ε(t) goes through the bifurcation of the deterministic system, which occurs at ε=0. In the limit of weak noise an approximate analytic expression generalizing earlier work of Suzuki [Phys. Lett. A 67, 339 (1978); Prog. Theor. Phys. (Kyoto) Suppl. 64, 402 (1978)] is obtained for the time-dependent distribution function P(A,t). The results compare favorably with a numerical simulation of the stochastic equation for the case of a linear ramp (both increasing and decreasing) and for a periodic time dependence of ε(t). The procedure can be generalized to an arbitrary deterministic part ∂ tA =D(A,t)+bar f(t), but the deterministic equation may then have to be solved numerically
Stability of Nonlinear Neutral Stochastic Functional Differential Equations
Directory of Open Access Journals (Sweden)
Minggao Xue
2010-01-01
Full Text Available Neutral stochastic functional differential equations (NSFDEs have recently been studied intensively. The well-known conditions imposed for the existence and uniqueness and exponential stability of the global solution are the local Lipschitz condition and the linear growth condition. Therefore, the existing results cannot be applied to many important nonlinear NSFDEs. The main aim of this paper is to remove the linear growth condition and establish a Khasminskii-type test for nonlinear NSFDEs. New criteria not only cover a wide class of highly nonlinear NSFDEs but they can also be verified much more easily than the classical criteria. Finally, several examples are given to illustrate main results.
On Exponential Hedging and Related Quadratic Backward Stochastic Differential Equations
International Nuclear Information System (INIS)
Sekine, Jun
2006-01-01
The dual optimization problem for the exponential hedging problem is addressed with a cone constraint. Without boundedness conditions on the terminal payoff and the drift of the Ito-type controlled process, the backward stochastic differential equation, which has a quadratic growth term in the drift, is derived as a necessary and sufficient condition for optimality via a variational method and dynamic programming. Further, solvable situations are given, in which the value and the optimizer are expressed in closed forms with the help of the Clark-Haussmann-Ocone formula
Infinite time interval backward stochastic differential equations with continuous coefficients.
Zong, Zhaojun; Hu, Feng
2016-01-01
In this paper, we study the existence theorem for [Formula: see text] [Formula: see text] solutions to a class of 1-dimensional infinite time interval backward stochastic differential equations (BSDEs) under the conditions that the coefficients are continuous and have linear growths. We also obtain the existence of a minimal solution. Furthermore, we study the existence and uniqueness theorem for [Formula: see text] [Formula: see text] solutions of infinite time interval BSDEs with non-uniformly Lipschitz coefficients. It should be pointed out that the assumptions of this result is weaker than that of Theorem 3.1 in Zong (Turkish J Math 37:704-718, 2013).
xSPDE: Extensible software for stochastic equations
Directory of Open Access Journals (Sweden)
Simon Kiesewetter
2016-01-01
Full Text Available We introduce an extensible software toolbox, xSPDE, for solving ordinary and partial stochastic differential equations. The toolbox makes extensive use of vector and parallel methods. Inputs are exceptionally simple, to reduce the learning curve, with default options for all of the many input parameters. The code calculates functional means, correlations and spectra, checks for errors in both time-step and sampling, and provides several choices of algorithm. Most aspects of the code, including the numerical algorithm, have a modular functional design to allow user modifications.
Variance estimates for transport in stochastic media by means of the master equation
International Nuclear Information System (INIS)
Pautz, S. D.; Franke, B. C.; Prinja, A. K.
2013-01-01
The master equation has been used to examine properties of transport in stochastic media. It has been shown previously that not only may the Levermore-Pomraning (LP) model be derived from the master equation for a description of ensemble-averaged transport quantities, but also that equations describing higher-order statistical moments may be obtained. We examine in greater detail the equations governing the second moments of the distribution of the angular fluxes, from which variances may be computed. We introduce a simple closure for these equations, as well as several models for estimating the variances of derived transport quantities. We revisit previous benchmarks for transport in stochastic media in order to examine the error of these new variance models. We find, not surprisingly, that the errors in these variance estimates are at least as large as the corresponding estimates of the average, and sometimes much larger. We also identify patterns in these variance estimates that may help guide the construction of more accurate models. (authors)
Set-Valued Stochastic Equation with Set-Valued Square Integrable Martingale
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Li Jun-Gang
2017-01-01
Full Text Available In this paper, we shall introduce the stochastic integral of a stochastic process with respect to set-valued square integrable martingale. Then we shall give the Aumann integral measurable theorem, and give the set-valued stochastic Lebesgue integral and set-valued square integrable martingale integral equation. The existence and uniqueness of solution to set-valued stochastic integral equation are proved. The discussion will be useful in optimal control and mathematical finance in psychological factors.
Deriving average soliton equations with a perturbative method
International Nuclear Information System (INIS)
Ballantyne, G.J.; Gough, P.T.; Taylor, D.P.
1995-01-01
The method of multiple scales is applied to periodically amplified, lossy media described by either the nonlinear Schroedinger (NLS) equation or the Korteweg--de Vries (KdV) equation. An existing result for the NLS equation, derived in the context of nonlinear optical communications, is confirmed. The method is then applied to the KdV equation and the result is confirmed numerically
Nonparametric estimation of stochastic differential equations with sparse Gaussian processes.
García, Constantino A; Otero, Abraham; Félix, Paulo; Presedo, Jesús; Márquez, David G
2017-08-01
The application of stochastic differential equations (SDEs) to the analysis of temporal data has attracted increasing attention, due to their ability to describe complex dynamics with physically interpretable equations. In this paper, we introduce a nonparametric method for estimating the drift and diffusion terms of SDEs from a densely observed discrete time series. The use of Gaussian processes as priors permits working directly in a function-space view and thus the inference takes place directly in this space. To cope with the computational complexity that requires the use of Gaussian processes, a sparse Gaussian process approximation is provided. This approximation permits the efficient computation of predictions for the drift and diffusion terms by using a distribution over a small subset of pseudosamples. The proposed method has been validated using both simulated data and real data from economy and paleoclimatology. The application of the method to real data demonstrates its ability to capture the behavior of complex systems.
Stochastic modeling of mode interactions via linear parabolized stability equations
Ran, Wei; Zare, Armin; Hack, M. J. Philipp; Jovanovic, Mihailo
2017-11-01
Low-complexity approximations of the Navier-Stokes equations have been widely used in the analysis of wall-bounded shear flows. In particular, the parabolized stability equations (PSE) and Floquet theory have been employed to capture the evolution of primary and secondary instabilities in spatially-evolving flows. We augment linear PSE with Floquet analysis to formally treat modal interactions and the evolution of secondary instabilities in the transitional boundary layer via a linear progression. To this end, we leverage Floquet theory by incorporating the primary instability into the base flow and accounting for different harmonics in the flow state. A stochastic forcing is introduced into the resulting linear dynamics to model the effect of nonlinear interactions on the evolution of modes. We examine the H-type transition scenario to demonstrate how our approach can be used to model nonlinear effects and capture the growth of the fundamental and subharmonic modes observed in direct numerical simulations and experiments.
Population density equations for stochastic processes with memory kernels
Lai, Yi Ming; de Kamps, Marc
2017-06-01
We present a method for solving population density equations (PDEs)-a mean-field technique describing homogeneous populations of uncoupled neurons—where the populations can be subject to non-Markov noise for arbitrary distributions of jump sizes. The method combines recent developments in two different disciplines that traditionally have had limited interaction: computational neuroscience and the theory of random networks. The method uses a geometric binning scheme, based on the method of characteristics, to capture the deterministic neurodynamics of the population, separating the deterministic and stochastic process cleanly. We can independently vary the choice of the deterministic model and the model for the stochastic process, leading to a highly modular numerical solution strategy. We demonstrate this by replacing the master equation implicit in many formulations of the PDE formalism by a generalization called the generalized Montroll-Weiss equation—a recent result from random network theory—describing a random walker subject to transitions realized by a non-Markovian process. We demonstrate the method for leaky- and quadratic-integrate and fire neurons subject to spike trains with Poisson and gamma-distributed interspike intervals. We are able to model jump responses for both models accurately to both excitatory and inhibitory input under the assumption that all inputs are generated by one renewal process.
Polynomial asymptotic stability of damped stochastic differential equations
Directory of Open Access Journals (Sweden)
John Appleby
2004-08-01
Full Text Available The paper studies the polynomial convergence of solutions of a scalar nonlinear It\\^{o} stochastic differential equation\\[dX(t = -f(X(t\\,dt + \\sigma(t\\,dB(t\\] where it is known, {\\it a priori}, that $\\lim_{t\\rightarrow\\infty} X(t=0$, a.s. The intensity of the stochastic perturbation $\\sigma$ is a deterministic, continuous and square integrable function, which tends to zero more quickly than a polynomially decaying function. The function $f$ obeys $\\lim_{x\\rightarrow 0}\\mbox{sgn}(xf(x/|x|^\\beta = a$, for some $\\beta>1$, and $a>0$.We study two asymptotic regimes: when $\\sigma$ tends to zero sufficiently quickly the polynomial decay rate of solutions is the same as for the deterministic equation (when $\\sigma\\equiv0$. When $\\sigma$ decays more slowly, a weaker almost sure polynomial upper bound on the decay rate of solutions is established. Results which establish the necessity for $\\sigma$ to decay polynomially in order to guarantee the almost sure polynomial decay of solutions are also proven.
Energy Technology Data Exchange (ETDEWEB)
Wang, Lijin, E-mail: ljwang@ucas.ac.cn [School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049 (China)
2016-06-08
The stochastic protein kinetic equations can be stiff for certain parameters, which makes their numerical simulation rely on very small time step sizes, resulting in large computational cost and accumulated round-off errors. For such situation, we provide a method of reducing stiffness of the stochastic protein kinetic equation by means of a kind of variable transformation. Theoretical and numerical analysis show effectiveness of this method. Its generalization to a more general class of stochastic differential equation models is also discussed.
Müller, Eike H; Scheichl, Rob; Shardlow, Tony
2015-04-08
This paper applies several well-known tricks from the numerical treatment of deterministic differential equations to improve the efficiency of the multilevel Monte Carlo (MLMC) method for stochastic differential equations (SDEs) and especially the Langevin equation. We use modified equations analysis as an alternative to strong-approximation theory for the integrator, and we apply this to introduce MLMC for Langevin-type equations with integrators based on operator splitting. We combine this with extrapolation and investigate the use of discrete random variables in place of the Gaussian increments, which is a well-known technique for the weak approximation of SDEs. We show that, for small-noise problems, discrete random variables can lead to an increase in efficiency of almost two orders of magnitude for practical levels of accuracy.
Fractional Stochastic Differential Equations Satisfying Fluctuation-Dissipation Theorem
Li, Lei; Liu, Jian-Guo; Lu, Jianfeng
2017-10-01
We propose in this work a fractional stochastic differential equation (FSDE) model consistent with the over-damped limit of the generalized Langevin equation model. As a result of the `fluctuation-dissipation theorem', the differential equations driven by fractional Brownian noise to model memory effects should be paired with Caputo derivatives, and this FSDE model should be understood in an integral form. We establish the existence of strong solutions for such equations and discuss the ergodicity and convergence to Gibbs measure. In the linear forcing regime, we show rigorously the algebraic convergence to Gibbs measure when the `fluctuation-dissipation theorem' is satisfied, and this verifies that satisfying `fluctuation-dissipation theorem' indeed leads to the correct physical behavior. We further discuss possible approaches to analyze the ergodicity and convergence to Gibbs measure in the nonlinear forcing regime, while leave the rigorous analysis for future works. The FSDE model proposed is suitable for systems in contact with heat bath with power-law kernel and subdiffusion behaviors.
Inviscid limit of stochastic damped 2D Navier–Stokes equations
International Nuclear Information System (INIS)
Bessaih, Hakima; Ferrario, Benedetta
2014-01-01
We consider the inviscid limit of the stochastic damped 2D Navier–Stokes equations. We prove that, when the viscosity vanishes, the stationary solution of the stochastic damped Navier–Stokes equations converges to a stationary solution of the stochastic damped Euler equation and that the rate of dissipation of enstrophy converges to zero. In particular, this limit obeys an enstrophy balance. The rates are computed with respect to a limit measure of the unique invariant measure of the stochastic damped Navier–Stokes equations. (paper)
DEFF Research Database (Denmark)
Duun-Henriksen, Anne Katrine; Schmidt, S.; Nørgaard, K.
2013-01-01
extension incorporating exercise effects on insulin and glucose dynamics. Our model is constructed as a stochastic state space model consisting of a set of stochastic differential equations (SDEs). In a stochastic state space model, the residual error is split into random measurement error...
Stochastic weighted particle methods for population balance equations
International Nuclear Information System (INIS)
Patterson, Robert I.A.; Wagner, Wolfgang; Kraft, Markus
2011-01-01
Highlights: → Weight transfer functions for Monte Carlo simulation of coagulation. → Efficient support for single-particle growth processes. → Comparisons to analytic solutions and soot formation problems. → Better numerical accuracy for less common particles. - Abstract: A class of coagulation weight transfer functions is constructed, each member of which leads to a stochastic particle algorithm for the numerical treatment of population balance equations. These algorithms are based on systems of weighted computational particles and the weight transfer functions are constructed such that the number of computational particles does not change during coagulation events. The algorithms also facilitate the simulation of physical processes that change single particles, such as growth, or other surface reactions. Four members of the algorithm family have been numerically validated by comparison to analytic solutions to simple problems. Numerical experiments have been performed for complex laminar premixed flame systems in which members of the class of stochastic weighted particle methods were compared to each other and to a direct simulation algorithm. Two of the weighted algorithms have been shown to offer performance advantages over the direct simulation algorithm in situations where interest is focused on the larger particles in a system. The extent of this advantage depends on the particular system and on the quantities of interest.
Stochastic Ocean Predictions with Dynamically-Orthogonal Primitive Equations
Subramani, D. N.; Haley, P., Jr.; Lermusiaux, P. F. J.
2017-12-01
The coastal ocean is a prime example of multiscale nonlinear fluid dynamics. Ocean fields in such regions are complex and intermittent with unstationary heterogeneous statistics. Due to the limited measurements, there are multiple sources of uncertainties, including the initial conditions, boundary conditions, forcing, parameters, and even the model parameterizations and equations themselves. For efficient and rigorous quantification and prediction of these uncertainities, the stochastic Dynamically Orthogonal (DO) PDEs for a primitive equation ocean modeling system with a nonlinear free-surface are derived and numerical schemes for their space-time integration are obtained. Detailed numerical studies with idealized-to-realistic regional ocean dynamics are completed. These include consistency checks for the numerical schemes and comparisons with ensemble realizations. As an illustrative example, we simulate the 4-d multiscale uncertainty in the Middle Atlantic/New York Bight region during the months of Jan to Mar 2017. To provide intitial conditions for the uncertainty subspace, uncertainties in the region were objectively analyzed using historical data. The DO primitive equations were subsequently integrated in space and time. The probability distribution function (pdf) of the ocean fields is compared to in-situ, remote sensing, and opportunity data collected during the coincident POSYDON experiment. Results show that our probabilistic predictions had skill and are 3- to 4- orders of magnitude faster than classic ensemble schemes.
A stochastic differential equation analysis of cerebrospinal fluid dynamics.
Raman, Kalyan
2011-01-18
Clinical measurements of intracranial pressure (ICP) over time show fluctuations around the deterministic time path predicted by a classic mathematical model in hydrocephalus research. Thus an important issue in mathematical research on hydrocephalus remains unaddressed--modeling the effect of noise on CSF dynamics. Our objective is to mathematically model the noise in the data. The classic model relating the temporal evolution of ICP in pressure-volume studies to infusions is a nonlinear differential equation based on natural physical analogies between CSF dynamics and an electrical circuit. Brownian motion was incorporated into the differential equation describing CSF dynamics to obtain a nonlinear stochastic differential equation (SDE) that accommodates the fluctuations in ICP. The SDE is explicitly solved and the dynamic probabilities of exceeding critical levels of ICP under different clinical conditions are computed. A key finding is that the probabilities display strong threshold effects with respect to noise. Above the noise threshold, the probabilities are significantly influenced by the resistance to CSF outflow and the intensity of the noise. Fluctuations in the CSF formation rate increase fluctuations in the ICP and they should be minimized to lower the patient's risk. The nonlinear SDE provides a scientific methodology for dynamic risk management of patients. The dynamic output of the SDE matches the noisy ICP data generated by the actual intracranial dynamics of patients better than the classic model used in prior research.
A stochastic differential equation analysis of cerebrospinal fluid dynamics
Directory of Open Access Journals (Sweden)
Raman Kalyan
2011-01-01
Full Text Available Abstract Background Clinical measurements of intracranial pressure (ICP over time show fluctuations around the deterministic time path predicted by a classic mathematical model in hydrocephalus research. Thus an important issue in mathematical research on hydrocephalus remains unaddressed--modeling the effect of noise on CSF dynamics. Our objective is to mathematically model the noise in the data. Methods The classic model relating the temporal evolution of ICP in pressure-volume studies to infusions is a nonlinear differential equation based on natural physical analogies between CSF dynamics and an electrical circuit. Brownian motion was incorporated into the differential equation describing CSF dynamics to obtain a nonlinear stochastic differential equation (SDE that accommodates the fluctuations in ICP. Results The SDE is explicitly solved and the dynamic probabilities of exceeding critical levels of ICP under different clinical conditions are computed. A key finding is that the probabilities display strong threshold effects with respect to noise. Above the noise threshold, the probabilities are significantly influenced by the resistance to CSF outflow and the intensity of the noise. Conclusions Fluctuations in the CSF formation rate increase fluctuations in the ICP and they should be minimized to lower the patient's risk. The nonlinear SDE provides a scientific methodology for dynamic risk management of patients. The dynamic output of the SDE matches the noisy ICP data generated by the actual intracranial dynamics of patients better than the classic model used in prior research.
Stochastic Galerkin methods for the steady-state Navier–Stokes equations
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Sousedík, Bedřich, E-mail: sousedik@umbc.edu [Department of Mathematics and Statistics, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250 (United States); Elman, Howard C., E-mail: elman@cs.umd.edu [Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, College Park, MD 20742 (United States)
2016-07-01
We study the steady-state Navier–Stokes equations in the context of stochastic finite element discretizations. Specifically, we assume that the viscosity is a random field given in the form of a generalized polynomial chaos expansion. For the resulting stochastic problem, we formulate the model and linearization schemes using Picard and Newton iterations in the framework of the stochastic Galerkin method, and we explore properties of the resulting stochastic solutions. We also propose a preconditioner for solving the linear systems of equations arising at each step of the stochastic (Galerkin) nonlinear iteration and demonstrate its effectiveness for solving a set of benchmark problems.
Variational and potential formulation for stochastic partial differential equations
International Nuclear Information System (INIS)
Munoz S, A G; Ojeda, J; Sierra D, P; Soldovieri, T
2006-01-01
Recently there has been interest in finding a potential formulation for stochastic partial differential equations (SPDEs). The rationale behind this idea lies in obtaining all the dynamical information of the system under study from one single expression. In this letter we formally provide a general Lagrangian formalism for SPDEs using the Hojman et al method. We show that it is possible to write the corresponding effective potential starting from an s-equivalent Lagrangian, and that this potential is able to reproduce all the dynamics of the system once a special differential operator has been applied. This procedure can be used to study the complete time evolution and spatial inhomogeneities of the system under consideration, and is also suitable for the statistical mechanics description of the problem. (letter to the editor)
Frank, T D; Beek, P J
2001-08-01
Recently, Küchler and Mensch [Stochastics Stochastics Rep. 40, 23 (1992)] derived exact stationary probability densities for linear stochastic delay differential equations. This paper presents an alternative derivation of these solutions by means of the Fokker-Planck approach introduced by Guillouzic [Phys. Rev. E 59, 3970 (1999); 61, 4906 (2000)]. Applications of this approach, which is argued to have greater generality, are discussed in the context of stochastic models for population growth and tracking movements.
Some Additional Remarks on the Cumulant Expansion for Linear Stochastic Differential Equations
Roerdink, J.B.T.M.
1984-01-01
We summarize our previous results on cumulant expansions for linear stochastic differential equations with correlated multipliclative and additive noise. The application of the general formulas to equations with statistically independent multiplicative and additive noise is reconsidered in detail,
Some additional remarks on the cumulant expansion for linear stochastic differential equations
Roerdink, J.B.T.M.
1984-01-01
We summarize our previous results on cumular expasions for linear stochastic differential equations with correlated multipliclative and additive noise. The application of the general formulas to equations with statistically independent multiplicative and additive noise is reconsidered in detail,
Effective average action for gauge theories and exact evolution equations
International Nuclear Information System (INIS)
Reuter, M.; Wetterich, C.
1993-11-01
We propose a new nonperturbative evolution equation for Yang-Mills theories. It describes the scale dependence of an effective action. The running of the nonabelian gauge coupling in arbitrary dimension is computed. (orig.)
Gaudreault, Mathieu; Drolet, François; Viñals, Jorge
2010-11-01
Analytical expressions for pitchfork and Hopf bifurcation thresholds are given for a nonlinear stochastic differential delay equation with feedback. Our results assume that the delay time τ is small compared to other characteristic time scales, not a significant limitation close to the bifurcation line. A pitchfork bifurcation line is found, the location of which depends on the conditional average , where x(t) is the dynamical variable. This conditional probability incorporates the combined effect of fluctuation correlations and delayed feedback. We also find a Hopf bifurcation line which is obtained by a multiple scale expansion around the oscillatory solution near threshold. We solve the Fokker-Planck equation associated with the slowly varying amplitudes and use it to determine the threshold location. In both cases, the predicted bifurcation lines are in excellent agreement with a direct numerical integration of the governing equations. Contrary to the known case involving no delayed feedback, we show that the stochastic bifurcation lines are shifted relative to the deterministic limit and hence that the interaction between fluctuation correlations and delay affect the stability of the solutions of the model equation studied.
Exponential p-stability of impulsive stochastic differential equations with delays
International Nuclear Information System (INIS)
Yang Zhiguo; Xu Daoyi; Xiang Li
2006-01-01
In this Letter, we establish a method to study the exponential p-stability of the zero solution of impulsive stochastic differential equations with delays. By establishing an L-operator inequality and using the properties of M-cone and stochastic analysis technique, we obtain some new conditions ensuring the exponential p-stability of the zero solution of impulsive stochastic differential equations with delays. Two illustrative examples have been provided to show the effectiveness of our results
Chadha, Alka; Bora, Swaroop Nandan
2017-11-01
This paper studies the existence, uniqueness, and exponential stability in mean square for the mild solution of neutral second order stochastic partial differential equations with infinite delay and Poisson jumps. By utilizing the Banach fixed point theorem, first the existence and uniqueness of the mild solution of neutral second order stochastic differential equations is established. Then, the mean square exponential stability for the mild solution of the stochastic system with Poisson jumps is obtained with the help of an established integral inequality.
On solutions of neutral stochastic delay Volterra equations with singular kernels
Directory of Open Access Journals (Sweden)
Xiaotai Wu
2012-08-01
Full Text Available In this paper, existence, uniqueness and continuity of the adapted solutions for neutral stochastic delay Volterra equations with singular kernels are discussed. In addition, continuous dependence on the initial date is also investigated. Finally, stochastic Volterra equation with the kernel of fractional Brownian motion is studied to illustrate the effectiveness of our results.
EXPONENTIAL ERGODICITY FOR STOCHASTIC BURGERS AND 2D NAVIER-STOKES EQUATIONS
Goldys, B
2004-01-01
It is shown that transition measures of the stochastic Navier-Stokes equation in dimension 2 converge exponentially fast to the corresponding invariant measures in the distance of total variation. As a corollary we obtain the existence of spectral gap for a related semigroup obtained by a sort of ground state trasformation. Analogous results are proved for the stochastic Burgers equation.
Hyperbolic white noise functional solutions of Wick-type stochastic compound KdV-Burgers equations
International Nuclear Information System (INIS)
Han Xiu; Xie Yingchao
2009-01-01
Variable coefficient and Wick-type stochastic compound KdV-Burgers equations are investigated. By using white noise analysis, Hermite transform and the hyperbolic function method, we obtain a number of Wick versions of hyperbolic white noise functional solutions and hyperbolic function solutions for Wick-type stochastic and variable coefficient compound KdV-Burgers equations, respectively.
Stochastic solutions to the Schrodinger equation for fermions
International Nuclear Information System (INIS)
Arnow, D.M.
1981-01-01
An exact stochastic method has been developed for generating the antisymmetric eigensolution of lowest index and its associated eigenvalue for the Schrodinger wave equation in 3N dimensions. The method is called the Green's function Monte Carlo method for fermions (FGFMC) because it is based on a Monte Carlo solution to the integral form of the Schrodinger equation (using Green's function) and because it is the fermion class of particles in physics which require antisymmetric solutions. The solution consists of two sets of 3N-dimensional points, [R/sub j/ + ] and [R/sub j/ - ], distributed by density functions psi + and psi - , whose difference, psi + -psi - , is proportional to the eigensolution, psi/sub F/. The FGFMC method is successfully applied to a one dimensional problem and a nine dimensional problem, the results of which are presented here. These results demonstrate that this method can be successfully applied to small physical problems on medium-scale computing machines. The key to this success was the transformation of the problem from exponential to linear cost as a function of accuracy. The strong dependence on dimensionality, however, currently results in an exponential cost as a function of problem size, and this, until overcome, imposes a severe barrier to calculations on large systems
Horowitz, Jordan M
2015-07-28
The stochastic thermodynamics of a dilute, well-stirred mixture of chemically reacting species is built on the stochastic trajectories of reaction events obtained from the chemical master equation. However, when the molecular populations are large, the discrete chemical master equation can be approximated with a continuous diffusion process, like the chemical Langevin equation or low noise approximation. In this paper, we investigate to what extent these diffusion approximations inherit the stochastic thermodynamics of the chemical master equation. We find that a stochastic-thermodynamic description is only valid at a detailed-balanced, equilibrium steady state. Away from equilibrium, where there is no consistent stochastic thermodynamics, we show that one can still use the diffusive solutions to approximate the underlying thermodynamics of the chemical master equation.
Structure and properties of Hughston's stochastic extension of the Schroedinger equation
International Nuclear Information System (INIS)
Adler, Stephen L.; Horwitz, Lawrence P.
2000-01-01
Hughston has recently proposed a stochastic extension of the Schroedinger equation, expressed as a stochastic differential equation on projective Hilbert space. We derive new projective Hilbert space identities, which we use to give a general proof that Hughston's equation leads to state vector collapse to energy eigenstates, with collapse probabilities given by the quantum mechanical probabilities computed from the initial state. We discuss the relation of Hughston's equation to earlier work on norm-preserving stochastic equations, and show that Hughston's equation can be written as a manifestly unitary stochastic evolution equation for the pure state density matrix. We discuss the behavior of systems constructed as direct products of independent subsystems, and briefly address the question of whether an energy-based approach, such as Hughston's, suffices to give an objective interpretation of the measurement process in quantum mechanics. (c) 2000 American Institute of Physics
DEFF Research Database (Denmark)
Klim, Søren; Mortensen, Stig Bousgaard; Kristensen, Niels Rode
2009-01-01
are often partly ignored in PK/PD modelling although violating the hypothesis for many standard statistical tests. This article presents a package for the statistical program R that is able to handle SDEs in a mixed-effects setting. The estimation method implemented is the FOCE1 approximation......The extension from ordinary to stochastic differential equations (SDEs) in pharmacokinetic and pharmacodynamic (PK/PD) modelling is an emerging field and has been motivated in a number of articles [N.R. Kristensen, H. Madsen, S.H. Ingwersen, Using stochastic differential equations for PK/PD model...... development, J. Pharmacokinet. Pharmacodyn. 32 (February(l)) (2005) 109-141; C.W. Tornoe, R.V Overgaard, H. Agerso, H.A. Nielsen, H. Madsen, E.N. Jonsson, Stochastic differential equations in NONMEM: implementation, application, and comparison with ordinary differential equations, Pharm. Res. 22 (August(8...
Climate Change and Integrodifference Equations in a Stochastic Environment.
Bouhours, Juliette; Lewis, Mark A
2016-09-01
Climate change impacts population distributions, forcing some species to migrate poleward if they are to survive and keep up with the suitable habitat that is shifting with the temperature isoclines. Previous studies have analysed whether populations have the capacity to keep up with shifting temperature isoclines, and have mathematically determined the combination of growth and dispersal that is needed to achieve this. However, the rate of isocline movement can be highly variable, with much uncertainty associated with yearly shifts. The same is true for population growth rates. Growth rates can be variable and uncertain, even within suitable habitats for growth. In this paper, we reanalyse the question of population persistence in the context of the uncertainty and variability in isocline shifts and rates of growth. Specifically, we employ a stochastic integrodifference equation model on a patch of suitable habitat that shifts poleward at a random rate. We derive a metric describing the asymptotic growth rate of the linearised operator of the stochastic model. This metric yields a threshold criterion for population persistence. We demonstrate that the variability in the yearly shift and in the growth rate has a significant negative effect on the persistence in the sense that it decreases the threshold criterion for population persistence. Mathematically, we show how the persistence metric can be connected to the principal eigenvalue problem for a related integral operator, at least for the case where isocline shifting speed is deterministic. Analysis of dynamics for the case where the dispersal kernel is Gaussian leads to the existence of a critical shifting speed, above which the population will go extinct, and below which the population will persist. This leads to clear bounds on rate of environmental change if the population is to persist. Finally, we illustrate our different results for butterfly population using numerical simulations and demonstrate how
Large Wilson loop averages from the Schwinger-Dyson equation
International Nuclear Information System (INIS)
Xue Shesheng
1987-01-01
Using Schwinger-Dyson equations for the large Wilson loop in abelian lattice gauge theories, we evaluate the vacuum expectation values of the Wilson loop of sizes 1x2, 2x2, 2x3, and so on, from which the string tension is extracted. (orig.)
Liu Yang; Yao Xiong; Xiao-jiao Tong
2017-01-01
We construct a new two-stage stochastic model of supply chain with multiple factories and distributors for perishable product. By introducing a second-order stochastic dominance (SSD) constraint, we can describe the preference consistency of the risk taker while minimizing the expected cost of company. To solve this problem, we convert it into a one-stage stochastic model equivalently; then we use sample average approximation (SAA) method to approximate the expected values of the underlying r...
International Nuclear Information System (INIS)
Hsiang, J.-T.; Hu, B.L.
2015-01-01
The existence and uniqueness of a steady state for nonequilibrium systems (NESS) is a fundamental subject and a main theme of research in statistical mechanics for decades. For Gaussian systems, such as a chain of classical harmonic oscillators connected at each end to a heat bath, and for classical anharmonic oscillators under specified conditions, definitive answers exist in the form of proven theorems. Answering this question for quantum many-body systems poses a challenge for the present. In this work we address this issue by deriving the stochastic equations for the reduced system with self-consistent backaction from the two baths, calculating the energy flow from one bath to the chain to the other bath, and exhibiting a power balance relation in the total (chain + baths) system which testifies to the existence of a NESS in this system at late times. Its insensitivity to the initial conditions of the chain corroborates to its uniqueness. The functional method we adopt here entails the use of the influence functional, the coarse-grained and stochastic effective actions, from which one can derive the stochastic equations and calculate the average values of physical variables in open quantum systems. This involves both taking the expectation values of quantum operators of the system and the distributional averages of stochastic variables stemming from the coarse-grained environment. This method though formal in appearance is compact and complete. It can also easily accommodate perturbative techniques and diagrammatic methods from field theory. Taken all together it provides a solid platform for carrying out systematic investigations into the nonequilibrium dynamics of open quantum systems and quantum thermodynamics. -- Highlights: •Nonequilibrium steady state (NESS) for interacting quantum many-body systems. •Derivation of stochastic equations for quantum oscillator chain with two heat baths. •Explicit calculation of the energy flow from one bath to the
International Nuclear Information System (INIS)
Gambetta, Jay; Wiseman, H.M.
2002-01-01
Do stochastic Schroedinger equations, also known as unravelings, have a physical interpretation? In the Markovian limit, where the system on average obeys a master equation, the answer is yes. Markovian stochastic Schroedinger equations generate quantum trajectories for the system state conditioned on continuously monitoring the bath. For a given master equation, there are many different unravelings, corresponding to different sorts of measurement on the bath. In this paper we address the non-Markovian case, and in particular the sort of stochastic Schroedinger equation introduced by Strunz, Diosi, and Gisin [Phys. Rev. Lett. 82, 1801 (1999)]. Using a quantum-measurement theory approach, we rederive their unraveling that involves complex-valued Gaussian noise. We also derive an unraveling involving real-valued Gaussian noise. We show that in the Markovian limit, these two unravelings correspond to heterodyne and homodyne detection, respectively. Although we use quantum-measurement theory to define these unravelings, we conclude that the stochastic evolution of the system state is not a true quantum trajectory, as the identity of the state through time is a fiction
Stochastic Growth Theory of Spatially-Averaged Distributions of Langmuir Fields in Earth's Foreshock
Boshuizen, Christopher R.; Cairns, Iver H.; Robinson, P. A.
2001-01-01
Langmuir-like waves in the foreshock of Earth are characteristically bursty and irregular, and are the subject of a number of recent studies. Averaged over the foreshock, it is observed that the probability distribution is power-law P(bar)(log E) in the wave field E with the bar denoting this averaging over position, In this paper it is shown that stochastic growth theory (SGT) can explain a power-law spatially-averaged distributions P(bar)(log E), when the observed power-law variations of the mean and standard deviation of log E with position are combined with the log normal statistics predicted by SGT at each location.
Approximate Dual Averaging Method for Multiagent Saddle-Point Problems with Stochastic Subgradients
Directory of Open Access Journals (Sweden)
Deming Yuan
2014-01-01
Full Text Available This paper considers the problem of solving the saddle-point problem over a network, which consists of multiple interacting agents. The global objective function of the problem is a combination of local convex-concave functions, each of which is only available to one agent. Our main focus is on the case where the projection steps are calculated approximately and the subgradients are corrupted by some stochastic noises. We propose an approximate version of the standard dual averaging method and show that the standard convergence rate is preserved, provided that the projection errors decrease at some appropriate rate and the noises are zero-mean and have bounded variance.
A New Control Paradigm for Stochastic Differential Equations
Schmid, Matthias J. A.
This study presents a novel comprehensive approach to the control of dynamic systems under uncertainty governed by stochastic differential equations (SDEs). Large Deviations (LD) techniques are employed to arrive at a control law for a large class of nonlinear systems minimizing sample path deviations. Thereby, a paradigm shift is suggested from point-in-time to sample path statistics on function spaces. A suitable formal control framework which leverages embedded Freidlin-Wentzell theory is proposed and described in detail. This includes the precise definition of the control objective and comprises an accurate discussion of the adaptation of the Freidlin-Wentzell theorem to the particular situation. The new control design is enabled by the transformation of an ill-posed control objective into a well-conditioned sequential optimization problem. A direct numerical solution process is presented using quadratic programming, but the emphasis is on the development of a closed-form expression reflecting the asymptotic deviation probability of a particular nominal path. This is identified as the key factor in the success of the new paradigm. An approach employing the second variation and the differential curvature of the effective action is suggested for small deviation channels leading to the Jacobi field of the rate function and the subsequently introduced Jacobi field performance measure. This closed-form solution is utilized in combination with the supplied parametrization of the objective space. For the first time, this allows for an LD based control design applicable to a large class of nonlinear systems. Thus, Minimum Large Deviations (MLD) control is effectively established in a comprehensive structured framework. The construction of the new paradigm is completed by an optimality proof for the Jacobi field performance measure, an interpretive discussion, and a suggestion for efficient implementation. The potential of the new approach is exhibited by its extension
International Nuclear Information System (INIS)
Granita; Bahar, A.
2015-01-01
This paper discusses on linear birth and death with immigration and emigration (BIDE) process to stochastic differential equation (SDE) model. Forward Kolmogorov equation in continuous time Markov chain (CTMC) with a central-difference approximation was used to find Fokker-Planckequation corresponding to a diffusion process having the stochastic differential equation of BIDE process. The exact solution, mean and variance function of BIDE process was found
Energy Technology Data Exchange (ETDEWEB)
Granita, E-mail: granitafc@gmail.com [Dept. Mathematical Education, State Islamic University of Sultan Syarif Kasim Riau, 28293 Indonesia and Dept. of Mathematical Science, Universiti Teknologi Malaysia, 81310,Johor (Malaysia); Bahar, A. [Dept. of Mathematical Science, Universiti Teknologi Malaysia, 81310,Johor Malaysia and UTM Center for Industrial and Applied Mathematics (UTM-CIAM) (Malaysia)
2015-03-09
This paper discusses on linear birth and death with immigration and emigration (BIDE) process to stochastic differential equation (SDE) model. Forward Kolmogorov equation in continuous time Markov chain (CTMC) with a central-difference approximation was used to find Fokker-Planckequation corresponding to a diffusion process having the stochastic differential equation of BIDE process. The exact solution, mean and variance function of BIDE process was found.
Characteristics of phase-averaged equations for modulated wave groups
Klopman, G.; Petit, H.A.H.; Battjes, J.A.
2000-01-01
The project concerns the influence of long waves on coastal morphology. The modelling of the combined motion of the long waves and short waves in the horizontal plane is done by phase-averaging over the short wave motion and using intra-wave modelling for the long waves, see e.g. Roelvink (1993).
A stochastic differential equation model of diurnal cortisol patterns
Brown, E. N.; Meehan, P. M.; Dempster, A. P.
2001-01-01
Circadian modulation of episodic bursts is recognized as the normal physiological pattern of diurnal variation in plasma cortisol levels. The primary physiological factors underlying these diurnal patterns are the ultradian timing of secretory events, circadian modulation of the amplitude of secretory events, infusion of the hormone from the adrenal gland into the plasma, and clearance of the hormone from the plasma by the liver. Each measured plasma cortisol level has an error arising from the cortisol immunoassay. We demonstrate that all of these three physiological principles can be succinctly summarized in a single stochastic differential equation plus measurement error model and show that physiologically consistent ranges of the model parameters can be determined from published reports. We summarize the model parameters in terms of the multivariate Gaussian probability density and establish the plausibility of the model with a series of simulation studies. Our framework makes possible a sensitivity analysis in which all model parameters are allowed to vary simultaneously. The model offers an approach for simultaneously representing cortisol's ultradian, circadian, and kinetic properties. Our modeling paradigm provides a framework for simulation studies and data analysis that should be readily adaptable to the analysis of other endocrine hormone systems.
Controlled Nonlinear Stochastic Delay Equations: Part I: Modeling and Approximations
International Nuclear Information System (INIS)
Kushner, Harold J.
2012-01-01
This two-part paper deals with “foundational” issues that have not been previously considered in the modeling and numerical optimization of nonlinear stochastic delay systems. There are new classes of models, such as those with nonlinear functions of several controls (such as products), each with is own delay, controlled random Poisson measure driving terms, admissions control with delayed retrials, and others. There are two basic and interconnected themes for these models. The first, dealt with in this part, concerns the definition of admissible control. The classical definition of an admissible control as a nonanticipative relaxed control is inadequate for these models and needs to be extended. This is needed for the convergence proofs of numerical approximations for optimal controls as well as to have a well-defined model. It is shown that the new classes of admissible controls do not enlarge the range of the value functions, is closed (together with the associated paths) under weak convergence, and is approximatable by ordinary controls. The second theme, dealt with in Part II, concerns transportation equation representations, and their role in the development of numerical algorithms with much reduced memory and computational requirements.
Solving Langevin equation with the stochastic algebraically correlated noise
International Nuclear Information System (INIS)
Ploszajczak, M.; Srokowski, T.
1996-01-01
Long time tail in the velocity and force autocorrelation function has been found recently in the molecular dynamics simulations of the peripheral collisions of ions. Simulation of those slowly decaying correlations in the stochastic transport theory requires the development of new methods of generating stochastic force of arbitrarily long correlation times. The Markovian process and the multidimensional Kangaroo process which permit describing various algebraic correlated stochastic processes are proposed. (author)
Stability of Equilibrium Points of Fractional Difference Equations with Stochastic Perturbations
Directory of Open Access Journals (Sweden)
Shaikhet Leonid
2008-01-01
Full Text Available It is supposed that the fractional difference equation , has an equilibrium point and is exposed to additive stochastic perturbations type of that are directly proportional to the deviation of the system state from the equilibrium point . It is shown that known results in the theory of stability of stochastic difference equations that were obtained via V. Kolmanovskii and L. Shaikhet general method of Lyapunov functionals construction can be successfully used for getting of sufficient conditions for stability in probability of equilibrium points of the considered stochastic fractional difference equation. Numerous graphical illustrations of stability regions and trajectories of solutions are plotted.
Wang, Xiaohu; Lu, Kening; Wang, Bixiang
2018-01-01
In this paper, we study the Wong-Zakai approximations given by a stationary process via the Wiener shift and their associated long term behavior of the stochastic reaction-diffusion equation driven by a white noise. We first prove the existence and uniqueness of tempered pullback attractors for the Wong-Zakai approximations of stochastic reaction-diffusion equation. Then, we show that the attractors of Wong-Zakai approximations converges to the attractor of the stochastic reaction-diffusion equation for both additive and multiplicative noise.
Klim, Søren; Mortensen, Stig Bousgaard; Kristensen, Niels Rode; Overgaard, Rune Viig; Madsen, Henrik
2009-06-01
The extension from ordinary to stochastic differential equations (SDEs) in pharmacokinetic and pharmacodynamic (PK/PD) modelling is an emerging field and has been motivated in a number of articles [N.R. Kristensen, H. Madsen, S.H. Ingwersen, Using stochastic differential equations for PK/PD model development, J. Pharmacokinet. Pharmacodyn. 32 (February(1)) (2005) 109-141; C.W. Tornøe, R.V. Overgaard, H. Agersø, H.A. Nielsen, H. Madsen, E.N. Jonsson, Stochastic differential equations in NONMEM: implementation, application, and comparison with ordinary differential equations, Pharm. Res. 22 (August(8)) (2005) 1247-1258; R.V. Overgaard, N. Jonsson, C.W. Tornøe, H. Madsen, Non-linear mixed-effects models with stochastic differential equations: implementation of an estimation algorithm, J. Pharmacokinet. Pharmacodyn. 32 (February(1)) (2005) 85-107; U. Picchini, S. Ditlevsen, A. De Gaetano, Maximum likelihood estimation of a time-inhomogeneous stochastic differential model of glucose dynamics, Math. Med. Biol. 25 (June(2)) (2008) 141-155]. PK/PD models are traditionally based ordinary differential equations (ODEs) with an observation link that incorporates noise. This state-space formulation only allows for observation noise and not for system noise. Extending to SDEs allows for a Wiener noise component in the system equations. This additional noise component enables handling of autocorrelated residuals originating from natural variation or systematic model error. Autocorrelated residuals are often partly ignored in PK/PD modelling although violating the hypothesis for many standard statistical tests. This article presents a package for the statistical program R that is able to handle SDEs in a mixed-effects setting. The estimation method implemented is the FOCE(1) approximation to the population likelihood which is generated from the individual likelihoods that are approximated using the Extended Kalman Filter's one-step predictions.
Stochastic optimal control in infinite dimension dynamic programming and HJB equations
Fabbri, Giorgio; Święch, Andrzej
2017-01-01
Providing an introduction to stochastic optimal control in infinite dimension, this book gives a complete account of the theory of second-order HJB equations in infinite-dimensional Hilbert spaces, focusing on its applicability to associated stochastic optimal control problems. It features a general introduction to optimal stochastic control, including basic results (e.g. the dynamic programming principle) with proofs, and provides examples of applications. A complete and up-to-date exposition of the existing theory of viscosity solutions and regular solutions of second-order HJB equations in Hilbert spaces is given, together with an extensive survey of other methods, with a full bibliography. In particular, Chapter 6, written by M. Fuhrman and G. Tessitore, surveys the theory of regular solutions of HJB equations arising in infinite-dimensional stochastic control, via BSDEs. The book is of interest to both pure and applied researchers working in the control theory of stochastic PDEs, and in PDEs in infinite ...
Seif, Dariush; Ghoniem, Nasr M.
2014-12-01
A rate theory model based on the theory of nonlinear stochastic differential equations (SDEs) is developed to estimate the time-dependent size distribution of helium bubbles in metals under irradiation. Using approaches derived from Itô's calculus, rate equations for the first five moments of the size distribution in helium-vacancy space are derived, accounting for the stochastic nature of the atomic processes involved. In the first iteration of the model, the distribution is represented as a bivariate Gaussian distribution. The spread of the distribution about the mean is obtained by white-noise terms in the second-order moments, driven by fluctuations in the general absorption and emission of point defects by bubbles, and fluctuations stemming from collision cascades. This statistical model for the reconstruction of the distribution by its moments is coupled to a previously developed reduced-set, mean-field, rate theory model. As an illustrative case study, the model is applied to a tungsten plasma facing component under irradiation. Our findings highlight the important role of stochastic atomic fluctuations on the evolution of helium-vacancy cluster size distributions. It is found that when the average bubble size is small (at low dpa levels), the relative spread of the distribution is large and average bubble pressures may be very large. As bubbles begin to grow in size, average bubble pressures decrease, and stochastic fluctuations have a lessened effect. The distribution becomes tighter as it evolves in time, corresponding to a more uniform bubble population. The model is formulated in a general way, capable of including point defect drift due to internal temperature and/or stress gradients. These arise during pulsed irradiation, and also during steady irradiation as a result of externally applied or internally generated non-homogeneous stress fields. Discussion is given into how the model can be extended to include full spatial resolution and how the
International Nuclear Information System (INIS)
Seif, Dariush; Ghoniem, Nasr M.
2014-01-01
A rate theory model based on the theory of nonlinear stochastic differential equations (SDEs) is developed to estimate the time-dependent size distribution of helium bubbles in metals under irradiation. Using approaches derived from Itô’s calculus, rate equations for the first five moments of the size distribution in helium–vacancy space are derived, accounting for the stochastic nature of the atomic processes involved. In the first iteration of the model, the distribution is represented as a bivariate Gaussian distribution. The spread of the distribution about the mean is obtained by white-noise terms in the second-order moments, driven by fluctuations in the general absorption and emission of point defects by bubbles, and fluctuations stemming from collision cascades. This statistical model for the reconstruction of the distribution by its moments is coupled to a previously developed reduced-set, mean-field, rate theory model. As an illustrative case study, the model is applied to a tungsten plasma facing component under irradiation. Our findings highlight the important role of stochastic atomic fluctuations on the evolution of helium–vacancy cluster size distributions. It is found that when the average bubble size is small (at low dpa levels), the relative spread of the distribution is large and average bubble pressures may be very large. As bubbles begin to grow in size, average bubble pressures decrease, and stochastic fluctuations have a lessened effect. The distribution becomes tighter as it evolves in time, corresponding to a more uniform bubble population. The model is formulated in a general way, capable of including point defect drift due to internal temperature and/or stress gradients. These arise during pulsed irradiation, and also during steady irradiation as a result of externally applied or internally generated non-homogeneous stress fields. Discussion is given into how the model can be extended to include full spatial resolution and how the
Energy Technology Data Exchange (ETDEWEB)
Seif, Dariush, E-mail: dariush.seif@iwm-extern.fraunhofer.de [Fraunhofer Institut für Werkstoffmechanik, Freiburg 79108 (Germany); Department of Mechanical and Aerospace Engineering, University of California, Los Angeles, CA 90095-1597 (United States); Ghoniem, Nasr M. [Department of Mechanical and Aerospace Engineering, University of California, Los Angeles, CA 90095-1597 (United States)
2014-12-15
A rate theory model based on the theory of nonlinear stochastic differential equations (SDEs) is developed to estimate the time-dependent size distribution of helium bubbles in metals under irradiation. Using approaches derived from Itô’s calculus, rate equations for the first five moments of the size distribution in helium–vacancy space are derived, accounting for the stochastic nature of the atomic processes involved. In the first iteration of the model, the distribution is represented as a bivariate Gaussian distribution. The spread of the distribution about the mean is obtained by white-noise terms in the second-order moments, driven by fluctuations in the general absorption and emission of point defects by bubbles, and fluctuations stemming from collision cascades. This statistical model for the reconstruction of the distribution by its moments is coupled to a previously developed reduced-set, mean-field, rate theory model. As an illustrative case study, the model is applied to a tungsten plasma facing component under irradiation. Our findings highlight the important role of stochastic atomic fluctuations on the evolution of helium–vacancy cluster size distributions. It is found that when the average bubble size is small (at low dpa levels), the relative spread of the distribution is large and average bubble pressures may be very large. As bubbles begin to grow in size, average bubble pressures decrease, and stochastic fluctuations have a lessened effect. The distribution becomes tighter as it evolves in time, corresponding to a more uniform bubble population. The model is formulated in a general way, capable of including point defect drift due to internal temperature and/or stress gradients. These arise during pulsed irradiation, and also during steady irradiation as a result of externally applied or internally generated non-homogeneous stress fields. Discussion is given into how the model can be extended to include full spatial resolution and how the
Kleinert, H.; Zatloukal, V.
2013-11-01
The statistics of rare events, the so-called black-swan events, is governed by non-Gaussian distributions with heavy power-like tails. We calculate the Green functions of the associated Fokker-Planck equations and solve the related stochastic differential equations. We also discuss the subject in the framework of path integration.
An estimator for the relative entropy rate of path measures for stochastic differential equations
Energy Technology Data Exchange (ETDEWEB)
Opper, Manfred, E-mail: manfred.opper@tu-berlin.de
2017-02-01
We address the problem of estimating the relative entropy rate (RER) for two stochastic processes described by stochastic differential equations. For the case where the drift of one process is known analytically, but one has only observations from the second process, we use a variational bound on the RER to construct an estimator.
Zollanvari, Amin; Dougherty, Edward R
2016-12-01
In classification, prior knowledge is incorporated in a Bayesian framework by assuming that the feature-label distribution belongs to an uncertainty class of feature-label distributions governed by a prior distribution. A posterior distribution is then derived from the prior and the sample data. An optimal Bayesian classifier (OBC) minimizes the expected misclassification error relative to the posterior distribution. From an application perspective, prior construction is critical. The prior distribution is formed by mapping a set of mathematical relations among the features and labels, the prior knowledge, into a distribution governing the probability mass across the uncertainty class. In this paper, we consider prior knowledge in the form of stochastic differential equations (SDEs). We consider a vector SDE in integral form involving a drift vector and dispersion matrix. Having constructed the prior, we develop the optimal Bayesian classifier between two models and examine, via synthetic experiments, the effects of uncertainty in the drift vector and dispersion matrix. We apply the theory to a set of SDEs for the purpose of differentiating the evolutionary history between two species.
Parameter estimation in a simple stochastic differential equation for phytoplankton modelling
DEFF Research Database (Denmark)
Møller, Jan Kloppenborg; Madsen, Henrik; Carstensen, Jacob
2011-01-01
The use of stochastic differential equations (SDEs) for simulation of aquatic ecosystems has attracted increasing attention in recent years. The SDE setting also provides the opportunity for statistical estimation of ecosystem parameters. We present an estimation procedure, based on Kalman...
Stability of numerical method for semi-linear stochastic pantograph differential equations
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Yu Zhang
2016-01-01
Full Text Available Abstract As a particular expression of stochastic delay differential equations, stochastic pantograph differential equations have been widely used in nonlinear dynamics, quantum mechanics, and electrodynamics. In this paper, we mainly study the stability of analytical solutions and numerical solutions of semi-linear stochastic pantograph differential equations. Some suitable conditions for the mean-square stability of an analytical solution are obtained. Then we proved the general mean-square stability of the exponential Euler method for a numerical solution of semi-linear stochastic pantograph differential equations, that is, if an analytical solution is stable, then the exponential Euler method applied to the system is mean-square stable for arbitrary step-size h > 0 $h>0$ . Numerical examples further illustrate the obtained theoretical results.
EXISTENCE OF SOLUTION TO NONLINEAR SECOND ORDER NEUTRAL STOCHASTIC DIFFERENTIAL EQUATIONS WITH DELAY
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
This paper is concerned with nonlinear second order neutral stochastic differential equations with delay in a Hilbert space. Sufficient conditions for the existence of solution to the system are obtained by Picard iterations.
A stochastic collocation method for the second order wave equation with a discontinuous random speed
Motamed, Mohammad; Nobile, Fabio; Tempone, Raul
2012-01-01
In this paper we propose and analyze a stochastic collocation method for solving the second order wave equation with a random wave speed and subjected to deterministic boundary and initial conditions. The speed is piecewise smooth in the physical
Bessaih, Hakima; Efendiev, Yalchin; Maris, Florin
2015-01-01
The evolution Stokes equation in a domain containing periodically distributed obstacles subject to Fourier boundary condition on the boundaries is considered. We assume that the dynamic is driven by a stochastic perturbation on the interior
Non-Intrusive Solution of Stochastic and Parametric Equations
Matthies, Hermann
2015-01-07
Many problems depend on parameters, which may be a finite set of numerical values, or mathematically more complicated objects like for example processes or fields. We address the situation where we have an equation which depends on parameters; stochastic equations are a special case of such parametric problems where the parameters are elements from a probability space. One common way to represent this dependability on parameters is by evaluating the state (or solution) of the system under investigation for different values of the parameters. But often one wants to evaluate the solution quickly for a new set of parameters where it has not been sampled. In this situation it may be advantageous to express the parameter dependent solution with an approximation which allows for rapid evaluation of the solution. Such approximations are also called proxy or surrogate models, response functions, or emulators. All these methods may be seen as functional approximations—representations of the solution by an “easily computable” function of the parameters, as opposed to pure samples. The most obvious methods of approximation used are based on interpolation, in this context often labelled as collocation. In the frequent situation where one has a “solver” for the equation for a given parameter value, i.e. a software component or a program, it is evident that this can be used to independently—if desired in parallel—solve for all the parameter values which subsequently may be used either for the interpolation or in the quadrature for the projection. Such methods are therefore uncoupled for each parameter value, and they additionally often carry the label “non-intrusive”. Without much argument all other methods— which produce a coupled system of equations–are almost always labelled as “intrusive”, meaning that one cannot use the original solver. We want to show here that this not necessarily the case. Another approach is to choose some other projection onto
Non-Intrusive Solution of Stochastic and Parametric Equations
Matthies, Hermann
2015-01-01
Many problems depend on parameters, which may be a finite set of numerical values, or mathematically more complicated objects like for example processes or fields. We address the situation where we have an equation which depends on parameters; stochastic equations are a special case of such parametric problems where the parameters are elements from a probability space. One common way to represent this dependability on parameters is by evaluating the state (or solution) of the system under investigation for different values of the parameters. But often one wants to evaluate the solution quickly for a new set of parameters where it has not been sampled. In this situation it may be advantageous to express the parameter dependent solution with an approximation which allows for rapid evaluation of the solution. Such approximations are also called proxy or surrogate models, response functions, or emulators. All these methods may be seen as functional approximations—representations of the solution by an “easily computable” function of the parameters, as opposed to pure samples. The most obvious methods of approximation used are based on interpolation, in this context often labelled as collocation. In the frequent situation where one has a “solver” for the equation for a given parameter value, i.e. a software component or a program, it is evident that this can be used to independently—if desired in parallel—solve for all the parameter values which subsequently may be used either for the interpolation or in the quadrature for the projection. Such methods are therefore uncoupled for each parameter value, and they additionally often carry the label “non-intrusive”. Without much argument all other methods— which produce a coupled system of equations–are almost always labelled as “intrusive”, meaning that one cannot use the original solver. We want to show here that this not necessarily the case. Another approach is to choose some other projection onto
Numerical solution of second-order stochastic differential equations with Gaussian random parameters
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Rahman Farnoosh
2014-07-01
Full Text Available In this paper, we present the numerical solution of ordinary differential equations (or SDEs, from each orderespecially second-order with time-varying and Gaussian random coefficients. We indicate a complete analysisfor second-order equations in specially case of scalar linear second-order equations (damped harmonicoscillators with additive or multiplicative noises. Making stochastic differential equations system from thisequation, it could be approximated or solved numerically by different numerical methods. In the case oflinear stochastic differential equations system by Computing fundamental matrix of this system, it could becalculated based on the exact solution of this system. Finally, this stochastic equation is solved by numericallymethod like E.M. and Milstein. Also its Asymptotic stability and statistical concepts like expectationand variance of solutions are discussed.
International Nuclear Information System (INIS)
Wang Bo; Song Ruili
2009-01-01
We consider a more general wealth process with a drift coefficient which is Lipschitz continuous and the portfolio process with convex constraint. We convert the problem of hedging American contingent claims into the problem of minimal solution of backward stochastic differential equation with stopping time. We adopt the penalization method for constructing the minimal solution of stochastic differential equations and obtain the upper hedging price of American contingent claims.
Directory of Open Access Journals (Sweden)
Hua Yang
2012-01-01
Full Text Available We are concerned with the stochastic differential delay equations with Poisson jump and Markovian switching (SDDEsPJMSs. Most SDDEsPJMSs cannot be solved explicitly as stochastic differential equations. Therefore, numerical solutions have become an important issue in the study of SDDEsPJMSs. The key contribution of this paper is to investigate the strong convergence between the true solutions and the numerical solutions to SDDEsPJMSs when the drift and diffusion coefficients are Taylor approximations.
DEFF Research Database (Denmark)
Tornøe, Christoffer Wenzel; Overgaard, Rune Viig; Agerso, H.
2005-01-01
of noise: a measurement and a system noise term. The measurement noise represents uncorrelated error due to, for example, assay error while the system noise accounts for structural misspecifications, approximations of the dynamical model, and true random physiological fluctuations. Since the system noise...... degarelix. Conclusions. The EKF-based algorithm was successfully implemented in NONMEM for parameter estimation in population PK/PD models described by systems of SDEs. The example indicated that it was possible to pinpoint structural model deficiencies, and that valuable information may be obtained......Purpose. The objective of the present analysis was to explore the use of stochastic differential equations (SDEs) in population pharmacokinetic/pharmacodynamic (PK/PD) modeling. Methods. The intra-individual variability in nonlinear mixed-effects models based on SDEs is decomposed into two types...
Directory of Open Access Journals (Sweden)
Rice Sean H
2008-09-01
Full Text Available Abstract Background Evolution involves both deterministic and random processes, both of which are known to contribute to directional evolutionary change. A number of studies have shown that when fitness is treated as a random variable, meaning that each individual has a distribution of possible fitness values, then both the mean and variance of individual fitness distributions contribute to directional evolution. Unfortunately the most general mathematical description of evolution that we have, the Price equation, is derived under the assumption that both fitness and offspring phenotype are fixed values that are known exactly. The Price equation is thus poorly equipped to study an important class of evolutionary processes. Results I present a general equation for directional evolutionary change that incorporates both deterministic and stochastic processes and applies to any evolving system. This is essentially a stochastic version of the Price equation, but it is derived independently and contains terms with no analog in Price's formulation. This equation shows that the effects of selection are actually amplified by random variation in fitness. It also generalizes the known tendency of populations to be pulled towards phenotypes with minimum variance in fitness, and shows that this is matched by a tendency to be pulled towards phenotypes with maximum positive asymmetry in fitness. This equation also contains a term, having no analog in the Price equation, that captures cases in which the fitness of parents has a direct effect on the phenotype of their offspring. Conclusion Directional evolution is influenced by the entire distribution of individual fitness, not just the mean and variance. Though all moments of individuals' fitness distributions contribute to evolutionary change, the ways that they do so follow some general rules. These rules are invisible to the Price equation because it describes evolution retrospectively. An equally general
Directory of Open Access Journals (Sweden)
Mourad Kerboua
2014-12-01
Full Text Available We introduce a new notion called fractional stochastic nonlocal condition, and then we study approximate controllability of class of fractional stochastic nonlinear differential equations of Sobolev type in Hilbert spaces. We use Hölder's inequality, fixed point technique, fractional calculus, stochastic analysis and methods adopted directly from deterministic control problems for the main results. A new set of sufficient conditions is formulated and proved for the fractional stochastic control system to be approximately controllable. An example is given to illustrate the abstract results.
Application of Stochastic Partial Differential Equations to Reservoir Property Modelling
Potsepaev, R.; Farmer, C.L.
2010-01-01
in parametric space. In order to sample in physical space we introduce a stochastic elliptic PDE with tensor coefficients, where the tensor is related to correlation anisotropy and its variation is physical space.
Numerical simulation of stochastic point kinetic equation in the dynamical system of nuclear reactor
International Nuclear Information System (INIS)
Saha Ray, S.
2012-01-01
Highlights: ► In this paper stochastic neutron point kinetic equations have been analyzed. ► Euler–Maruyama method and Strong Taylor 1.5 order method have been discussed. ► These methods are applied for the solution of stochastic point kinetic equations. ► Comparison between the results of these methods and others are presented in tables. ► Graphs for neutron and precursor sample paths are also presented. -- Abstract: In the present paper, the numerical approximation methods, applied to efficiently calculate the solution for stochastic point kinetic equations () in nuclear reactor dynamics, are investigated. A system of Itô stochastic differential equations has been analyzed to model the neutron density and the delayed neutron precursors in a point nuclear reactor. The resulting system of Itô stochastic differential equations are solved over each time-step size. The methods are verified by considering different initial conditions, experimental data and over constant reactivities. The computational results indicate that the methods are simple and suitable for solving stochastic point kinetic equations. In this article, a numerical investigation is made in order to observe the random oscillations in neutron and precursor population dynamics in subcritical and critical reactors.
Narasimha Murthy, K. V.; Saravana, R.; Vijaya Kumar, K.
2018-04-01
The paper investigates the stochastic modelling and forecasting of monthly average maximum and minimum temperature patterns through suitable seasonal auto regressive integrated moving average (SARIMA) model for the period 1981-2015 in India. The variations and distributions of monthly maximum and minimum temperatures are analyzed through Box plots and cumulative distribution functions. The time series plot indicates that the maximum temperature series contain sharp peaks in almost all the years, while it is not true for the minimum temperature series, so both the series are modelled separately. The possible SARIMA model has been chosen based on observing autocorrelation function (ACF), partial autocorrelation function (PACF), and inverse autocorrelation function (IACF) of the logarithmic transformed temperature series. The SARIMA (1, 0, 0) × (0, 1, 1)12 model is selected for monthly average maximum and minimum temperature series based on minimum Bayesian information criteria. The model parameters are obtained using maximum-likelihood method with the help of standard error of residuals. The adequacy of the selected model is determined using correlation diagnostic checking through ACF, PACF, IACF, and p values of Ljung-Box test statistic of residuals and using normal diagnostic checking through the kernel and normal density curves of histogram and Q-Q plot. Finally, the forecasting of monthly maximum and minimum temperature patterns of India for the next 3 years has been noticed with the help of selected model.
Cotter, C J; Gottwald, G A; Holm, D D
2017-09-01
In Holm (Holm 2015 Proc. R. Soc. A 471 , 20140963. (doi:10.1098/rspa.2014.0963)), stochastic fluid equations were derived by employing a variational principle with an assumed stochastic Lagrangian particle dynamics. Here we show that the same stochastic Lagrangian dynamics naturally arises in a multi-scale decomposition of the deterministic Lagrangian flow map into a slow large-scale mean and a rapidly fluctuating small-scale map. We employ homogenization theory to derive effective slow stochastic particle dynamics for the resolved mean part, thereby obtaining stochastic fluid partial equations in the Eulerian formulation. To justify the application of rigorous homogenization theory, we assume mildly chaotic fast small-scale dynamics, as well as a centring condition. The latter requires that the mean of the fluctuating deviations is small, when pulled back to the mean flow.
Bessaih, Hakima
2015-04-01
The evolution Stokes equation in a domain containing periodically distributed obstacles subject to Fourier boundary condition on the boundaries is considered. We assume that the dynamic is driven by a stochastic perturbation on the interior of the domain and another stochastic perturbation on the boundaries of the obstacles. We represent the solid obstacles by holes in the fluid domain. The macroscopic (homogenized) equation is derived as another stochastic partial differential equation, defined in the whole non perforated domain. Here, the initial stochastic perturbation on the boundary becomes part of the homogenized equation as another stochastic force. We use the twoscale convergence method after extending the solution with 0 in the holes to pass to the limit. By Itô stochastic calculus, we get uniform estimates on the solution in appropriate spaces. In order to pass to the limit on the boundary integrals, we rewrite them in terms of integrals in the whole domain. In particular, for the stochastic integral on the boundary, we combine the previous idea of rewriting it on the whole domain with the assumption that the Brownian motion is of trace class. Due to the particular boundary condition dealt with, we get that the solution of the stochastic homogenized equation is not divergence free. However, it is coupled with the cell problem that has a divergence free solution. This paper represents an extension of the results of Duan and Wang (Comm. Math. Phys. 275:1508-1527, 2007), where a reaction diffusion equation with a dynamical boundary condition with a noise source term on both the interior of the domain and on the boundary was studied, and through a tightness argument and a pointwise two scale convergence method the homogenized equation was derived. © American Institute of Mathematical Sciences.
Nonperturbative renormalization group study of the stochastic Navier-Stokes equation.
Mejía-Monasterio, Carlos; Muratore-Ginanneschi, Paolo
2012-07-01
We study the renormalization group flow of the average action of the stochastic Navier-Stokes equation with power-law forcing. Using Galilean invariance, we introduce a nonperturbative approximation adapted to the zero-frequency sector of the theory in the parametric range of the Hölder exponent 4-2ε of the forcing where real-space local interactions are relevant. In any spatial dimension d, we observe the convergence of the resulting renormalization group flow to a unique fixed point which yields a kinetic energy spectrum scaling in agreement with canonical dimension analysis. Kolmogorov's -5/3 law is, thus, recovered for ε = 2 as also predicted by perturbative renormalization. At variance with the perturbative prediction, the -5/3 law emerges in the presence of a saturation in the ε dependence of the scaling dimension of the eddy diffusivity at ε = 3/2 when, according to perturbative renormalization, the velocity field becomes infrared relevant.
International Nuclear Information System (INIS)
Srihirun, B; Meleshko, S V; Schulz, E
2006-01-01
The definition of an admitted Lie group of transformations for stochastic differential equations has been already presented for equations with one-dimensional Brownian motion. The transformation of the dependent variables involves time as well, and it has been proven that Brownian motion is transformed to Brownian motion. In this paper, we will discuss this concept for stochastic differential equations involving multi-dimensional Brownian motion and present applications to a variety of stochastic differential equations
A stochastic differential equation framework for the timewise dynamics of turbulent velocities
DEFF Research Database (Denmark)
Barndorff-Nielsen, Ole Eiler; Schmiegel, Jürgen
2008-01-01
We discuss a stochastic differential equation as a modeling framework for the timewise dynamics of turbulent velocities. The equation is capable of capturing basic stylized facts of the statistics of temporal velocity increments. In particular, we focus on the evolution of the probability density...
White noise solutions to the stochastic mKdV equation
International Nuclear Information System (INIS)
Zhang Zhongjun; Wei Caimin
2009-01-01
In this paper, we present the white noise solutions of the stochastic mKdV equation via the Hermite transformation and variable-coefficient generalized projected Ricatti equation expansion method. These solutions include white noise solitary wave solutions, white noise soliton-like solutions and white noise trigonometric function solutions.
Role of statistical linearization in the solution of nonlinear stochastic equations
International Nuclear Information System (INIS)
Budgor, A.B.
1977-01-01
The solution of a generalized Langevin equation is referred to as a stochastic process. If the external forcing function is Gaussian white noise, the forward Kolmogarov equation yields the transition probability density function. Nonlinear problems must be handled by approximation procedures e.g., perturbation theories, eigenfunction expansions, and nonlinear optimization procedures. After some comments on the first two of these, attention is directed to the third, and the method of statistical linearization is used to demonstrate a relation to the former two. Nonlinear stochastic systems exhibiting sustained or forced oscillations and the centered nonlinear Schroedinger equation in the presence of Gaussian white noise excitation are considered as examples. 5 figures, 2 tables
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Eman Ali Hussain
2015-01-01
Full Text Available Absract In this project A new method for solving Stochastic Differential Equations SDEs deriving by Wiener process numerically will be construct and implement using Accelerated Genetic Algorithm AGA. An SDE is a differential equation in which one or more of the terms and hence the solutions itself is a stochastic process. Solving stochastic differential equations requires going away from the recognizable deterministic setting of ordinary and partial differential equations into a world where the evolution of a quantity has an inherent random component and where the expected behavior of this quantity can be described in terms of probability distributions. We applied our method on the Ito formula which is equivalent to the SDE to find approximation solution of the SDEs. Numerical experiments illustrate the behavior of the proposed method.
International Nuclear Information System (INIS)
Grigoriu, Mircea; Samorodnitsky, Gennady
2004-01-01
Two methods are considered for assessing the asymptotic stability of the trivial solution of linear stochastic differential equations driven by Poisson white noise, interpreted as the formal derivative of a compound Poisson process. The first method attempts to extend a result for diffusion processes satisfying linear stochastic differential equations to the case of linear equations with Poisson white noise. The developments for the method are based on Ito's formula for semimartingales and Lyapunov exponents. The second method is based on a geometric ergodic theorem for Markov chains providing a criterion for the asymptotic stability of the solution of linear stochastic differential equations with Poisson white noise. Two examples are presented to illustrate the use and evaluate the potential of the two methods. The examples demonstrate limitations of the first method and the generality of the second method
Treatment of constraints in the stochastic quantization method and covariantized Langevin equation
International Nuclear Information System (INIS)
Ikegami, Kenji; Kimura, Tadahiko; Mochizuki, Riuji
1993-01-01
We study the treatment of the constraints in the stochastic quantization method. We improve the treatment of the stochastic consistency condition proposed by Namiki et al. by suitably taking into account the Ito calculus. Then we obtain an improved Langevin equation and the Fokker-Planck equation which naturally leads to the correct path integral quantization of the constrained system as the stochastic equilibrium state. This treatment is applied to an O(N) non-linear σ model and it is shown that singular terms appearing in the improved Langevin equation cancel out the δ n (0) divergences in one loop order. We also ascertain that the above Langevin equation, rewritten in terms of independent variables, is actually equivalent to the one in the general-coordinate transformation covariant and vielbein-rotation invariant formalism. (orig.)
Large deviations for solutions to stochastic recurrence equations under Kesten's condition
DEFF Research Database (Denmark)
Buraczewski, Dariusz; Damek, Ewa; Mikosch, Thomas Valentin
2013-01-01
In this paper we prove large deviations results for partial sums constructed from the solution to a stochastic recurrence equation. We assume Kesten’s condition [17] under which the solution of the stochastic recurrence equation has a marginal distribution with power law tails, while the noise...... sequence of the equations can have light tails. The results of the paper are analogs of those obtained by A.V. and S.V. Nagaev [21, 22] in the case of partial sums of iid random variables. In the latter case, the large deviation probabilities of the partial sums are essentially determined by the largest...... step size of the partial sum. For the solution to a stochastic recurrence equation, the magnitude of the large deviation probabilities is again given by the tail of the maximum summand, but the exact asymptotic tail behavior is also influenced by clusters of extreme values, due to dependencies...
Zalaletdinov, R. M.
1998-04-01
The averaging problem in general relativity is briefly discussed. A new setting of the problem as that of macroscopic description of gravitation is proposed. A covariant space-time averaging procedure is described. The structure of the geometry of macroscopic space-time, which follows from averaging Cartan's structure equations, is described and the correlation tensors present in the theory are discussed. The macroscopic field equations (averaged Einstein's equations) derived in the framework of the approach are presented and their structure is analysed. The correspondence principle for macroscopic gravity is formulated and a definition of the stress-energy tensor for the macroscopic gravitational field is proposed. It is shown that the physical meaning of using Einstein's equations with a hydrodynamic stress-energy tensor in looking for cosmological models means neglecting all gravitational field correlations. The system of macroscopic gravity equations to be solved when the correlations are taken into consideration is given and described.
Lv, Qiming; Schneider, Manuel K; Pitchford, Jonathan W
2008-08-01
We study individual plant growth and size hierarchy formation in an experimental population of Arabidopsis thaliana, within an integrated analysis that explicitly accounts for size-dependent growth, size- and space-dependent competition, and environmental stochasticity. It is shown that a Gompertz-type stochastic differential equation (SDE) model, involving asymmetric competition kernels and a stochastic term which decreases with the logarithm of plant weight, efficiently describes individual plant growth, competition, and variability in the studied population. The model is evaluated within a Bayesian framework and compared to its deterministic counterpart, and to several simplified stochastic models, using distributional validation. We show that stochasticity is an important determinant of size hierarchy and that SDE models outperform the deterministic model if and only if structural components of competition (asymmetry; size- and space-dependence) are accounted for. Implications of these results are discussed in the context of plant ecology and in more general modelling situations.
Tornøe, Christoffer W; Overgaard, Rune V; Agersø, Henrik; Nielsen, Henrik A; Madsen, Henrik; Jonsson, E Niclas
2005-08-01
The objective of the present analysis was to explore the use of stochastic differential equations (SDEs) in population pharmacokinetic/pharmacodynamic (PK/PD) modeling. The intra-individual variability in nonlinear mixed-effects models based on SDEs is decomposed into two types of noise: a measurement and a system noise term. The measurement noise represents uncorrelated error due to, for example, assay error while the system noise accounts for structural misspecifications, approximations of the dynamical model, and true random physiological fluctuations. Since the system noise accounts for model misspecifications, the SDEs provide a diagnostic tool for model appropriateness. The focus of the article is on the implementation of the Extended Kalman Filter (EKF) in NONMEM for parameter estimation in SDE models. Various applications of SDEs in population PK/PD modeling are illustrated through a systematic model development example using clinical PK data of the gonadotropin releasing hormone (GnRH) antagonist degarelix. The dynamic noise estimates were used to track variations in model parameters and systematically build an absorption model for subcutaneously administered degarelix. The EKF-based algorithm was successfully implemented in NONMEM for parameter estimation in population PK/PD models described by systems of SDEs. The example indicated that it was possible to pinpoint structural model deficiencies, and that valuable information may be obtained by tracking unexplained variations in parameters.
Noise Analysis of Single-Ended Input Differential Amplifier using Stochastic Differential Equation
Tarun Kumar Rawat; Abhirup Lahiri; Ashish Gupta
2008-01-01
In this paper, we analyze the effect of noise in a single- ended input differential amplifier working at high frequencies. Both extrinsic and intrinsic noise are analyzed using time domain method employing techniques from stochastic calculus. Stochastic differential equations are used to obtain autocorrelation functions of the output noise voltage and other solution statistics like mean and variance. The analysis leads to important design implications and suggests changes in the device parame...
Stochastic Effects for the Reaction-Duffing Equation with Wick-Type Product
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Jin Hyuk Choi
2016-01-01
Full Text Available We construct new explicit solutions of the Wick-type stochastic reaction-Duffing equation arising from mathematical physics with the help of the white noise theory and the system technique. Based on these exact solutions, we also discuss the influences of stochastic effects for dynamical behaviors according to functions h1(t, h2(t, and Brownian motion B(t which are the solitary wave group velocities.
Antisymmetrized molecular dynamics of wave packets with stochastic incorporation of Vlasov equation
International Nuclear Information System (INIS)
Ono, Akira; Horiuchi, Hisashi.
1996-01-01
The first purpose of this report is to present an extended AMD model which can generally describe such minor branching processes by removing the restriction on the one-body distribution function. This is done not by generalizing the wave packets to arbitrary single-particle wave functions but by representing the diffused and/or deformed wave packet as an ensemble of Gaussian wave packets. In other words, stochastic displacements are given to the wave packets in phase space so that the ensemble-average of the time evolution of the one-body distribution function is essentially equivalent to the solution of Vlasov equation which does not have any restriction on the shape of wave packets. This new model is called AMD-V. Although AMD-V is equivalent to Vlasov equation in the instantaneous time evolution of the one-body distribution function for an AMD wave function, AMD-V describes the branching into channels and the fluctuation of the mean field which are caused by the spreading or the splitting of the single-particle wave function. The second purpose of this report is to show the drastic effect of this new stochastic process of wave packet splitting on the dynamics of heavy ion collisions, especially in the fragmentation mechanism. We take the 40 Ca + 40 Ca system at the incident energy 35 MeV/nucleon. It will be shown that the reproduction of data by the AMD-V calculation is surprisingly good. We will see that the effect of the wave packet diffusion is crucially important to remove the spurious binary feature of the AMD calculation and to enable the multi-fragment final state. (J.P.N.)
Derivation of a volume-averaged neutron diffusion equation; Atomos para el desarrollo de Mexico
Energy Technology Data Exchange (ETDEWEB)
Vazquez R, R.; Espinosa P, G. [UAM-Iztapalapa, Av. San Rafael Atlixco 186, Col. Vicentina, Mexico D.F. 09340 (Mexico); Morales S, Jaime B. [UNAM, Laboratorio de Analisis en Ingenieria de Reactores Nucleares, Paseo Cuauhnahuac 8532, Jiutepec, Morelos 62550 (Mexico)]. e-mail: rvr@xanum.uam.mx
2008-07-01
This paper presents a general theoretical analysis of the problem of neutron motion in a nuclear reactor, where large variations on neutron cross sections normally preclude the use of the classical neutron diffusion equation. A volume-averaged neutron diffusion equation is derived which includes correction terms to diffusion and nuclear reaction effects. A method is presented to determine closure-relationships for the volume-averaged neutron diffusion equation (e.g., effective neutron diffusivity). In order to describe the distribution of neutrons in a highly heterogeneous configuration, it was necessary to extend the classical neutron diffusion equation. Thus, the volume averaged diffusion equation include two corrections factor: the first correction is related with the absorption process of the neutron and the second correction is a contribution to the neutron diffusion, both parameters are related to neutron effects on the interface of a heterogeneous configuration. (Author)
Set-valued and fuzzy stochastic integral equations driven by semimartingales under Osgood condition
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Malinowski Marek T.
2015-01-01
Full Text Available We analyze the set-valued stochastic integral equations driven by continuous semimartingales and prove the existence and uniqueness of solutions to such equations in the framework of the hyperspace of nonempty, bounded, convex and closed subsets of the Hilbert space L2 (consisting of square integrable random vectors. The coefficients of the equations are assumed to satisfy the Osgood type condition that is a generalization of the Lipschitz condition. Continuous dependence of solutions with respect to data of the equation is also presented. We consider equations driven by semimartingale Z and equations driven by processes A;M from decomposition of Z, where A is a process of finite variation and M is a local martingale. These equations are not equivalent. Finally, we show that the analysis of the set-valued stochastic integral equations can be extended to a case of fuzzy stochastic integral equations driven by semimartingales under Osgood type condition. To obtain our results we use the set-valued and fuzzy Maruyama type approximations and Bihari’s inequality.
International Nuclear Information System (INIS)
McCarthy, S; Rachinskii, D
2011-01-01
We describe two Euler type numerical schemes obtained by discretisation of a stochastic differential equation which contains the Preisach memory operator. Equations of this type are of interest in areas such as macroeconomics and terrestrial hydrology where deterministic models containing the Preisach operator have been developed but do not fully encapsulate stochastic aspects of the area. A simple price dynamics model is presented as one motivating example for our studies. Some numerical evidence is given that the two numerical schemes converge to the same limit as the time step decreases. We show that the Preisach term introduces a damping effect which increases on the parts of the trajectory demonstrating a stronger upwards or downwards trend. The results are preliminary to a broader programme of research of stochastic differential equations with the Preisach hysteresis operator.
International Nuclear Information System (INIS)
Caraballo, T.; Kloeden, P.E.
2006-01-01
Under a one-sided dissipative Lipschitz condition on its drift, a stochastic evolution equation with additive noise of the reaction-diffusion type is shown to have a unique stochastic stationary solution which pathwise attracts all other solutions. A similar situation holds for each Galerkin approximation and each implicit Euler scheme applied to these Galerkin approximations. Moreover, the stationary solution of the Euler scheme converges pathwise to that of the Galerkin system as the stepsize tends to zero and the stationary solutions of the Galerkin systems converge pathwise to that of the evolution equation as the dimension increases. The analysis is carried out on random partial and ordinary differential equations obtained from their stochastic counterparts by subtraction of appropriate Ornstein-Uhlenbeck stationary solutions
Short-term Probabilistic Forecasting of Wind Speed Using Stochastic Differential Equations
DEFF Research Database (Denmark)
Iversen, Jan Emil Banning; Morales González, Juan Miguel; Møller, Jan Kloppenborg
2016-01-01
and uncertain nature. In this paper, we propose a modeling framework for wind speed that is based on stochastic differential equations. We show that stochastic differential equations allow us to naturally capture the time dependence structure of wind speed prediction errors (from 1 up to 24 hours ahead) and......It is widely accepted today that probabilistic forecasts of wind power production constitute valuable information for both wind power producers and power system operators to economically exploit this form of renewable energy, while mitigating the potential adverse effects related to its variable......, most importantly, to derive point and quantile forecasts, predictive distributions, and time-path trajectories (also referred to as scenarios or ensemble forecasts), all by one single stochastic differential equation model characterized by a few parameters....
On a stochastic Burgers equation with Dirichlet boundary conditions
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Ekaterina T. Kolkovska
2003-01-01
Full Text Available We consider the one-dimensional Burgers equation perturbed by a white noise term with Dirichlet boundary conditions and a non-Lipschitz coefficient. We obtain existence of a weak solution proving tightness for a sequence of polygonal approximations for the equation and solving a martingale problem for the weak limit.
Averaging of the Equations of the Standard Cosmological Model over Rapid Oscillations
Ignat'ev, Yu. G.; Samigullina, A. R.
2017-11-01
An averaging of the equations of the standard cosmological model (SCM) is carried out. It is shown that the main contribution to the macroscopic energy density of the scalar field comes from its microscopic oscillations with the Compton period. The effective macroscopic equation of state of the oscillations of the scalar field corresponds to the nonrelativistic limit.
Methods for solving the stochastic point reactor kinetic equations
International Nuclear Information System (INIS)
Quabili, E.R.; Karasulu, M.
1979-01-01
Two new methods are presented for analysis of the statistical properties of nonlinear outputs of a point reactor to stochastic non-white reactivity inputs. They are Bourret's approximation and logarithmic linearization. The results have been compared with the exact results, previously obtained in the case of Gaussian white reactivity input. It was found that when the reactivity noise has short correlation time, Bourret's approximation should be recommended because it yields results superior to those yielded by logarithmic linearization. When the correlation time is long, Bourret's approximation is not valid, but in that case, if one can assume the reactivity noise to be Gaussian, one may use the logarithmic linearization. (author)
Simulation of quantum dynamics based on the quantum stochastic differential equation.
Li, Ming
2013-01-01
The quantum stochastic differential equation derived from the Lindblad form quantum master equation is investigated. The general formulation in terms of environment operators representing the quantum state diffusion is given. The numerical simulation algorithm of stochastic process of direct photodetection of a driven two-level system for the predictions of the dynamical behavior is proposed. The effectiveness and superiority of the algorithm are verified by the performance analysis of the accuracy and the computational cost in comparison with the classical Runge-Kutta algorithm.
Weak Second Order Explicit Stabilized Methods for Stiff Stochastic Differential Equations
Abdulle, Assyr
2013-01-01
We introduce a new family of explicit integrators for stiff Itô stochastic differential equations (SDEs) of weak order two. These numerical methods belong to the class of one-step stabilized methods with extended stability domains and do not suffer from the step size reduction faced by standard explicit methods. The family is based on the standard second order orthogonal Runge-Kutta-Chebyshev (ROCK2) methods for deterministic problems. The convergence, meansquare, and asymptotic stability properties of the methods are analyzed. Numerical experiments, including applications to nonlinear SDEs and parabolic stochastic partial differential equations are presented and confirm the theoretical results. © 2013 Society for Industrial and Applied Mathematics.
The propagation of travelling waves for stochastic generalized KPP equations
International Nuclear Information System (INIS)
Elworthy, K.D.; Zhao, H.Z.
1993-09-01
We study the existence and propagation of approximate travelling waves of generalized KPP equations with seasonal multiplicative white noise perturbations of Ito type. Three regimes of perturbation are considered: weak, milk, and strong. We show that weak perturbations have little effect on the wave like solutions of the unperturbed equations while strong perturbations essentially destroy the wave and force the solutions to die down. For mild perturbations we show that there is a residual wave form but propagating at a different speed to that of the unperturbed equation. In the appendix J.G. Gaines illustrates these different regimes by computer simulations. (author). 27 refs, 13 figs
Lyapunov functionals and stability of stochastic difference equations
Shaikhet, Leonid
2011-01-01
This book offers a general method of Lyapunov functional construction which lets researchers analyze the degree to which the stability properties of differential equations are preserved in their difference analogues. Includes examples from physical systems.
International Nuclear Information System (INIS)
Levenshtam, V B
2006-01-01
We justify the averaging method for abstract parabolic equations with stationary principal part that contain non-linearities (subordinate to the principal part) some of whose terms are rapidly oscillating in time with zero mean and are proportional to the square root of the frequency of oscillation. Our interest in the exponent 1/2 is motivated by the fact that terms proportional to lower powers of the frequency have no influence on the average. For linear equations of the same type, we justify an algorithm for the study of the stability of solutions in the case when the stationary averaged problem has eigenvalues on the imaginary axis (the critical case)
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Liu Yang
2017-01-01
Full Text Available We construct a new two-stage stochastic model of supply chain with multiple factories and distributors for perishable product. By introducing a second-order stochastic dominance (SSD constraint, we can describe the preference consistency of the risk taker while minimizing the expected cost of company. To solve this problem, we convert it into a one-stage stochastic model equivalently; then we use sample average approximation (SAA method to approximate the expected values of the underlying random functions. A smoothing approach is proposed with which we can get the global solution and avoid introducing new variables and constraints. Meanwhile, we investigate the convergence of an optimal value from solving the transformed model and show that, with probability approaching one at exponential rate, the optimal value converges to its counterpart as the sample size increases. Numerical results show the effectiveness of the proposed algorithm and analysis.
Klyatskin, Valery I
2015-01-01
This monograph set presents a consistent and self-contained framework of stochastic dynamic systems with maximal possible completeness. Volume 1 presents the basic concepts, exact results, and asymptotic approximations of the theory of stochastic equations on the basis of the developed functional approach. This approach offers a possibility of both obtaining exact solutions to stochastic problems for a number of models of fluctuating parameters and constructing various asymptotic buildings. Ideas of statistical topography are used to discuss general issues of generating coherent structures from chaos with probability one, i.e., almost in every individual realization of random parameters. The general theory is illustrated with certain problems and applications of stochastic mathematical physics in various fields such as mechanics, hydrodynamics, magnetohydrodynamics, acoustics, optics, and radiophysics.
Krylov, N. V.; Priola, E.
2017-09-01
We show, among other things, how knowing Schauder or Sobolev-space estimates for the one-dimensional heat equation allows one to derive their multidimensional analogs for equations with coefficients depending only on the time variable with the same constants as in the case of the one-dimensional heat equation. The method is quite general and is based on using the Poisson stochastic process. It also applies to equations involving non-local operators. It looks like no other methods are available at this time and it is a very challenging problem to find a purely analytical approach to proving such results.
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Diem Dang Huan
2015-12-01
Full Text Available The current paper is concerned with the controllability of nonlocal second-order impulsive neutral stochastic functional integro-differential equations with infinite delay and Poisson jumps in Hilbert spaces. Using the theory of a strongly continuous cosine family of bounded linear operators, stochastic analysis theory and with the help of the Banach fixed point theorem, we derive a new set of sufficient conditions for the controllability of nonlocal second-order impulsive neutral stochastic functional integro-differential equations with infinite delay and Poisson jumps. Finally, an application to the stochastic nonlinear wave equation with infinite delay and Poisson jumps is given.
Deterministic and stochastic evolution equations for fully dispersive and weakly nonlinear waves
DEFF Research Database (Denmark)
Eldeberky, Y.; Madsen, Per A.
1999-01-01
and stochastic formulations are solved numerically for the case of cross shore motion of unidirectional waves and the results are verified against laboratory data for wave propagation over submerged bars and over a plane slope. Outside the surf zone the two model predictions are generally in good agreement......This paper presents a new and more accurate set of deterministic evolution equations for the propagation of fully dispersive, weakly nonlinear, irregular, multidirectional waves. The equations are derived directly from the Laplace equation with leading order nonlinearity in the surface boundary...... is significantly underestimated for larger wave numbers. In the present work we correct this inconsistency. In addition to the improved deterministic formulation, we present improved stochastic evolution equations in terms of the energy spectrum and the bispectrum for multidirectional waves. The deterministic...
On the interpretations of Langevin stochastic equation in different coordinate systems
International Nuclear Information System (INIS)
Martinez, E.; Lopez-Diaz, L.; Torres, L.; Alejos, O.
2004-01-01
The stochastic Langevin Landau-Lifshitz equation is usually utilized in micromagnetics formalism to account for thermal effects. Commonly, two different interpretations of the stochastic integrals can be made: Ito and Stratonovich. In this work, the Langevin-Landau-Lifshitz (LLL) equation is written in both Cartesian and Spherical coordinates. If Spherical coordinates are employed, the noise is additive, and therefore, Ito and Stratonovich solutions are equal. This is not the case when (LLL) equation is written in Cartesian coordinates. In this case, the Langevin equation must be interpreted in the Stratonovich sense in order to reproduce correct statistical results. Nevertheless, the statistics of the numerical results obtained from Euler-Ito and Euler-Stratonovich schemes are equivalent due to the additional numerical constraint imposed in Cartesian system after each time step, which itself assures that the magnitude of the magnetization is preserved
Maddix, Danielle C.; Sampaio, Luiz; Gerritsen, Margot
2018-05-01
The degenerate parabolic Generalized Porous Medium Equation (GPME) poses numerical challenges due to self-sharpening and its sharp corner solutions. For these problems, we show results for two subclasses of the GPME with differentiable k (p) with respect to p, namely the Porous Medium Equation (PME) and the superslow diffusion equation. Spurious temporal oscillations, and nonphysical locking and lagging have been reported in the literature. These issues have been attributed to harmonic averaging of the coefficient k (p) for small p, and arithmetic averaging has been suggested as an alternative. We show that harmonic averaging is not solely responsible and that an improved discretization can mitigate these issues. Here, we investigate the causes of these numerical artifacts using modified equation analysis. The modified equation framework can be used for any type of discretization. We show results for the second order finite volume method. The observed problems with harmonic averaging can be traced to two leading error terms in its modified equation. This is also illustrated numerically through a Modified Harmonic Method (MHM) that can locally modify the critical terms to remove the aforementioned numerical artifacts.
International Nuclear Information System (INIS)
Bouard, Anne de; Debussche, Arnaud
2006-01-01
In this article we analyze the error of a semidiscrete scheme for the stochastic nonlinear Schrodinger equation with power nonlinearity. We consider supercritical or subcritical nonlinearity and the equation can be either focusing or defocusing. Allowing sufficient spatial regularity we prove that the numerical scheme has strong order 1/2 in general and order 1 if the noise is additive. Furthermore, we also prove that the weak order is always 1
Advances in nonlinear partial differential equations and stochastics
Kawashima, S
1998-01-01
In the past two decades, there has been great progress in the theory of nonlinear partial differential equations. This book describes the progress, focusing on interesting topics in gas dynamics, fluid dynamics, elastodynamics etc. It contains ten articles, each of which discusses a very recent result obtained by the author. Some of these articles review related results.
A model for the stochastic origins of Schrodinger's equation
Davidson, Mark P.
2001-01-01
A model for the motion of a charged particle in the vacuum is presented which, although purely classical in concept, yields Schrodinger's equation as a solution. It suggests that the origins of the peculiar and nonclassical features of quantum mechanics are actually inherent in a statistical description of the radiative reactive force.
Xiao-Li Ding; Juan J. Nieto
2018-01-01
In this paper, we investigate analytical solutions of multi-time scale fractional stochastic differential equations driven by fractional Brownian motions. We firstly decompose homogeneous multi-time scale fractional stochastic differential equations driven by fractional Brownian motions into independent differential subequations, and give their analytical solutions. Then, we use the variation of constant parameters to obtain the solutions of nonhomogeneous multi-time scale fractional stochast...
Numerical schemes for dynamically orthogonal equations of stochastic fluid and ocean flows
International Nuclear Information System (INIS)
Ueckermann, M.P.; Lermusiaux, P.F.J.; Sapsis, T.P.
2013-01-01
The quantification of uncertainties is critical when systems are nonlinear and have uncertain terms in their governing equations or are constrained by limited knowledge of initial and boundary conditions. Such situations are common in multiscale, intermittent and non-homogeneous fluid and ocean flows. The dynamically orthogonal (DO) field equations provide an adaptive methodology to predict the probability density functions of such flows. The present work derives efficient computational schemes for the DO methodology applied to unsteady stochastic Navier–Stokes and Boussinesq equations, and illustrates and studies the numerical aspects of these schemes. Semi-implicit projection methods are developed for the mean and for the DO modes, and time-marching schemes of first to fourth order are used for the stochastic coefficients. Conservative second-order finite-volumes are employed in physical space with new advection schemes based on total variation diminishing methods. Other results include: (i) the definition of pseudo-stochastic pressures to obtain a number of pressure equations that is linear in the subspace size instead of quadratic; (ii) symmetric advection schemes for the stochastic velocities; (iii) the use of generalized inversion to deal with singular subspace covariances or deterministic modes; and (iv) schemes to maintain orthonormal modes at the numerical level. To verify our implementation and study the properties of our schemes and their variations, a set of stochastic flow benchmarks are defined including asymmetric Dirac and symmetric lock-exchange flows, lid-driven cavity flows, and flows past objects in a confined channel. Different Reynolds number and Grashof number regimes are employed to illustrate robustness. Optimal convergence under both time and space refinements is shown as well as the convergence of the probability density functions with the number of stochastic realizations.
On the small time asymptotics of 3D stochastic primitive equations
Dong, Zhao; Zhang, Rangrang
2017-01-01
In this paper, we establish a small time large deviation principle for the strong solution of 3D stochastic primitive equations driven by multiplicative noise. Both the small noise and the small, but highly nonlinear, unbounded nonlinear terms should be taken into consideration.
Parametric inference for stochastic differential equations: a smooth and match approach
Gugushvili, S.; Spreij, P.
2012-01-01
We study the problem of parameter estimation for a univariate discretely observed ergodic diffusion process given as a solution to a stochastic differential equation. The estimation procedure we propose consists of two steps. In the first step, which is referred to as a smoothing step, we smooth the
DEFF Research Database (Denmark)
Møller, Jan Kloppenborg; Philipsen, Kirsten Riber; Christiansen, Lasse Engbo
2012-01-01
In the present study, bacterial growth in a rich media is analysed in a Stochastic Differential Equation (SDE) framework. It is demonstrated that the SDE formulation and smoothened state estimates provide a systematic framework for data driven model improvements, using random walk hidden states...
Existence and Uniqueness of Solutions to the Stochastic Porous Media Equations of Saturated Flows
International Nuclear Information System (INIS)
Ciotir, Ioana
2010-01-01
This paper proves the existence and uniqueness of nonnegative solutions for the stochastic porous media equations with multiplicative noise, infinite jump and discontinuous diffusivity function relevant in description of saturation processes in underground water infiltration in a bounded domain of R 3 .
Directory of Open Access Journals (Sweden)
John A. D. Appleby
2008-01-01
Full Text Available This paper considers necessary and sufficient conditions for the solution of a stochastically and deterministically perturbed Volterra equation to converge exponentially to a nonequilibrium and nontrivial limit. Convergence in an almost sure and pth mean sense is obtained.
Weak Second Order Explicit Stabilized Methods for Stiff Stochastic Differential Equations
Abdulle, Assyr; Vilmart, Gilles; Zygalakis, Konstantinos C.
2013-01-01
We introduce a new family of explicit integrators for stiff Itô stochastic differential equations (SDEs) of weak order two. These numerical methods belong to the class of one-step stabilized methods with extended stability domains and do not suffer
On the history of a stochastic ansatz for solving the transport equation
International Nuclear Information System (INIS)
Williams, M.M.R.
2010-01-01
A very useful approximate tool for understanding the role of random material properties on solutions of the transport equation is described and its historical derivation given. The development of this stochastic tool, from its introduction by Randall, to its use in describing current problems involving dichotomic or pseudo-dichotomic Markov processes is discussed.
Statistical inference for discrete-time samples from affine stochastic delay differential equations
DEFF Research Database (Denmark)
Küchler, Uwe; Sørensen, Michael
2013-01-01
Statistical inference for discrete time observations of an affine stochastic delay differential equation is considered. The main focus is on maximum pseudo-likelihood estimators, which are easy to calculate in practice. A more general class of prediction-based estimating functions is investigated...
International Nuclear Information System (INIS)
Menezes, G.; Svaiter, N.F.
2006-04-01
We use the method of stochastic quantization in a topological field theory defined in an Euclidean space, assuming a Langevin equation with a memory kernel. We show that our procedure for the Abelian Chern-Simons theory converges regardless of the nature of the Chern-Simons coefficient. (author)
DEFF Research Database (Denmark)
Debrabant, Kristian; Samaey, Giovanni; Zieliński, Przemysław
2017-01-01
We present and analyse a micro-macro acceleration method for the Monte Carlo simulation of stochastic differential equations with separation between the (fast) time-scale of individual trajectories and the (slow) time-scale of the macroscopic function of interest. The algorithm combines short...
Davies, I M; Zhao, H
2004-01-01
We study the inviscid limit, $\\mu\\to 0$, of the stochastic viscous Burgers equation, for the velocity field $v^{\\mu}(x,t)$, $t>0$, $x\\in\\mathbb R^d$,\\frac{\\partial{v^{\\mu}}}{\\partial{t}} + (v^{\\mu}\\cdot\
International Nuclear Information System (INIS)
Frank, T D
2005-01-01
Stationary distributions of processes are derived that involve a time delay and are defined by a linear stochastic neutral delay differential equation. The distributions are Gaussian distributions. The variances of the Gaussian distributions are either monotonically increasing or decreasing functions of the time delays. The variances become infinite when fixed points of corresponding deterministic processes become unstable. (letter to the editor)
Large Deviations for Stochastic Tamed 3D Navier-Stokes Equations
International Nuclear Information System (INIS)
Roeckner, Michael; Zhang, Tusheng; Zhang Xicheng
2010-01-01
In this paper, using weak convergence method, we prove a large deviation principle of Freidlin-Wentzell type for the stochastic tamed 3D Navier-Stokes equations driven by multiplicative noise, which was investigated in (Roeckner and Zhang in Probab. Theory Relat. Fields 145(1-2), 211-267, 2009).
Lp Theory for Super-Parabolic Backward Stochastic Partial Differential Equations in the Whole Space
International Nuclear Information System (INIS)
Du Kai; Qiu, Jinniao; Tang Shanjian
2012-01-01
This paper is concerned with semi-linear backward stochastic partial differential equations (BSPDEs for short) of super-parabolic type. An L p -theory is given for the Cauchy problem of BSPDEs, separately for the case of p∈(1,2] and for the case of p∈(2,∞). A comparison theorem is also addressed.
DEFF Research Database (Denmark)
Knudsen, Thomas Skov
1997-01-01
For stochastic differential equations (SDEs) of the form dX(t) = b(X)(t)) dt + sigma(X(t))dW(t) where b and sigma are Lipschitz continuous, it is shown that if we consider a fixed sigma is an element of C-5, bounded and with bounded derivatives, the random field of solutions is pathwise locally...
Directory of Open Access Journals (Sweden)
Zhanhua Yu
2011-01-01
Full Text Available We study the almost surely asymptotic stability of exact solutions to neutral stochastic pantograph equations (NSPEs, and sufficient conditions are obtained. Based on these sufficient conditions, we show that the backward Euler method (BEM with variable stepsize can preserve the almost surely asymptotic stability. Numerical examples are demonstrated for illustration.
Modelling the heat dynamics of a building using stochastic differential equations
DEFF Research Database (Denmark)
Andersen, Klaus Kaae; Madsen, Henrik; Hansen, Lars Henrik
2000-01-01
estimation and model validation, while physical knowledge is used in forming the model structure. The suggested lumped parameter model is thus based on thermodynamics and formulated as a system of stochastic differential equations. Due to the continuous time formulation the parameters of the model...
Projection scheme for a reflected stochastic heat equation with additive noise
Higa, Arturo Kohatsu; Pettersson, Roger
2005-02-01
We consider a projection scheme as a numerical solution of a reflected stochastic heat equation driven by a space-time white noise. Convergence is obtained via a discrete contraction principle and known convergence results for numerical solutions of parabolic variational inequalities.
Energy Technology Data Exchange (ETDEWEB)
Webster, Clayton G [ORNL; Zhang, Guannan [ORNL; Gunzburger, Max D [ORNL
2012-10-01
Accurate predictive simulations of complex real world applications require numerical approximations to first, oppose the curse of dimensionality and second, converge quickly in the presence of steep gradients, sharp transitions, bifurcations or finite discontinuities in high-dimensional parameter spaces. In this paper we present a novel multi-dimensional multi-resolution adaptive (MdMrA) sparse grid stochastic collocation method, that utilizes hierarchical multiscale piecewise Riesz basis functions constructed from interpolating wavelets. The basis for our non-intrusive method forms a stable multiscale splitting and thus, optimal adaptation is achieved. Error estimates and numerical examples will used to compare the efficiency of the method with several other techniques.
Stochastic Computational Approach for Complex Nonlinear Ordinary Differential Equations
International Nuclear Information System (INIS)
Khan, Junaid Ali; Raja, Muhammad Asif Zahoor; Qureshi, Ijaz Mansoor
2011-01-01
We present an evolutionary computational approach for the solution of nonlinear ordinary differential equations (NLODEs). The mathematical modeling is performed by a feed-forward artificial neural network that defines an unsupervised error. The training of these networks is achieved by a hybrid intelligent algorithm, a combination of global search with genetic algorithm and local search by pattern search technique. The applicability of this approach ranges from single order NLODEs, to systems of coupled differential equations. We illustrate the method by solving a variety of model problems and present comparisons with solutions obtained by exact methods and classical numerical methods. The solution is provided on a continuous finite time interval unlike the other numerical techniques with comparable accuracy. With the advent of neuroprocessors and digital signal processors the method becomes particularly interesting due to the expected essential gains in the execution speed. (general)
Stochastic Averaging for Constrained Optimization With Application to Online Resource Allocation
Chen, Tianyi; Mokhtari, Aryan; Wang, Xin; Ribeiro, Alejandro; Giannakis, Georgios B.
2017-06-01
Existing approaches to resource allocation for nowadays stochastic networks are challenged to meet fast convergence and tolerable delay requirements. The present paper leverages online learning advances to facilitate stochastic resource allocation tasks. By recognizing the central role of Lagrange multipliers, the underlying constrained optimization problem is formulated as a machine learning task involving both training and operational modes, with the goal of learning the sought multipliers in a fast and efficient manner. To this end, an order-optimal offline learning approach is developed first for batch training, and it is then generalized to the online setting with a procedure termed learn-and-adapt. The novel resource allocation protocol permeates benefits of stochastic approximation and statistical learning to obtain low-complexity online updates with learning errors close to the statistical accuracy limits, while still preserving adaptation performance, which in the stochastic network optimization context guarantees queue stability. Analysis and simulated tests demonstrate that the proposed data-driven approach improves the delay and convergence performance of existing resource allocation schemes.
International Nuclear Information System (INIS)
Brenner, H.
1991-01-01
Macrotransport processes (generalized Taylor dispersion phenomena) constitute coarse-grained descriptions of comparable convective diffusive-reactive microtransport processes, the latter supposed governed by microscale linear constitutive equations and boundary conditions, but characterized by spatially nonuniform phenomenological coefficients. Following a brief review of existing applications of the theory, the author focuses - by way of background information-upon the original (and now classical) Taylor - Aris dispersion problem, involving the combined convective and molecular diffusive transport of a point-size Brownian solute molecule (tracer) suspended in a Poiseuille solvent flow within a circular tube. A series of elementary generalizations of this prototype problem to chromatographic-like solute transport processes in tubes is used to illustrate some novel statistical-physical features. These examples emphasize the fact that a solute molecule may, on average, move axially down the tube at a different mean velocity (either larger or smaller) than that of a solvent molecule. Moreover, this solute molecule may suffer axial dispersion about its mean velocity at a rate greatly exceeding that attributable to its axial molecular diffusion alone. Such chromatographic anomalies represent novel macroscale non-linearities originating from physicochemical interactions between spatially inhomogeneous convective-diffusive-reactive microtransport processes
Stochastic substitute for coupled rate equations in the modeling of highly ionized transient plasmas
International Nuclear Information System (INIS)
Eliezer, S.; Falquina, R.; Minguez, E.
1994-01-01
Plasmas produced by intense laser pulses incident on solid targets often do not satisfy the conditions for local thermodynamic equilibrium, and so cannot be modeled by transport equations relying on equations of state. A proper description involves an excessively large number of coupled rate equations connecting many quantum states of numerous species having different degrees of ionization. Here we pursue a recent suggestion to model the plasma by a few dominant states perturbed by a stochastic driving force. The driving force is taken to be a Poisson impulse process, giving a Langevin equation which is equivalent to a Fokker-Planck equation for the probability density governing the distribution of electron density. An approximate solution to the Langevin equation permits calculation of the characteristic relaxation rate. An exact stationary solution to the Fokker-Planck equation is given as a function of the strength of the stochastic driving force. This stationary solution is used, along with a Laplace transform, to convert the Fokker-Planck equation to one of Schroedinger type. We consider using the classical Hamiltonian formalism and the WKB method to obtain the time-dependent solution
Exponentially Stable Stationary Solutions for Stochastic Evolution Equations and Their Perturbation
International Nuclear Information System (INIS)
Caraballo, Tomas; Kloeden, Peter E.; Schmalfuss, Bjoern
2004-01-01
We consider the exponential stability of stochastic evolution equations with Lipschitz continuous non-linearities when zero is not a solution for these equations. We prove the existence of anon-trivial stationary solution which is exponentially stable, where the stationary solution is generated by the composition of a random variable and the Wiener shift. We also construct stationary solutions with the stronger property of attracting bounded sets uniformly. The existence of these stationary solutions follows from the theory of random dynamical systems and their attractors. In addition, we prove some perturbation results and formulate conditions for the existence of stationary solutions for semilinear stochastic partial differential equations with Lipschitz continuous non-linearities
International Nuclear Information System (INIS)
Lee, Kok Foong; Patterson, Robert I.A.; Wagner, Wolfgang; Kraft, Markus
2015-01-01
Graphical abstract: -- Highlights: •Problems concerning multi-compartment population balance equations are studied. •A class of fragmentation weight transfer functions is presented. •Three stochastic weighted algorithms are compared against the direct simulation algorithm. •The numerical errors of the stochastic solutions are assessed as a function of fragmentation rate. •The algorithms are applied to a multi-dimensional granulation model. -- Abstract: This paper introduces stochastic weighted particle algorithms for the solution of multi-compartment population balance equations. In particular, it presents a class of fragmentation weight transfer functions which are constructed such that the number of computational particles stays constant during fragmentation events. The weight transfer functions are constructed based on systems of weighted computational particles and each of it leads to a stochastic particle algorithm for the numerical treatment of population balance equations. Besides fragmentation, the algorithms also consider physical processes such as coagulation and the exchange of mass with the surroundings. The numerical properties of the algorithms are compared to the direct simulation algorithm and an existing method for the fragmentation of weighted particles. It is found that the new algorithms show better numerical performance over the two existing methods especially for systems with significant amount of large particles and high fragmentation rates.
Energy Technology Data Exchange (ETDEWEB)
Lee, Kok Foong [Department of Chemical Engineering and Biotechnology, University of Cambridge, New Museums Site, Pembroke Street, Cambridge CB2 3RA (United Kingdom); Patterson, Robert I.A.; Wagner, Wolfgang [Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117 Berlin (Germany); Kraft, Markus, E-mail: mk306@cam.ac.uk [Department of Chemical Engineering and Biotechnology, University of Cambridge, New Museums Site, Pembroke Street, Cambridge CB2 3RA (United Kingdom); School of Chemical and Biomedical Engineering, Nanyang Technological University, 62 Nanyang Drive, Singapore, 637459 (Singapore)
2015-12-15
Graphical abstract: -- Highlights: •Problems concerning multi-compartment population balance equations are studied. •A class of fragmentation weight transfer functions is presented. •Three stochastic weighted algorithms are compared against the direct simulation algorithm. •The numerical errors of the stochastic solutions are assessed as a function of fragmentation rate. •The algorithms are applied to a multi-dimensional granulation model. -- Abstract: This paper introduces stochastic weighted particle algorithms for the solution of multi-compartment population balance equations. In particular, it presents a class of fragmentation weight transfer functions which are constructed such that the number of computational particles stays constant during fragmentation events. The weight transfer functions are constructed based on systems of weighted computational particles and each of it leads to a stochastic particle algorithm for the numerical treatment of population balance equations. Besides fragmentation, the algorithms also consider physical processes such as coagulation and the exchange of mass with the surroundings. The numerical properties of the algorithms are compared to the direct simulation algorithm and an existing method for the fragmentation of weighted particles. It is found that the new algorithms show better numerical performance over the two existing methods especially for systems with significant amount of large particles and high fragmentation rates.
Stochastic solutions of Navier-Stokes equations: an experimental evidence.
Djurek, Ivan; Djurek, Danijel; Petosić, Antonio
2010-12-01
An electrodynamic loudspeaker has been operated in anharmonic regime indicated by the nonlinear ordinary differential equation when spring constant γ in restoring term, as well as, viscoelasticity of the membrane material, increases with displacement. For driving currents in the range of 2.8-3.3 A, doubling of the vibration period appears, while for currents in the range of 3.3-3.6 A, multiple sequences of subharmonic vibrations begin with f/4 and 3f/4. An application of currents higher than 3.6 A results in a spectrum, characteristic for the chaotic state. The loudspeaker was then operated in a closed chamber, and subharmonic vibrations disappeared by an evacuation. Subsequent injection of air revoked them again at ∼ 120 mbar (Re(')=476) when air viscous forces dominate the Morse convection. At 430 mbar (Re=538) single vibration state was restored, and the phenomenon is in an agreement with prediction of the five mode truncation procedure applied to the Navier-Stokes equations describing a two-dimensional incompressible fluid. © 2010 American Institute of Physics.
Fractional diffusion equation with distributed-order material derivative. Stochastic foundations
International Nuclear Information System (INIS)
Magdziarz, M; Teuerle, M
2017-01-01
In this paper, we present the stochastic foundations of fractional dynamics driven by the fractional material derivative of distributed-order type. Before stating our main result, we present the stochastic scenario which underlies the dynamics given by the fractional material derivative. Then we introduce the Lévy walk process of distributed-order type to establish our main result, which is the scaling limit of the considered process. It appears that the probability density function of the scaling limit process fulfills, in a weak sense, the fractional diffusion equation with the material derivative of distributed-order type. (paper)
Hutzenthaler, Martin
2015-01-01
Many stochastic differential equations (SDEs) in the literature have a superlinearly growing nonlinearity in their drift or diffusion coefficient. Unfortunately, moments of the computationally efficient Euler-Maruyama approximation method diverge for these SDEs in finite time. This article develops a general theory based on rare events for studying integrability properties such as moment bounds for discrete-time stochastic processes. Using this approach, the authors establish moment bounds for fully and partially drift-implicit Euler methods and for a class of new explicit approximation method
Probabilistic Forecast of Wind Power Generation by Stochastic Differential Equation Models
Elkantassi, Soumaya
2017-04-01
Reliable forecasting of wind power generation is crucial to optimal control of costs in generation of electricity with respect to the electricity demand. Here, we propose and analyze stochastic wind power forecast models described by parametrized stochastic differential equations, which introduce appropriate fluctuations in numerical forecast outputs. We use an approximate maximum likelihood method to infer the model parameters taking into account the time correlated sets of data. Furthermore, we study the validity and sensitivity of the parameters for each model. We applied our models to Uruguayan wind power production as determined by historical data and corresponding numerical forecasts for the period of March 1 to May 31, 2016.
International Nuclear Information System (INIS)
Papiez, L.; Moskvin, V.; Tulovsky, V.
2001-01-01
The process of angular-spatial evolution of multiple scattering of charged particles can be described by a special case of Boltzmann integro-differential equation called Lewis equation. The underlying stochastic process for this evolution is the compound Poisson process on the surface of the unit sphere. The significant portion of events that constitute compound Poisson process that describes multiple scattering have diffusional character. This property allows to analyze the process of angular-spatial evolution of multiple scattering of charged particles as combination of soft and hard collision processes and compute appropriately its transition densities. These computations provide a method of the approximate solution to the Lewis equation. (orig.)
Stochastic differential equations for quantum dynamics of spin-boson networks
International Nuclear Information System (INIS)
Mandt, Stephan; Sadri, Darius; Houck, Andrew A; Türeci, Hakan E
2015-01-01
A popular approach in quantum optics is to map a master equation to a stochastic differential equation, where quantum effects manifest themselves through noise terms. We generalize this approach based on the positive-P representation to systems involving spin, in particular networks or lattices of interacting spins and bosons. We test our approach on a driven dimer of spins and photons, compare it to the master equation, and predict a novel dynamic phase transition in this system. Our numerical approach has scaling advantages over existing methods, but typically requires regularization in terms of drive and dissipation. (paper)
Coarse-mesh discretized low-order quasi-diffusion equations for subregion averaged scalar fluxes
International Nuclear Information System (INIS)
Anistratov, D. Y.
2004-01-01
In this paper we develop homogenization procedure and discretization for the low-order quasi-diffusion equations on coarse grids for core-level reactor calculations. The system of discretized equations of the proposed method is formulated in terms of the subregion averaged group scalar fluxes. The coarse-mesh solution is consistent with a given fine-mesh discretization of the transport equation in the sense that it preserves a set of average values of the fine-mesh transport scalar flux over subregions of coarse-mesh cells as well as the surface currents, and eigenvalue. The developed method generates numerical solution that mimics the large-scale behavior of the transport solution within assemblies. (authors)
Jimenez, M. Navarro; Le Maî tre, O. P.; Knio, Omar
2017-01-01
A Galerkin polynomial chaos (PC) method was recently proposed to perform variance decomposition and sensitivity analysis in stochastic differential equations (SDEs), driven by Wiener noise and involving uncertain parameters. The present paper extends the PC method to nonintrusive approaches enabling its application to more complex systems hardly amenable to stochastic Galerkin projection methods. We also discuss parallel implementations and the variance decomposition of the derived quantity of interest within the framework of nonintrusive approaches. In particular, a novel hybrid PC-sampling-based strategy is proposed in the case of nonsmooth quantities of interest (QoIs) but smooth SDE solution. Numerical examples are provided that illustrate the decomposition of the variance of QoIs into contributions arising from the uncertain parameters, the inherent stochastic forcing, and joint effects. The simulations are also used to support a brief analysis of the computational complexity of the method, providing insight on the types of problems that would benefit from the present developments.
Jimenez, M. Navarro
2017-04-18
A Galerkin polynomial chaos (PC) method was recently proposed to perform variance decomposition and sensitivity analysis in stochastic differential equations (SDEs), driven by Wiener noise and involving uncertain parameters. The present paper extends the PC method to nonintrusive approaches enabling its application to more complex systems hardly amenable to stochastic Galerkin projection methods. We also discuss parallel implementations and the variance decomposition of the derived quantity of interest within the framework of nonintrusive approaches. In particular, a novel hybrid PC-sampling-based strategy is proposed in the case of nonsmooth quantities of interest (QoIs) but smooth SDE solution. Numerical examples are provided that illustrate the decomposition of the variance of QoIs into contributions arising from the uncertain parameters, the inherent stochastic forcing, and joint effects. The simulations are also used to support a brief analysis of the computational complexity of the method, providing insight on the types of problems that would benefit from the present developments.
Yosida approximations of stochastic differential equations in infinite dimensions and applications
Govindan, T E
2016-01-01
This research monograph brings together, for the first time, the varied literature on Yosida approximations of stochastic differential equations (SDEs) in infinite dimensions and their applications into a single cohesive work. The author provides a clear and systematic introduction to the Yosida approximation method and justifies its power by presenting its applications in some practical topics such as stochastic stability and stochastic optimal control. The theory assimilated spans more than 35 years of mathematics, but is developed slowly and methodically in digestible pieces. The book begins with a motivational chapter that introduces the reader to several different models that play recurring roles throughout the book as the theory is unfolded, and invites readers from different disciplines to see immediately that the effort required to work through the theory that follows is worthwhile. From there, the author presents the necessary prerequisite material, and then launches the reader into the main discussi...
Tamellini, L.; Le Maî tre, O.; Nouy, A.
2014-01-01
In this paper we consider a proper generalized decomposition method to solve the steady incompressible Navier-Stokes equations with random Reynolds number and forcing term. The aim of such a technique is to compute a low-cost reduced basis approximation of the full stochastic Galerkin solution of the problem at hand. A particular algorithm, inspired by the Arnoldi method for solving eigenproblems, is proposed for an efficient greedy construction of a deterministic reduced basis approximation. This algorithm decouples the computation of the deterministic and stochastic components of the solution, thus allowing reuse of preexisting deterministic Navier-Stokes solvers. It has the remarkable property of only requiring the solution of m uncoupled deterministic problems for the construction of an m-dimensional reduced basis rather than M coupled problems of the full stochastic Galerkin approximation space, with m l M (up to one order of magnitudefor the problem at hand in this work). © 2014 Society for Industrial and Applied Mathematics.
Khan, Sami Ullah; Ali, Ishtiaq
2018-03-01
Explicit solutions to delay differential equation (DDE) and stochastic delay differential equation (SDDE) can rarely be obtained, therefore numerical methods are adopted to solve these DDE and SDDE. While on the other hand due to unstable nature of both DDE and SDDE numerical solutions are also not straight forward and required more attention. In this study, we derive an efficient numerical scheme for DDE and SDDE based on Legendre spectral-collocation method, which proved to be numerical methods that can significantly speed up the computation. The method transforms the given differential equation into a matrix equation by means of Legendre collocation points which correspond to a system of algebraic equations with unknown Legendre coefficients. The efficiency of the proposed method is confirmed by some numerical examples. We found that our numerical technique has a very good agreement with other methods with less computational effort.
A Stochastic Differential Equation Model for the Spread of HIV amongst People Who Inject Drugs
Directory of Open Access Journals (Sweden)
Yanfeng Liang
2016-01-01
Full Text Available We introduce stochasticity into the deterministic differential equation model for the spread of HIV amongst people who inject drugs (PWIDs studied by Greenhalgh and Hay (1997. This was based on the original model constructed by Kaplan (1989 which analyses the behaviour of HIV/AIDS amongst a population of PWIDs. We derive a stochastic differential equation (SDE for the fraction of PWIDs who are infected with HIV at time. The stochasticity is introduced using the well-known standard technique of parameter perturbation. We first prove that the resulting SDE for the fraction of infected PWIDs has a unique solution in (0, 1 provided that some infected PWIDs are initially present and next construct the conditions required for extinction and persistence. Furthermore, we show that there exists a stationary distribution for the persistence case. Simulations using realistic parameter values are then constructed to illustrate and support our theoretical results. Our results provide new insight into the spread of HIV amongst PWIDs. The results show that the introduction of stochastic noise into a model for the spread of HIV amongst PWIDs can cause the disease to die out in scenarios where deterministic models predict disease persistence.
A Stochastic Differential Equation Model for the Spread of HIV amongst People Who Inject Drugs.
Liang, Yanfeng; Greenhalgh, David; Mao, Xuerong
2016-01-01
We introduce stochasticity into the deterministic differential equation model for the spread of HIV amongst people who inject drugs (PWIDs) studied by Greenhalgh and Hay (1997). This was based on the original model constructed by Kaplan (1989) which analyses the behaviour of HIV/AIDS amongst a population of PWIDs. We derive a stochastic differential equation (SDE) for the fraction of PWIDs who are infected with HIV at time. The stochasticity is introduced using the well-known standard technique of parameter perturbation. We first prove that the resulting SDE for the fraction of infected PWIDs has a unique solution in (0, 1) provided that some infected PWIDs are initially present and next construct the conditions required for extinction and persistence. Furthermore, we show that there exists a stationary distribution for the persistence case. Simulations using realistic parameter values are then constructed to illustrate and support our theoretical results. Our results provide new insight into the spread of HIV amongst PWIDs. The results show that the introduction of stochastic noise into a model for the spread of HIV amongst PWIDs can cause the disease to die out in scenarios where deterministic models predict disease persistence.
Directory of Open Access Journals (Sweden)
Xiao-Li Ding
2018-01-01
Full Text Available In this paper, we investigate analytical solutions of multi-time scale fractional stochastic differential equations driven by fractional Brownian motions. We firstly decompose homogeneous multi-time scale fractional stochastic differential equations driven by fractional Brownian motions into independent differential subequations, and give their analytical solutions. Then, we use the variation of constant parameters to obtain the solutions of nonhomogeneous multi-time scale fractional stochastic differential equations driven by fractional Brownian motions. Finally, we give three examples to demonstrate the applicability of our obtained results.
Conditional stability in determination of initial data for stochastic parabolic equations
International Nuclear Information System (INIS)
Yuan, Ganghua
2017-01-01
In this paper, we solve two kinds of inverse problems in determination of the initial data for stochastic parabolic equations. One is determination of the initial data by lateral boundary observation on arbitrary portion of the boundary, the second one is determination of the initial data by internal observation in a subregion inside the domain. We obtain conditional stability for the two kinds of inverse problems. To prove the results, we estimate the initial data by a terminal observation near the initial time, then we estimate this terminal observation by lateral boundary observation on arbitrary portion of the boundary or internal observation in a subregion inside the domain. To achieve those goals, we derive several new Carleman estimates for stochastic parabolic equations in this paper. (paper)
Conditional stability in determination of initial data for stochastic parabolic equations
Yuan, Ganghua
2017-03-01
In this paper, we solve two kinds of inverse problems in determination of the initial data for stochastic parabolic equations. One is determination of the initial data by lateral boundary observation on arbitrary portion of the boundary, the second one is determination of the initial data by internal observation in a subregion inside the domain. We obtain conditional stability for the two kinds of inverse problems. To prove the results, we estimate the initial data by a terminal observation near the initial time, then we estimate this terminal observation by lateral boundary observation on arbitrary portion of the boundary or internal observation in a subregion inside the domain. To achieve those goals, we derive several new Carleman estimates for stochastic parabolic equations in this paper.
Fractional Order Stochastic Differential Equation with Application in European Option Pricing
Directory of Open Access Journals (Sweden)
Qing Li
2014-01-01
Full Text Available Memory effect is an important phenomenon in financial systems, and a number of research works have been carried out to study the long memory in the financial markets. In recent years, fractional order ordinary differential equation is used as an effective instrument for describing the memory effect in complex systems. In this paper, we establish a fractional order stochastic differential equation (FSDE model to describe the effect of trend memory in financial pricing. We, then, derive a European option pricing formula based on the FSDE model and prove the existence of the trend memory (i.e., the mean value function in the option pricing formula when the Hurst index is between 0.5 and 1. In addition, we make a comparison analysis between our proposed model, the classic Black-Scholes model, and the stochastic model with fractional Brownian motion. Numerical results suggest that our model leads to more accurate and lower standard deviation in the empirical study.
Workshop on quantum stochastic differential equations for the quantum simulation of physical systems
2016-09-22
that would be complimentary to the efforts at ARL. One the other hand, topological quantum field theories have a dual application to topological...Witten provided a path-integral definition of the Jones polynomial using a three-dimensional Chern-Simons quantum field theory (QFT) based on a non...topology, quantum field theory , quantum stochastic differential equations, quantum computing REPORT DOCUMENTATION PAGE 11. SPONSOR/MONITOR’S REPORT
Directory of Open Access Journals (Sweden)
Xiaolin Zhu
2014-01-01
Full Text Available This paper studies the T-stability of the Heun method and balanced method for solving stochastic differential delay equations (SDDEs. Two T-stable conditions of the Heun method are obtained for two kinds of linear SDDEs. Moreover, two conditions under which the balanced method is T-stable are obtained for two kinds of linear SDDEs. Some numerical examples verify the theoretical results proposed.
Invariant measure for the stochastic Navier-Stokes equations in unbounded 2D domains
Czech Academy of Sciences Publication Activity Database
Brzezniak, Z.; Motyl, E.; Ondreját, Martin
2017-01-01
Roč. 45, č. 5 (2017), s. 3145-3201 ISSN 0091-1798 R&D Projects: GA ČR(CZ) GA15-08819S Institutional support: RVO:67985556 Keywords : Invariant measure * bw-Feller semigroup * stochastic Navier–Stokes equation Subject RIV: BA - General Mathematics OBOR OECD: Statistics and probability Impact factor: 1.940, year: 2016 http://library.utia.cas.cz/separaty/2017/SI/ondrejat-0478383.pdf
Picard Approximation of Stochastic Differential Equations and Application to LIBOR Models
DEFF Research Database (Denmark)
Papapantoleon, Antonis; Skovmand, David
The aim of this work is to provide fast and accurate approximation schemes for the Monte Carlo pricing of derivatives in LIBOR market models. Standard methods can be applied to solve the stochastic differential equations of the successive LIBOR rates but the methods are generally slow. Our...... exponential to quadratic using truncated expansions of the product terms. We include numerical illustrations of the accuracy and speed of our method pricing caplets, swaptions and forward rate agreements....
Mélykúti, Bence
2010-01-01
The Chemical Langevin Equation (CLE), which is a stochastic differential equation driven by a multidimensional Wiener process, acts as a bridge between the discrete stochastic simulation algorithm and the deterministic reaction rate equation when simulating (bio)chemical kinetics. The CLE model is valid in the regime where molecular populations are abundant enough to assume their concentrations change continuously, but stochastic fluctuations still play a major role. The contribution of this work is that we observe and explore that the CLE is not a single equation, but a parametric family of equations, all of which give the same finite-dimensional distribution of the variables. On the theoretical side, we prove that as many Wiener processes are sufficient to formulate the CLE as there are independent variables in the equation, which is just the rank of the stoichiometric matrix. On the practical side, we show that in the case where there are m1 pairs of reversible reactions and m2 irreversible reactions there is another, simple formulation of the CLE with only m1 + m2 Wiener processes, whereas the standard approach uses 2 m1 + m2. We demonstrate that there are considerable computational savings when using this latter formulation. Such transformations of the CLE do not cause a loss of accuracy and are therefore distinct from model reduction techniques. We illustrate our findings by considering alternative formulations of the CLE for a human ether a-go-go related gene ion channel model and the Goldbeter-Koshland switch. © 2010 American Institute of Physics.
Rackauckas, Christopher; Nie, Qing
2017-01-01
Adaptive time-stepping with high-order embedded Runge-Kutta pairs and rejection sampling provides efficient approaches for solving differential equations. While many such methods exist for solving deterministic systems, little progress has been made for stochastic variants. One challenge in developing adaptive methods for stochastic differential equations (SDEs) is the construction of embedded schemes with direct error estimates. We present a new class of embedded stochastic Runge-Kutta (SRK) methods with strong order 1.5 which have a natural embedding of strong order 1.0 methods. This allows for the derivation of an error estimate which requires no additional function evaluations. Next we derive a general method to reject the time steps without losing information about the future Brownian path termed Rejection Sampling with Memory (RSwM). This method utilizes a stack data structure to do rejection sampling, costing only a few floating point calculations. We show numerically that the methods generate statistically-correct and tolerance-controlled solutions. Lastly, we show that this form of adaptivity can be applied to systems of equations, and demonstrate that it solves a stiff biological model 12.28x faster than common fixed timestep algorithms. Our approach only requires the solution to a bridging problem and thus lends itself to natural generalizations beyond SDEs.
Bound states of quarks calculated with stochastic integration of the Bethe-Salpeter equation
International Nuclear Information System (INIS)
Salomon, M.
1992-07-01
We have computed the masses, wave functions and sea quark content of mesons in their ground state by integrating the Bethe-Salpeter equation with a stochastic algorithm. This method allows the inclusion of a large set of diagrams. Inspection of the kernel of the equation shows that q-q-bar pairs with similar constituent masses in a singlet spin state exhibit a high bound state which is not present in other pairs. The pion, kaon and eta belongs to this category. 19 refs., 2 figs., 2 tabs
Directory of Open Access Journals (Sweden)
Bixiang Wang
2013-08-01
Full Text Available We prove the existence and uniqueness of random attractors for the p-Laplace equation driven simultaneously by non-autonomous deterministic and stochastic forcing. The nonlinearity of the equation is allowed to have a polynomial growth rate of any order which may be greater than p. We further establish the upper semicontinuity of random attractors as the intensity of noise approaches zero. In addition, we show the pathwise periodicity of random attractors when all non-autonomous deterministic forcing terms are time periodic.
International Nuclear Information System (INIS)
Rupšys, P.
2015-01-01
A system of stochastic differential equations (SDE) with mixed-effects parameters and multivariate normal copula density function were used to develop tree height model for Scots pine trees in Lithuania. A two-step maximum likelihood parameter estimation method is used and computational guidelines are given. After fitting the conditional probability density functions to outside bark diameter at breast height, and total tree height, a bivariate normal copula distribution model was constructed. Predictions from the mixed-effects parameters SDE tree height model calculated during this research were compared to the regression tree height equations. The results are implemented in the symbolic computational language MAPLE
GPELab, a Matlab toolbox to solve Gross-Pitaevskii equations II: Dynamics and stochastic simulations
Antoine, Xavier; Duboscq, Romain
2015-08-01
GPELab is a free Matlab toolbox for modeling and numerically solving large classes of systems of Gross-Pitaevskii equations that arise in the physics of Bose-Einstein condensates. The aim of this second paper, which follows (Antoine and Duboscq, 2014), is to first present the various pseudospectral schemes available in GPELab for computing the deterministic and stochastic nonlinear dynamics of Gross-Pitaevskii equations (Antoine, et al., 2013). Next, the corresponding GPELab functions are explained in detail. Finally, some numerical examples are provided to show how the code works for the complex dynamics of BEC problems.
Energy Technology Data Exchange (ETDEWEB)
Rupšys, P. [Aleksandras Stulginskis University, Studenų g. 11, Akademija, Kaunas district, LT – 53361 Lithuania (Lithuania)
2015-10-28
A system of stochastic differential equations (SDE) with mixed-effects parameters and multivariate normal copula density function were used to develop tree height model for Scots pine trees in Lithuania. A two-step maximum likelihood parameter estimation method is used and computational guidelines are given. After fitting the conditional probability density functions to outside bark diameter at breast height, and total tree height, a bivariate normal copula distribution model was constructed. Predictions from the mixed-effects parameters SDE tree height model calculated during this research were compared to the regression tree height equations. The results are implemented in the symbolic computational language MAPLE.
Grössing, Gerhard
2002-04-01
The Klein-Gordon equation is shown to be equivalent to coupled partial differential equations for a sub-quantum Brownian movement of a “particle”, which is both passively affected by, and actively affecting, a diffusion process of its generally nonlocal environment. This indicates circularly causal, or “cybernetic”, relationships between “particles” and their surroundings. Moreover, in the relativistic domain, the original stochastic theory of Nelson is shown to hold as a limiting case only, i.e., for a vanishing quantum potential.
Sign reversals of the output autocorrelation function for the stochastic Bernoulli-Verhulst equation
Energy Technology Data Exchange (ETDEWEB)
Lumi, N., E-mail: Neeme.Lumi@tlu.ee; Mankin, R., E-mail: Romi.Mankin@tlu.ee [Institute of Mathematics and Natural Sciences, Tallinn University, 29 Narva Road, 10120 Tallinn (Estonia)
2015-10-28
We consider a stochastic Bernoulli-Verhulst equation as a model for population growth processes. The effect of fluctuating environment on the carrying capacity of a population is modeled as colored dichotomous noise. Relying on the composite master equation an explicit expression for the stationary autocorrelation function (ACF) of population sizes is found. On the basis of this expression a nonmonotonic decay of the ACF by increasing lag-time is shown. Moreover, in a certain regime of the noise parameters the ACF demonstrates anticorrelation as well as related sign reversals at some values of the lag-time. The conditions for the appearance of this highly unexpected effect are also discussed.
From quantum stochastic differential equations to Gisin-Percival state diffusion
Parthasarathy, K. R.; Usha Devi, A. R.
2017-08-01
Starting from the quantum stochastic differential equations of Hudson and Parthasarathy [Commun. Math. Phys. 93, 301 (1984)] and exploiting the Wiener-Itô-Segal isomorphism between the boson Fock reservoir space Γ (L2(R+ ) ⊗(Cn⊕Cn ) ) and the Hilbert space L2(μ ) , where μ is the Wiener probability measure of a complex n-dimensional vector-valued standard Brownian motion {B (t ) ,t ≥0 } , we derive a non-linear stochastic Schrödinger equation describing a classical diffusion of states of a quantum system, driven by the Brownian motion B. Changing this Brownian motion by an appropriate Girsanov transformation, we arrive at the Gisin-Percival state diffusion equation [N. Gisin and J. Percival, J. Phys. A 167, 315 (1992)]. This approach also yields an explicit solution of the Gisin-Percival equation, in terms of the Hudson-Parthasarathy unitary process and a randomized Weyl displacement process. Irreversible dynamics of system density operators described by the well-known Gorini-Kossakowski-Sudarshan-Lindblad master equation is unraveled by coarse-graining over the Gisin-Percival quantum state trajectories.
Energy Technology Data Exchange (ETDEWEB)
Mekkaoui, Abdessamad [IEK-4 Forschungszentrum Juelich 52428 (Germany)
2013-07-01
A method to derive stochastic differential equations for intermittent plasma density dynamics in magnetic fusion edge plasma is presented. It uses a measured first four moments (mean, variance, Skewness and Kurtosis) and the correlation time of turbulence to write a Pearson equation for the probability distribution function of fluctuations. The Fokker-Planck equation is then used to derive a Langevin equation for the plasma density fluctuations. A theoretical expectations are used as a constraints to fix the nonlinearity structure of the stochastic differential equation. In particular when the quadratically nonlinear dynamics is assumed, then it is shown that the plasma density is driven by a multiplicative Wiener process and evolves on the turbulence correlation time scale, while the linear growth is quadratically damped by the fluctuation level. Strong criteria for statistical discrimination of experimental time series are proposed as an alternative to the Kurtosis-Skewness scaling. This scaling is broadly used in contemporary literature to characterize edge turbulence, but it is inappropriate because a large family of distributions could share this scaling. Strong criteria allow us to focus on the relevant candidate distribution and approach a nonlinear structure of edge turbulence model.
International Nuclear Information System (INIS)
Brett, Tobias; Galla, Tobias
2014-01-01
We present a heuristic derivation of Gaussian approximations for stochastic chemical reaction systems with distributed delay. In particular, we derive the corresponding chemical Langevin equation. Due to the non-Markovian character of the underlying dynamics, these equations are integro-differential equations, and the noise in the Gaussian approximation is coloured. Following on from the chemical Langevin equation, a further reduction leads to the linear-noise approximation. We apply the formalism to a delay variant of the celebrated Brusselator model, and show how it can be used to characterise noise-driven quasi-cycles, as well as noise-triggered spiking. We find surprisingly intricate dependence of the typical frequency of quasi-cycles on the delay period
Brett, Tobias; Galla, Tobias
2014-03-28
We present a heuristic derivation of Gaussian approximations for stochastic chemical reaction systems with distributed delay. In particular, we derive the corresponding chemical Langevin equation. Due to the non-Markovian character of the underlying dynamics, these equations are integro-differential equations, and the noise in the Gaussian approximation is coloured. Following on from the chemical Langevin equation, a further reduction leads to the linear-noise approximation. We apply the formalism to a delay variant of the celebrated Brusselator model, and show how it can be used to characterise noise-driven quasi-cycles, as well as noise-triggered spiking. We find surprisingly intricate dependence of the typical frequency of quasi-cycles on the delay period.
Stochastic modeling of stock price process induced from the conjugate heat equation
Paeng, Seong-Hun
2015-02-01
Currency can be considered as a ruler for values of commodities. Then the price is the measured value by the ruler. We can suppose that inflation and variation of exchange rate are caused by variation of the scale of the ruler. In geometry, variation of the scale means that the metric is time-dependent. The conjugate heat equation is the modified heat equation which satisfies the heat conservation law for the time-dependent metric space. We propose a new model of stock prices by using the stochastic process whose transition probability is determined by the kernel of the conjugate heat equation. Our model of stock prices shows how the volatility term is affected by inflation and exchange rate. This model modifies the Black-Scholes equation in light of inflation and exchange rate.
DEFF Research Database (Denmark)
Köyluoglu, H.U.; Nielsen, Søren R.K.; Cakmak, A.S.
1994-01-01
perturbation method using stochastic differential equations. The joint statistical moments entering the perturbation solution are determined by considering an augmented dynamic system with state variables made up of the displacement and velocity vector and their first and second derivatives with respect......The paper deals with the first and second order statistical moments of the response of linear systems with random parameters subject to random excitation modelled as white-noise multiplied by an envelope function with random parameters. The method of analysis is basically a second order...... to the random parameters of the problem. Equations for partial derivatives are obtained from the partial differentiation of the equations of motion. The zero time-lag joint statistical moment equations for the augmented state vector are derived from the Itô differential formula. General formulation is given...
International Nuclear Information System (INIS)
Malescio, G.
1981-04-01
The two-dimensional Fokker-Planck equation describing the ion motion in a coherent lower hybrid wave above the stochasticity threshold is analytically solved. An expression is given for the steady state power dissipation
High Weak Order Methods for Stochastic Differential Equations Based on Modified Equations
Abdulle, Assyr; Cohen, David; Vilmart, Gilles; Zygalakis, Konstantinos C.
2012-01-01
© 2012 Society for Industrial and Applied Mathematics. Inspired by recent advances in the theory of modified differential equations, we propose a new methodology for constructing numerical integrators with high weak order for the time integration
International Nuclear Information System (INIS)
Guatteri, Giuseppina; Tessitore, Gianmario
2008-01-01
We study the Riccati equation arising in a class of quadratic optimal control problems with infinite dimensional stochastic differential state equation and infinite horizon cost functional. We allow the coefficients, both in the state equation and in the cost, to be random.In such a context backward stochastic Riccati equations are backward stochastic differential equations in the whole positive real axis that involve quadratic non-linearities and take values in a non-Hilbertian space. We prove existence of a minimal non-negative solution and, under additional assumptions, its uniqueness. We show that such a solution allows to perform the synthesis of the optimal control and investigate its attractivity properties. Finally the case where the coefficients are stationary is addressed and an example concerning a controlled wave equation in random media is proposed
McDonald, A. David; Sandal, Leif Kristoffer
1998-01-01
Estimation of parameters in the drift and diffusion terms of stochastic differential equations involves simulation and generally requires substantial data sets. We examine a method that can be applied when available time series are limited to less than 20 observations per replication. We compare and contrast parameter estimation for linear and nonlinear first-order stochastic differential equations using two criterion functions: one based on a Chi-square statistic, put forward by Hurn and Lin...
Stochastic models with power-law tails the equation X = AX + B
Buraczewski, Dariusz; Mikosch, Thomas
2016-01-01
In this monograph the authors give a systematic approach to the probabilistic properties of the fixed point equation X=AX+B. A probabilistic study of the stochastic recurrence equation X_t=A_tX_{t-1}+B_t for real- and matrix-valued random variables A_t, where (A_t,B_t) constitute an iid sequence, is provided. The classical theory for these equations, including the existence and uniqueness of a stationary solution, the tail behavior with special emphasis on power law behavior, moments and support, is presented. The authors collect recent asymptotic results on extremes, point processes, partial sums (central limit theory with special emphasis on infinite variance stable limit theory), large deviations, in the univariate and multivariate cases, and they further touch on the related topics of smoothing transforms, regularly varying sequences and random iterative systems. The text gives an introduction to the Kesten-Goldie theory for stochastic recurrence equations of the type X_t=A_tX_{t-1}+B_t. It provides the c...
Energy Technology Data Exchange (ETDEWEB)
Harko, Tiberiu [University College London, Department of Mathematics, London (United Kingdom); Leung, Chun Sing [Polytechnic University, Department of Applied Mathematics, Hong Kong (China); Mocanu, Gabriela [Babes-Bolyai University, Faculty of Physics, Cluj-Napoca (Romania)
2014-05-15
We consider a description of the stochastic oscillations of the general relativistic accretion disks around compact astrophysical objects interacting with their external medium based on a generalized Langevin equation with colored noise and on the fluctuation-dissipation theorems. The former accounts for the general memory and retarded effects of the frictional force. The presence of the memory effects influences the response of the disk to external random interactions, and it modifies the dynamical behavior of the disk, as well as the energy dissipation processes. The generalized Langevin equation of the motion of the disk in the vertical direction is studied numerically, and the vertical displacements, velocities, and luminosities of the stochastically perturbed disks are explicitly obtained for both the Schwarzschild and the Kerr cases. The power spectral distribution of the disk luminosity is also obtained. As a possible astrophysical application of the formalism we investigate the possibility that the intra-day variability of the active galactic nuclei may be due to the stochastic disk instabilities. The perturbations due to colored/nontrivially correlated noise induce a complicated disk dynamics, which could explain some astrophysical observational features related to disk variability. (orig.)
International Nuclear Information System (INIS)
Harko, Tiberiu; Leung, Chun Sing; Mocanu, Gabriela
2014-01-01
We consider a description of the stochastic oscillations of the general relativistic accretion disks around compact astrophysical objects interacting with their external medium based on a generalized Langevin equation with colored noise and on the fluctuation-dissipation theorems. The former accounts for the general memory and retarded effects of the frictional force. The presence of the memory effects influences the response of the disk to external random interactions, and it modifies the dynamical behavior of the disk, as well as the energy dissipation processes. The generalized Langevin equation of the motion of the disk in the vertical direction is studied numerically, and the vertical displacements, velocities, and luminosities of the stochastically perturbed disks are explicitly obtained for both the Schwarzschild and the Kerr cases. The power spectral distribution of the disk luminosity is also obtained. As a possible astrophysical application of the formalism we investigate the possibility that the intra-day variability of the active galactic nuclei may be due to the stochastic disk instabilities. The perturbations due to colored/nontrivially correlated noise induce a complicated disk dynamics, which could explain some astrophysical observational features related to disk variability. (orig.)
Harko, Tiberiu; Leung, Chun Sing; Mocanu, Gabriela
2014-05-01
We consider a description of the stochastic oscillations of the general relativistic accretion disks around compact astrophysical objects interacting with their external medium based on a generalized Langevin equation with colored noise and on the fluctuation-dissipation theorems. The former accounts for the general memory and retarded effects of the frictional force. The presence of the memory effects influences the response of the disk to external random interactions, and it modifies the dynamical behavior of the disk, as well as the energy dissipation processes. The generalized Langevin equation of the motion of the disk in the vertical direction is studied numerically, and the vertical displacements, velocities, and luminosities of the stochastically perturbed disks are explicitly obtained for both the Schwarzschild and the Kerr cases. The power spectral distribution of the disk luminosity is also obtained. As a possible astrophysical application of the formalism we investigate the possibility that the intra-day variability of the active galactic nuclei may be due to the stochastic disk instabilities. The perturbations due to colored/nontrivially correlated noise induce a complicated disk dynamics, which could explain some astrophysical observational features related to disk variability.
Average consensus and gossip algorithms in networks with stochastic asymmetric communications
Guerreiro Tome Antunes, D.J.; Silvestre, D.; Silvestre, C.J.
2011-01-01
We consider that a set of distributed agents desire to reach consensus on the average of their initial state values, while communicating with neighboring agents through a shared medium. This communication medium allows only one agent to transmit unidirectionally at a given time, which is true, e.g.,
International Nuclear Information System (INIS)
Kushner, Harold J.
2012-01-01
This is the second part of a work dealing with key issues that have not been addressed in the modeling and numerical optimization of nonlinear stochastic delay systems. We consider new classes of models, such as those with nonlinear functions of several controls (such as products), each with is own delay, controlled random Poisson measure driving terms, admissions control with delayed retrials, and others. Part I was concerned with issues concerning the class of admissible controls and their approximations, since the classical definitions are inadequate for our models. This part is concerned with transportation equation representations and their approximations. Such representations of nonlinear stochastic delay models have been crucial in the development of numerical algorithms with much reduced memory and computational requirements. The representations for the new models are not obvious and are developed. They also provide a template for the adaptation of the Markov chain approximation numerical methods.
Numerical resolution of N-dimensional Fokker-Plank stochastic equations
International Nuclear Information System (INIS)
Garcia-Olivares, A.; Muoz, A.
1992-01-01
This document describes the use of a library of programs able to solve stochastic Fokker-Planck equations in a N-dimensional space. the input data are essentially: (i) the initial distribution of the stochastic variable, (ii) the drift and fluctuation coefficients as a function of the state (which can be obtained from the transition probabilities between neighboring states) and (iii) some parameters controlling the run. A last version of the library accepts sources and sinks defined in the states space. The output is the temporal evolution of the probability distribution in the space defined by a N-dimensional grid. Some applications and readings in Synergetics, Self-Organization, transport phenomena, Ecology and other fields are suggested. If the probability distribution is interpreted as a distribution of particles then the codes can be used to solve the N-dimensional problem of advection-diffusion. (author) 21 fig. 16 ref
Numerical Resolution of N-dimensional Fokker-Planck stochastic equations
International Nuclear Information System (INIS)
Garcia-Olivares, R. A.; Munoz Roldan, A.
1992-01-01
This document describes the use of a library of programs able to solve stochastic Fokker-Planck equations in a N-dimensional space. The input data are essentially: (i) the initial distribution of the stochastic variable, (ii) the drift and fluctuation coefficients as a function of the state (which can be obtained from the transition probabilities between neighboring states) and (iii) some parameters controlling the run. A last version of the library accepts sources and sinks defined in the states space. The output is the temporal evolution of the probability distribution in the space defined by a N-dimensional grid. Some applications and readings in Synergetic, Self-Organization, transport phenomena, Ecology and other fields are suggested. If the probability distribution is interpreted as a distribution of particles then the codes can be used to solve the N-dimensional problem of advection-diffusion. (Author) 16 refs
A Posteriori Error Analysis of Stochastic Differential Equations Using Polynomial Chaos Expansions
Butler, T.; Dawson, C.; Wildey, T.
2011-01-01
We develop computable a posteriori error estimates for linear functionals of a solution to a general nonlinear stochastic differential equation with random model/source parameters. These error estimates are based on a variational analysis applied to stochastic Galerkin methods for forward and adjoint problems. The result is a representation for the error estimate as a polynomial in the random model/source parameter. The advantage of this method is that we use polynomial chaos representations for the forward and adjoint systems to cheaply produce error estimates by simple evaluation of a polynomial. By comparison, the typical method of producing such estimates requires repeated forward/adjoint solves for each new choice of random parameter. We present numerical examples showing that there is excellent agreement between these methods. © 2011 Society for Industrial and Applied Mathematics.
International Nuclear Information System (INIS)
Olsson, Magnus; Perninge, Magnus; Soeder, Lennart
2010-01-01
The inclusion of wind power into power systems has a significant impact on the demand for real-time balancing power due to the stochastic nature of wind power production. The overall aim of this paper is to present probabilistic models of the impact of large-scale integration of wind power on the continuous demand in MW for real-time balancing power. This is important not only for system operators, but also for producers and consumers since they in most systems through various market solutions provide balancing power. Since there can occur situations where the wind power variations cancel out other types of deviations in the system, models on an hourly basis are not sufficient. Therefore the developed model is in continuous time and is based on stochastic differential equations (SDE). The model can be used within an analytical framework or in Monte Carlo simulations. (author)
International Nuclear Information System (INIS)
Imai, Hajime; Ohtsuki, Yukiyoshi; Kono, Hirohiko
2015-01-01
Highlights: • Stochastic Liouville–von Neumann equation is applied to energy transfer dynamics. • Noise generation methods for dealing with exciton in FMO complexes are proposed. • Structured spectral densities could better support coherent population dynamics. - Abstract: A stochastic Liouville–von Neumann approach to solving a spin-boson model is applied to electronic energy transfer in Fenna–Matthews–Olson (FMO) complexes as a case study of the dynamics in biological systems. We modify a noise generation method to treat an experimentally obtained highly structured spectral density. By considering the population dynamics in a two-site system with a model structured spectral density, we numerically observe two kinds of coherent motions associated with inter-site coupling and system–bath coupling, the latter of which is mainly attributed to the peak structure of the spectral density
Higher-order Solution of Stochastic Diffusion equation with Nonlinear Losses Using WHEP technique
El-Beltagy, Mohamed A.; Al-Mulla, Noah
2014-01-01
Using Wiener-Hermite expansion with perturbation (WHEP) technique in the solution of the stochastic partial differential equations (SPDEs) has the advantage of converting the problem to a system of deterministic equations that can be solved efficiently using the standard deterministic numerical methods [1]. The Wiener-Hermite expansion is the only known expansion that handles the white/colored noise exactly. The main statistics, such as the mean, covariance, and higher order statistical moments, can be calculated by simple formulae involving only the deterministic Wiener-Hermite coefficients. In this poster, the WHEP technique is used to solve the 2D diffusion equation with nonlinear losses and excited with white noise. The solution will be obtained numerically and will be validated and compared with the analytical solution that can be obtained from any symbolic mathematics package such as Mathematica.
Higher-order Solution of Stochastic Diffusion equation with Nonlinear Losses Using WHEP technique
El-Beltagy, Mohamed A.
2014-01-06
Using Wiener-Hermite expansion with perturbation (WHEP) technique in the solution of the stochastic partial differential equations (SPDEs) has the advantage of converting the problem to a system of deterministic equations that can be solved efficiently using the standard deterministic numerical methods [1]. The Wiener-Hermite expansion is the only known expansion that handles the white/colored noise exactly. The main statistics, such as the mean, covariance, and higher order statistical moments, can be calculated by simple formulae involving only the deterministic Wiener-Hermite coefficients. In this poster, the WHEP technique is used to solve the 2D diffusion equation with nonlinear losses and excited with white noise. The solution will be obtained numerically and will be validated and compared with the analytical solution that can be obtained from any symbolic mathematics package such as Mathematica.
Recognition of Equations Using a Two-Dimensional Stochastic Context-Free Grammar
Chou, Philip A.
1989-11-01
We propose using two-dimensional stochastic context-free grammars for image recognition, in a manner analogous to using hidden Markov models for speech recognition. The value of the approach is demonstrated in a system that recognizes printed, noisy equations. The system uses a two-dimensional probabilistic version of the Cocke-Younger-Kasami parsing algorithm to find the most likely parse of the observed image, and then traverses the corresponding parse tree in accordance with translation formats associated with each production rule, to produce eqn I troff commands for the imaged equation. In addition, it uses two-dimensional versions of the Inside/Outside and Baum re-estimation algorithms for learning the parameters of the grammar from a training set of examples. Parsing the image of a simple noisy equation currently takes about one second of cpu time on an Alliant FX/80.
On the XFEL Schrödinger Equation: Highly Oscillatory Magnetic Potentials and Time Averaging
Antonelli, Paolo
2014-01-14
We analyse a nonlinear Schrödinger equation for the time-evolution of the wave function of an electron beam, interacting selfconsistently through a Hartree-Fock nonlinearity and through the repulsive Coulomb interaction of an atomic nucleus. The electrons are supposed to move under the action of a time dependent, rapidly periodically oscillating electromagnetic potential. This can be considered a simplified effective single particle model for an X-ray free electron laser. We prove the existence and uniqueness for the Cauchy problem and the convergence of wave-functions to corresponding solutions of a Schrödinger equation with a time-averaged Coulomb potential in the high frequency limit for the oscillations of the electromagnetic potential. © 2014 Springer-Verlag Berlin Heidelberg.
A Stochastic Model of Space-Time Variability of Tropical Rainfall: I. Statistics of Spatial Averages
Kundu, Prasun K.; Bell, Thomas L.; Lau, William K. M. (Technical Monitor)
2002-01-01
Global maps of rainfall are of great importance in connection with modeling of the earth s climate. Comparison between the maps of rainfall predicted by computer-generated climate models with observation provides a sensitive test for these models. To make such a comparison, one typically needs the total precipitation amount over a large area, which could be hundreds of kilometers in size over extended periods of time of order days or months. This presents a difficult problem since rain varies greatly from place to place as well as in time. Remote sensing methods using ground radar or satellites detect rain over a large area by essentially taking a series of snapshots at infrequent intervals and indirectly deriving the average rain intensity within a collection of pixels , usually several kilometers in size. They measure area average of rain at a particular instant. Rain gauges, on the other hand, record rain accumulation continuously in time but only over a very small area tens of centimeters across, say, the size of a dinner plate. They measure only a time average at a single location. In making use of either method one needs to fill in the gaps in the observation - either the gaps in the area covered or the gaps in time of observation. This involves using statistical models to obtain information about the rain that is missed from what is actually detected. This paper investigates such a statistical model and validates it with rain data collected over the tropical Western Pacific from ship borne radars during TOGA COARE (Tropical Oceans Global Atmosphere Coupled Ocean-Atmosphere Response Experiment). The model incorporates a number of commonly observed features of rain. While rain varies rapidly with location and time, the variability diminishes when averaged over larger areas or longer periods of time. Moreover, rain is patchy in nature - at any instant on the average only a certain fraction of the observed pixels contain rain. The fraction of area covered by
Energy Technology Data Exchange (ETDEWEB)
Yang, Xuetao; Zhu, Quanxin, E-mail: zqx22@126.com [School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing 210023, Jiangsu (China)
2015-12-15
In this paper, we are mainly concerned with a class of stochastic neutral functional differential equations of Sobolev-type with Poisson jumps. Under two different sets of conditions, we establish the existence of the mild solution by applying the Leray-Schauder alternative theory and the Sadakovskii’s fixed point theorem, respectively. Furthermore, we use the Bihari’s inequality to prove the Osgood type uniqueness. Also, the mean square exponential stability is investigated by applying the Gronwall inequality. Finally, two examples are given to illustrate the theory results.
Tail estimates for stochastic fixed point equations via nonlinear renewal theory
DEFF Research Database (Denmark)
Collamore, Jeffrey F.; Vidyashankar, Anand N.
2013-01-01
estimate P(V>u)~Cu^{-r} as u tends to infinity, and also present a corresponding Lundberg-type upper bound. To this end, we introduce a novel dual change of measure on a random time interval and analyze the path properties, using nonlinear renewal theory, of the Markov chain resulting from the forward...... iteration of the given stochastic fixed point equation. In the process, we establish several new results in the realm of nonlinear renewal theory for these processes. As a consequence of our techniques, we also establish a new characterization of the extremal index. Finally, we provide some extensions...... of our methods to Markov-driven processes....
Nonlinear stochastic heat equations with cubic nonlinearities and additive Q-regular noise in R^1
Directory of Open Access Journals (Sweden)
Henri Schurz
2010-09-01
Full Text Available Semilinear stochastic heat equations perturbed by cubic-type nonlinearities and additive space-time noise with homogeneous boundary conditions are discussed in R^1. The space-time noise is supposed to be Gaussian in time and possesses a Fourier expansion in space along the eigenfunctions of underlying Lapace operators. We follow the concept of approximate strong (classical Fourier solutions. The existence of unique continuous L^2-bounded solutions is proved. Furthermore, we present a procedure for its numerical approximation based on nonstandard methods (linear-implicit and justify their stability and consistency. The behavior of related total energy functional turns out to be crucial in the presented analysis.
Emergent user behavior on Twitter modelled by a stochastic differential equation.
Mollgaard, Anders; Mathiesen, Joachim
2015-01-01
Data from the social-media site, Twitter, is used to study the fluctuations in tweet rates of brand names. The tweet rates are the result of a strongly correlated user behavior, which leads to bursty collective dynamics with a characteristic 1/f noise. Here we use the aggregated "user interest" in a brand name to model collective human dynamics by a stochastic differential equation with multiplicative noise. The model is supported by a detailed analysis of the tweet rate fluctuations and it reproduces both the exact bursty dynamics found in the data and the 1/f noise.
Tuning of Controller for Type 1 Diabetes Treatment with Stochastic Differential Equations
DEFF Research Database (Denmark)
Duun-Henriksen, Anne Katrine; Boiroux, Dimitri; Schmidt, Signe
2012-01-01
due to the noise corrupted observations from the CGM. In this paper we present a method to estimate the optimal Kalman gain in the controller based on stochastic differential equation modeling. With this model type we could estimate the process noise and observation noise separately based on data from......People with type 1 diabetes need several insulin injections every day to keep their blood glucose level in the normal range and thereby avoiding the acute and long term complications of diabetes. One of the recent treatments consists of a pump injecting insulin into the subcutaneous layer combined...
Rare event simulation for stochastic fixed point equations related to the smoothing transform
DEFF Research Database (Denmark)
Collamore, Jeffrey F.; Vidyashankar, Anand N.; Xu, Jie
2013-01-01
In several applications arising in computer science, cascade theory, and other applied areas, it is of interest to evaluate the tail probabilities of non-homogeneous stochastic fixed point equations. Recently, techniques have been developed for the related linear recursions, yielding tail estimates...... and importance sampling methods for these recursions. However, such methods do not routinely generalize to non-homogeneous recursions. Drawing on techniques from the weighted branching process literature, we present a consistent, strongly efficient importance sampling algorithm for estimating the tail...
Gundersen, Kenneth; Kvaløy, Jan Terje; Eftestøl, Trygve; Kramer-Johansen, Jo
2015-10-15
For patients undergoing cardiopulmonary resuscitation (CPR) and being in a shockable rhythm, the coarseness of the electrocardiogram (ECG) signal is an indicator of the state of the patient. In the current work, we show how mixed effects stochastic differential equations (SDE) models, commonly used in pharmacokinetic and pharmacodynamic modelling, can be used to model the relationship between CPR quality measurements and ECG coarseness. This is a novel application of mixed effects SDE models to a setting quite different from previous applications of such models and where using such models nicely solves many of the challenges involved in analysing the available data. Copyright © 2015 John Wiley & Sons, Ltd.
Lima, L. S.; Miranda, L. L. B.
2018-01-01
We have used the Itô's stochastic differential equation for the double well with additive white noise as a mathematical model for price dynamics of the financial market. We have presented a model which allows us to test within the same framework the comparative explanatory power of rational agents versus irrational agents, with respect to the facts of financial markets. We have obtained the mean price in terms of the β parameter that represents the force of the randomness term of the model.
Energy Technology Data Exchange (ETDEWEB)
Barvik, I [International Centre for Theoretical Physics, Trieste (Italy); Polasek, M [Charles Univ., Prague (Czech Republic). Inst. of Physics; Herman, P [Pedagogical Univ., Hradec Kralove (Czech Republic)
1995-08-01
We used the formal stochastic Liouville equations within Haken-Strobl-Reineker parametrization for the description of the influence of the bath on the memory functions entering the GME for a dimer and a linear trimer with a trap (here modeled as a sink). The often used inclusion of the noncoherent regime in the MF by an exponentially damped prefactor (after Kenkre`s prescription) does not hold for finite systems. The analytical form of the MF is changed more pronouncely and the influence of the sink in the center of the trimer runs parallel with the influence of the bath in destroying the coherence. (author). 60 refs.
Large deviation tail estimates and related limit laws for stochastic fixed point equations
DEFF Research Database (Denmark)
Collamore, Jeffrey F.; Vidyashankar, Anand N.
2013-01-01
We study the forward and backward recursions generated by a stochastic fixed point equation (SFPE) of the form $V \\stackrel{d}{=} A\\max\\{V, D\\}+B$, where $(A, B, D) \\in (0, \\infty)\\times {\\mathbb R}^2$, for both the stationary and explosive cases. In the stationary case (when ${\\bf E} [\\log \\: A......] explosive case (when ${\\bf E} [\\log \\: A] > 0)$, we establish a central limit theorem for the forward recursion generated by the SFPE, namely the process $V_n= A_n \\max\\{V_{n-1...
Energy Technology Data Exchange (ETDEWEB)
Angstmann, C.N.; Donnelly, I.C. [School of Mathematics and Statistics, UNSW Australia, Sydney NSW 2052 (Australia); Henry, B.I., E-mail: B.Henry@unsw.edu.au [School of Mathematics and Statistics, UNSW Australia, Sydney NSW 2052 (Australia); Jacobs, B.A. [School of Computer Science and Applied Mathematics, University of the Witwatersrand, Johannesburg, Private Bag 3, Wits 2050 (South Africa); DST–NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS) (South Africa); Langlands, T.A.M. [Department of Mathematics and Computing, University of Southern Queensland, Toowoomba QLD 4350 (Australia); Nichols, J.A. [School of Mathematics and Statistics, UNSW Australia, Sydney NSW 2052 (Australia)
2016-02-15
We have introduced a new explicit numerical method, based on a discrete stochastic process, for solving a class of fractional partial differential equations that model reaction subdiffusion. The scheme is derived from the master equations for the evolution of the probability density of a sum of discrete time random walks. We show that the diffusion limit of the master equations recovers the fractional partial differential equation of interest. This limiting procedure guarantees the consistency of the numerical scheme. The positivity of the solution and stability results are simply obtained, provided that the underlying process is well posed. We also show that the method can be applied to standard reaction–diffusion equations. This work highlights the broader applicability of using discrete stochastic processes to provide numerical schemes for partial differential equations, including fractional partial differential equations.
Energy Technology Data Exchange (ETDEWEB)
Calloo, A.; Vidal, J.F.; Le Tellier, R.; Rimpault, G., E-mail: ansar.calloo@cea.fr, E-mail: jean-francois.vidal@cea.fr, E-mail: romain.le-tellier@cea.fr, E-mail: gerald.rimpault@cea.fr [CEA, DEN, DER/SPRC/LEPh, Saint-Paul-lez-Durance (France)
2011-07-01
This paper deals with the solving of the multigroup integro-differential form of the transport equation for fine energy group structure. In that case, multigroup transfer cross sections display strongly peaked shape for light scatterers and the current Legendre polynomial expansion is not well-suited to represent them. Furthermore, even if considering an exact scattering cross sections representation, the scattering source in the discrete ordinates method (also known as the Sn method) being calculated by sampling the angular flux at given directions, may be wrongly computed due to lack of angular support for the angular flux. Hence, following the work of Gerts and Matthews, an angular finite volume solver has been developed for 2D Cartesian geometries. It integrates the multigroup transport equation over discrete volume elements obtained by meshing the unit sphere with a product grid over the polar and azimuthal coordinates and by considering the integrated flux per solid angle element. The convergence of this method has been compared to the S{sub n} method for a highly anisotropic benchmark. Besides, piecewise-average scattering cross sections have been produced for non-bound Hydrogen atoms using a free gas model for thermal neutrons. LWR lattice calculations comparing Legendre representations of the Hydrogen scattering multigroup cross section at various orders and piecewise-average cross sections for this same atom are carried out (while keeping a Legendre representation for all other isotopes). (author)
International Nuclear Information System (INIS)
Calloo, A.; Vidal, J.F.; Le Tellier, R.; Rimpault, G.
2011-01-01
This paper deals with the solving of the multigroup integro-differential form of the transport equation for fine energy group structure. In that case, multigroup transfer cross sections display strongly peaked shape for light scatterers and the current Legendre polynomial expansion is not well-suited to represent them. Furthermore, even if considering an exact scattering cross sections representation, the scattering source in the discrete ordinates method (also known as the Sn method) being calculated by sampling the angular flux at given directions, may be wrongly computed due to lack of angular support for the angular flux. Hence, following the work of Gerts and Matthews, an angular finite volume solver has been developed for 2D Cartesian geometries. It integrates the multigroup transport equation over discrete volume elements obtained by meshing the unit sphere with a product grid over the polar and azimuthal coordinates and by considering the integrated flux per solid angle element. The convergence of this method has been compared to the S_n method for a highly anisotropic benchmark. Besides, piecewise-average scattering cross sections have been produced for non-bound Hydrogen atoms using a free gas model for thermal neutrons. LWR lattice calculations comparing Legendre representations of the Hydrogen scattering multigroup cross section at various orders and piecewise-average cross sections for this same atom are carried out (while keeping a Legendre representation for all other isotopes). (author)
International Nuclear Information System (INIS)
Frank, T.D.
2007-01-01
We present a generalized Kramers-Moyal expansion for stochastic differential equations with single and multiple delays. In particular, we show that the delay Fokker-Planck equation derived earlier in the literature is a special case of the proposed Kramers-Moyal expansion. Applications for bond pricing and a self-inhibitory neuron model are discussed
Backward-stochastic-differential-equation approach to modeling of gene expression.
Shamarova, Evelina; Chertovskih, Roman; Ramos, Alexandre F; Aguiar, Paulo
2017-03-01
In this article, we introduce a backward method to model stochastic gene expression and protein-level dynamics. The protein amount is regarded as a diffusion process and is described by a backward stochastic differential equation (BSDE). Unlike many other SDE techniques proposed in the literature, the BSDE method is backward in time; that is, instead of initial conditions it requires the specification of end-point ("final") conditions, in addition to the model parametrization. To validate our approach we employ Gillespie's stochastic simulation algorithm (SSA) to generate (forward) benchmark data, according to predefined gene network models. Numerical simulations show that the BSDE method is able to correctly infer the protein-level distributions that preceded a known final condition, obtained originally from the forward SSA. This makes the BSDE method a powerful systems biology tool for time-reversed simulations, allowing, for example, the assessment of the biological conditions (e.g., protein concentrations) that preceded an experimentally measured event of interest (e.g., mitosis, apoptosis, etc.).
Albert, Carlo; Ulzega, Simone; Stoop, Ruedi
2016-04-01
Parameter inference is a fundamental problem in data-driven modeling. Given observed data that is believed to be a realization of some parameterized model, the aim is to find parameter values that are able to explain the observed data. In many situations, the dominant sources of uncertainty must be included into the model for making reliable predictions. This naturally leads to stochastic models. Stochastic models render parameter inference much harder, as the aim then is to find a distribution of likely parameter values. In Bayesian statistics, which is a consistent framework for data-driven learning, this so-called posterior distribution can be used to make probabilistic predictions. We propose a novel, exact, and very efficient approach for generating posterior parameter distributions for stochastic differential equation models calibrated to measured time series. The algorithm is inspired by reinterpreting the posterior distribution as a statistical mechanics partition function of an object akin to a polymer, where the measurements are mapped on heavier beads compared to those of the simulated data. To arrive at distribution samples, we employ a Hamiltonian Monte Carlo approach combined with a multiple time-scale integration. A separation of time scales naturally arises if either the number of measurement points or the number of simulation points becomes large. Furthermore, at least for one-dimensional problems, we can decouple the harmonic modes between measurement points and solve the fastest part of their dynamics analytically. Our approach is applicable to a wide range of inference problems and is highly parallelizable.
Higher-order stochastic differential equations and the positive Wigner function
Drummond, P. D.
2017-12-01
General higher-order stochastic processes that correspond to any diffusion-type tensor of higher than second order are obtained. The relationship of multivariate higher-order stochastic differential equations with tensor decomposition theory and tensor rank is explained. Techniques for generating the requisite complex higher-order noise are proved to exist either using polar coordinates and γ distributions, or from products of Gaussian variates. This method is shown to allow the calculation of the dynamics of the Wigner function, after it is extended to a complex phase space. The results are illustrated physically through dynamical calculations of the positive Wigner distribution for three-mode parametric downconversion, widely used in quantum optics. The approach eliminates paradoxes arising from truncation of the higher derivative terms in Wigner function time evolution. Anomalous results of negative populations and vacuum scattering found in truncated Wigner quantum simulations in quantum optics and Bose-Einstein condensate dynamics are shown not to occur with this type of stochastic theory.
Evaluation of stochastic differential equation approximation of ion channel gating models.
Bruce, Ian C
2009-04-01
Fox and Lu derived an algorithm based on stochastic differential equations for approximating the kinetics of ion channel gating that is simpler and faster than "exact" algorithms for simulating Markov process models of channel gating. However, the approximation may not be sufficiently accurate to predict statistics of action potential generation in some cases. The objective of this study was to develop a framework for analyzing the inaccuracies and determining their origin. Simulations of a patch of membrane with voltage-gated sodium and potassium channels were performed using an exact algorithm for the kinetics of channel gating and the approximate algorithm of Fox & Lu. The Fox & Lu algorithm assumes that channel gating particle dynamics have a stochastic term that is uncorrelated, zero-mean Gaussian noise, whereas the results of this study demonstrate that in many cases the stochastic term in the Fox & Lu algorithm should be correlated and non-Gaussian noise with a non-zero mean. The results indicate that: (i) the source of the inaccuracy is that the Fox & Lu algorithm does not adequately describe the combined behavior of the multiple activation particles in each sodium and potassium channel, and (ii) the accuracy does not improve with increasing numbers of channels.
Anak Gisen, Jacqueline Isabella; Nijzink, Remko C.; Savenije, Hubert H. G.
2014-05-01
Dispersion mathematical representation of tidal mixing between sea water and fresh water in The definition of dispersion somehow remains unclear as it is not directly measurable. The role of dispersion is only meaningful if it is related to the appropriate temporal and spatial scale of mixing, which are identified as the tidal period, tidal excursion (longitudinal), width of estuary (lateral) and mixing depth (vertical). Moreover, the mixing pattern determines the salt intrusion length in an estuary. If a physically based description of the dispersion is defined, this would allow the analytical solution of the salt intrusion problem. The objective of this study is to develop a predictive equation for estimating the dispersion coefficient at tidal average (TA) condition, which can be applied in the salt intrusion model to predict the salinity profile for any estuary during different events. Utilizing available data of 72 measurements in 27 estuaries (including 6 recently studied estuaries in Malaysia), regressions analysis has been performed with various combinations of dimensionless parameters . The predictive dispersion equations have been developed for two different locations, at the mouth D0TA and at the inflection point D1TA (where the convergence length changes). Regressions have been carried out with two separated datasets: 1) more reliable data for calibration; and 2) less reliable data for validation. The combination of dimensionless ratios that give the best performance is selected as the final outcome which indicates that the dispersion coefficient is depending on the tidal excursion, tidal range, tidal velocity amplitude, friction and the Richardson Number. A limitation of the newly developed equation is that the friction is generally unknown. In order to compensate this problem, further analysis has been performed adopting the hydraulic model of Cai et. al. (2012) to estimate the friction and depth. Keywords: dispersion, alluvial estuaries, mixing, salt
On the well-posedness of the stochastic Allen–Cahn equation in two dimensions
International Nuclear Information System (INIS)
Ryser, Marc D.; Nigam, Nilima; Tupper, Paul F.
2012-01-01
White noise-driven nonlinear stochastic partial differential equations (SPDEs) of parabolic type are frequently used to model physical systems in space dimensions d = 1, 2, 3. Whereas existence and uniqueness of weak solutions to these equations are well established in one dimension, the situation is different for d ⩾ 2. Despite their popularity in the applied sciences, higher dimensional versions of these SPDE models are generally assumed to be ill-posed by the mathematics community. We study this discrepancy on the specific example of the two dimensional Allen–Cahn equation driven by additive white noise. Since it is unclear how to define the notion of a weak solution to this equation, we regularize the noise and introduce a family of approximations. Based on heuristic arguments and numerical experiments, we conjecture that these approximations exhibit divergent behavior in the continuum limit. The results strongly suggest that shrinking the mesh size in simulations of the two-dimensional white noise-driven Allen–Cahn equation does not lead to the recovery of a physically meaningful limit.
Directory of Open Access Journals (Sweden)
Fakhrodin Mohammadi
2017-10-01
Full Text Available Stochastic fractional differential equations (SFDEs have been used for modeling many physical problems in the fields of turbulance, heterogeneous, flows and matrials, viscoelasticity and electromagnetic theory. In this paper, an efficient wavelet Galerkin method based on the second kind Chebyshev wavelets are proposed for approximate solution of SFDEs. In this approach, operational matrices of the second kind Chebyshev wavelets are used for reducing SFDEs to a linear system of algebraic equations that can be solved easily. Convergence and error analysis of the proposed method is considered. Some numerical examples are performed to confirm the applicability and efficiency of the proposed method.
XMDS2: Fast, scalable simulation of coupled stochastic partial differential equations
Dennis, Graham R.; Hope, Joseph J.; Johnsson, Mattias T.
2013-01-01
XMDS2 is a cross-platform, GPL-licensed, open source package for numerically integrating initial value problems that range from a single ordinary differential equation up to systems of coupled stochastic partial differential equations. The equations are described in a high-level XML-based script, and the package generates low-level optionally parallelised C++ code for the efficient solution of those equations. It combines the advantages of high-level simulations, namely fast and low-error development, with the speed, portability and scalability of hand-written code. XMDS2 is a complete redesign of the XMDS package, and features support for a much wider problem space while also producing faster code. Program summaryProgram title: XMDS2 Catalogue identifier: AENK_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AENK_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: GNU General Public License, version 2 No. of lines in distributed program, including test data, etc.: 872490 No. of bytes in distributed program, including test data, etc.: 45522370 Distribution format: tar.gz Programming language: Python and C++. Computer: Any computer with a Unix-like system, a C++ compiler and Python. Operating system: Any Unix-like system; developed under Mac OS X and GNU/Linux. RAM: Problem dependent (roughly 50 bytes per grid point) Classification: 4.3, 6.5. External routines: The external libraries required are problem-dependent. Uses FFTW3 Fourier transforms (used only for FFT-based spectral methods), dSFMT random number generation (used only for stochastic problems), MPI message-passing interface (used only for distributed problems), HDF5, GNU Scientific Library (used only for Bessel-based spectral methods) and a BLAS implementation (used only for non-FFT-based spectral methods). Nature of problem: General coupled initial-value stochastic partial differential equations. Solution method: Spectral method
Ancey, C.; Bohorquez, P.; Heyman, J.
2015-12-01
The advection-diffusion equation is one of the most widespread equations in physics. It arises quite often in the context of sediment transport, e.g., for describing time and space variations in the particle activity (the solid volume of particles in motion per unit streambed area). Phenomenological laws are usually sufficient to derive this equation and interpret its terms. Stochastic models can also be used to derive it, with the significant advantage that they provide information on the statistical properties of particle activity. These models are quite useful when sediment transport exhibits large fluctuations (typically at low transport rates), making the measurement of mean values difficult. Among these stochastic models, the most common approach consists of random walk models. For instance, they have been used to model the random displacement of tracers in rivers. Here we explore an alternative approach, which involves monitoring the evolution of the number of particles moving within an array of cells of finite length. Birth-death Markov processes are well suited to this objective. While the topic has been explored in detail for diffusion-reaction systems, the treatment of advection has received no attention. We therefore look into the possibility of deriving the advection-diffusion equation (with a source term) within the framework of birth-death Markov processes. We show that in the continuum limit (when the cell size becomes vanishingly small), we can derive an advection-diffusion equation for particle activity. Yet while this derivation is formally valid in the continuum limit, it runs into difficulty in practical applications involving cells or meshes of finite length. Indeed, within our stochastic framework, particle advection produces nonlocal effects, which are more or less significant depending on the cell size and particle velocity. Albeit nonlocal, these effects look like (local) diffusion and add to the intrinsic particle diffusion (dispersal due
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Sun Zhiyuan; Yu Xin; Liu Ying; Gao Yitian
2012-01-01
We investigate the dynamics of the bound vector solitons (BVSs) for the coupled nonlinear Schrödinger equations with the nonhomogenously stochastic perturbations added on their dispersion terms. Soliton switching (besides soliton breakup) can be observed between the two components of the BVSs. Rate of the maximum switched energy (absolute values) within the fixed propagation distance (about 10 periods of the BVSs) enhances in the sense of statistics when the amplitudes of stochastic perturbations increase. Additionally, it is revealed that the BVSs with enhanced coherence are more robust against the perturbations with nonhomogenous stochasticity. Diagram describing the approximate borders of the splitting and non-splitting areas is also given. Our results might be helpful in dynamics of the BVSs with stochastic noises in nonlinear optical fibers or with stochastic quantum fluctuations in Bose-Einstein condensates.
Sun, Zhi-Yuan; Gao, Yi-Tian; Yu, Xin; Liu, Ying
2012-12-01
We investigate the dynamics of the bound vector solitons (BVSs) for the coupled nonlinear Schrödinger equations with the nonhomogenously stochastic perturbations added on their dispersion terms. Soliton switching (besides soliton breakup) can be observed between the two components of the BVSs. Rate of the maximum switched energy (absolute values) within the fixed propagation distance (about 10 periods of the BVSs) enhances in the sense of statistics when the amplitudes of stochastic perturbations increase. Additionally, it is revealed that the BVSs with enhanced coherence are more robust against the perturbations with nonhomogenous stochasticity. Diagram describing the approximate borders of the splitting and non-splitting areas is also given. Our results might be helpful in dynamics of the BVSs with stochastic noises in nonlinear optical fibers or with stochastic quantum fluctuations in Bose-Einstein condensates.
Lin, Yen Ting; Chylek, Lily A; Lemons, Nathan W; Hlavacek, William S
2018-06-21
The chemical kinetics of many complex systems can be concisely represented by reaction rules, which can be used to generate reaction events via a kinetic Monte Carlo method that has been termed network-free simulation. Here, we demonstrate accelerated network-free simulation through a novel approach to equation-free computation. In this process, variables are introduced that approximately capture system state. Derivatives of these variables are estimated using short bursts of exact stochastic simulation and finite differencing. The variables are then projected forward in time via a numerical integration scheme, after which a new exact stochastic simulation is initialized and the whole process repeats. The projection step increases efficiency by bypassing the firing of numerous individual reaction events. As we show, the projected variables may be defined as populations of building blocks of chemical species. The maximal number of connected molecules included in these building blocks determines the degree of approximation. Equation-free acceleration of network-free simulation is found to be both accurate and efficient.
Alfonso, Lester; Zamora, Jose; Cruz, Pedro
2015-04-01
The stochastic approach to coagulation considers the coalescence process going in a system of a finite number of particles enclosed in a finite volume. Within this approach, the full description of the system can be obtained from the solution of the multivariate master equation, which models the evolution of the probability distribution of the state vector for the number of particles of a given mass. Unfortunately, due to its complexity, only limited results were obtained for certain type of kernels and monodisperse initial conditions. In this work, a novel numerical algorithm for the solution of the multivariate master equation for stochastic coalescence that works for any type of kernels and initial conditions is introduced. The performance of the method was checked by comparing the numerically calculated particle mass spectrum with analytical solutions obtained for the constant and sum kernels, with an excellent correspondence between the analytical and numerical solutions. In order to increase the speedup of the algorithm, software parallelization techniques with OpenMP standard were used, along with an implementation in order to take advantage of new accelerator technologies. Simulations results show an important speedup of the parallelized algorithms. This study was funded by a grant from Consejo Nacional de Ciencia y Tecnologia de Mexico SEP-CONACYT CB-131879. The authors also thanks LUFAC® Computacion SA de CV for CPU time and all the support provided.
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Vidal-Codina, F.; Nguyen, N.C.; Giles, M.B.; Peraire, J.
2015-01-01
We present a model and variance reduction method for the fast and reliable computation of statistical outputs of stochastic elliptic partial differential equations. Our method consists of three main ingredients: (1) the hybridizable discontinuous Galerkin (HDG) discretization of elliptic partial differential equations (PDEs), which allows us to obtain high-order accurate solutions of the governing PDE; (2) the reduced basis method for a new HDG discretization of the underlying PDE to enable real-time solution of the parameterized PDE in the presence of stochastic parameters; and (3) a multilevel variance reduction method that exploits the statistical correlation among the different reduced basis approximations and the high-fidelity HDG discretization to accelerate the convergence of the Monte Carlo simulations. The multilevel variance reduction method provides efficient computation of the statistical outputs by shifting most of the computational burden from the high-fidelity HDG approximation to the reduced basis approximations. Furthermore, we develop a posteriori error estimates for our approximations of the statistical outputs. Based on these error estimates, we propose an algorithm for optimally choosing both the dimensions of the reduced basis approximations and the sizes of Monte Carlo samples to achieve a given error tolerance. We provide numerical examples to demonstrate the performance of the proposed method
Gupta, Chinmaya; López, José Manuel; Azencott, Robert; Bennett, Matthew R; Josić, Krešimir; Ott, William
2014-05-28
Delay is an important and ubiquitous aspect of many biochemical processes. For example, delay plays a central role in the dynamics of genetic regulatory networks as it stems from the sequential assembly of first mRNA and then protein. Genetic regulatory networks are therefore frequently modeled as stochastic birth-death processes with delay. Here, we examine the relationship between delay birth-death processes and their appropriate approximating delay chemical Langevin equations. We prove a quantitative bound on the error between the pathwise realizations of these two processes. Our results hold for both fixed delay and distributed delay. Simulations demonstrate that the delay chemical Langevin approximation is accurate even at moderate system sizes. It captures dynamical features such as the oscillatory behavior in negative feedback circuits, cross-correlations between nodes in a network, and spatial and temporal information in two commonly studied motifs of metastability in biochemical systems. Overall, these results provide a foundation for using delay stochastic differential equations to approximate the dynamics of birth-death processes with delay.
Energy Technology Data Exchange (ETDEWEB)
Gupta, Chinmaya; López, José Manuel; Azencott, Robert; Ott, William [Department of Mathematics, University of Houston, Houston, Texas 77004 (United States); Bennett, Matthew R. [Department of Biochemistry and Cell Biology, Rice University, Houston, Texas 77204, USA and Institute of Biosciences and Bioengineering, Rice University, Houston, Texas 77005 (United States); Josić, Krešimir [Department of Mathematics, University of Houston, Houston, Texas 77004 (United States); Department of Biology and Biochemistry, University of Houston, Houston, Texas 77204 (United States)
2014-05-28
Delay is an important and ubiquitous aspect of many biochemical processes. For example, delay plays a central role in the dynamics of genetic regulatory networks as it stems from the sequential assembly of first mRNA and then protein. Genetic regulatory networks are therefore frequently modeled as stochastic birth-death processes with delay. Here, we examine the relationship between delay birth-death processes and their appropriate approximating delay chemical Langevin equations. We prove a quantitative bound on the error between the pathwise realizations of these two processes. Our results hold for both fixed delay and distributed delay. Simulations demonstrate that the delay chemical Langevin approximation is accurate even at moderate system sizes. It captures dynamical features such as the oscillatory behavior in negative feedback circuits, cross-correlations between nodes in a network, and spatial and temporal information in two commonly studied motifs of metastability in biochemical systems. Overall, these results provide a foundation for using delay stochastic differential equations to approximate the dynamics of birth-death processes with delay.
International Nuclear Information System (INIS)
Gupta, Chinmaya; López, José Manuel; Azencott, Robert; Ott, William; Bennett, Matthew R.; Josić, Krešimir
2014-01-01
Delay is an important and ubiquitous aspect of many biochemical processes. For example, delay plays a central role in the dynamics of genetic regulatory networks as it stems from the sequential assembly of first mRNA and then protein. Genetic regulatory networks are therefore frequently modeled as stochastic birth-death processes with delay. Here, we examine the relationship between delay birth-death processes and their appropriate approximating delay chemical Langevin equations. We prove a quantitative bound on the error between the pathwise realizations of these two processes. Our results hold for both fixed delay and distributed delay. Simulations demonstrate that the delay chemical Langevin approximation is accurate even at moderate system sizes. It captures dynamical features such as the oscillatory behavior in negative feedback circuits, cross-correlations between nodes in a network, and spatial and temporal information in two commonly studied motifs of metastability in biochemical systems. Overall, these results provide a foundation for using delay stochastic differential equations to approximate the dynamics of birth-death processes with delay
Energy Technology Data Exchange (ETDEWEB)
Cui, Jianbo, E-mail: jianbocui@lsec.cc.ac.cn [Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences, Beijing, 100190 (China); Hong, Jialin, E-mail: hjl@lsec.cc.ac.cn [Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences, Beijing, 100190 (China); Liu, Zhihui, E-mail: liuzhihui@lsec.cc.ac.cn [Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences, Beijing, 100190 (China); Zhou, Weien, E-mail: weienzhou@nudt.edu.cn [College of Science, National University of Defense Technology, Changsha 410073 (China)
2017-08-01
We indicate that the nonlinear Schrödinger equation with white noise dispersion possesses stochastic symplectic and multi-symplectic structures. Based on these structures, we propose the stochastic symplectic and multi-symplectic methods, which preserve the continuous and discrete charge conservation laws, respectively. Moreover, we show that the proposed methods are convergent with temporal order one in probability. Numerical experiments are presented to verify our theoretical results.
International Nuclear Information System (INIS)
Cui, Jianbo; Hong, Jialin; Liu, Zhihui; Zhou, Weien
2017-01-01
We indicate that the nonlinear Schrödinger equation with white noise dispersion possesses stochastic symplectic and multi-symplectic structures. Based on these structures, we propose the stochastic symplectic and multi-symplectic methods, which preserve the continuous and discrete charge conservation laws, respectively. Moreover, we show that the proposed methods are convergent with temporal order one in probability. Numerical experiments are presented to verify our theoretical results.
Periodic solutions of Lienard differential equations via averaging theory of order two.
Llibre, Jaume; Novaes, Douglas D; Teixeira, Marco A
2015-01-01
For ε ≠ 0 sufficiently small we provide sufficient conditions for the existence of periodic solutions for the Lienard differential equations of the form x'' + f (x) x' + n2x + g (x) = ε2p1 (t) + ε3 p2(t), where n is a positive integer, f : ℝ → ℝ is a C 3 function, g : ℝ → ℝ is a C 4 function, and p i : ℝ → ℝ for i = 1, 2 are continuous 2π-periodic function. The main tool used in this paper is the averaging theory of second order. We also provide one application of the main result obtained.
Periodic solutions of Lienard differential equations via averaging theory of order two
Directory of Open Access Journals (Sweden)
JAUME LLIBRE
2015-12-01
Full Text Available Abstract For ε ≠ 0sufficiently small we provide sufficient conditions for the existence of periodic solutions for the Lienard differential equations of the form x ′′ + f ( x x ′ + n 2 x + g ( x = ε 2 p 1 ( t + ε 3 p 2 ( t , where n is a positive integer, f : ℝ → ℝis a C 3function, g : ℝ → ℝis a C 4function, and p i : ℝ → ℝfor i = 1 , 2are continuous 2 π–periodic function. The main tool used in this paper is the averaging theory of second order. We also provide one application of the main result obtained.
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Espinosa-Paredes, Gilberto
2010-01-01
The aim of this paper is to propose a framework to obtain a new formulation for multiphase flow conservation equations without length-scale restrictions, based on the non-local form of the averaged volume conservation equations. The simplification of the local averaging volume of the conservation equations to obtain practical equations is subject to the following length-scale restrictions: d << l << L, where d is the characteristic length of the dispersed phases, l is the characteristic length of the averaging volume, and L is the characteristic length of the physical system. If the foregoing inequality does not hold, or if the scale of the problem of interest is of the order of l, the averaging technique and therefore, the macroscopic theories of multiphase flow should be modified in order to include appropriate considerations and terms in the corresponding equations. In these cases the local form of the averaged volume conservation equations are not appropriate to describe the multiphase system. As an example of the conservation equations without length-scale restrictions, the natural circulation boiling water reactor was consider to study the non-local effects on the thermal-hydraulic core performance during steady-state and transient behaviors, and the results were compared with the classic local averaging volume conservation equations.
Scaling Relations and Self-Similarity of 3-Dimensional Reynolds-Averaged Navier-Stokes Equations.
Ercan, Ali; Kavvas, M Levent
2017-07-25
Scaling conditions to achieve self-similar solutions of 3-Dimensional (3D) Reynolds-Averaged Navier-Stokes Equations, as an initial and boundary value problem, are obtained by utilizing Lie Group of Point Scaling Transformations. By means of an open-source Navier-Stokes solver and the derived self-similarity conditions, we demonstrated self-similarity within the time variation of flow dynamics for a rigid-lid cavity problem under both up-scaled and down-scaled domains. The strength of the proposed approach lies in its ability to consider the underlying flow dynamics through not only from the governing equations under consideration but also from the initial and boundary conditions, hence allowing to obtain perfect self-similarity in different time and space scales. The proposed methodology can be a valuable tool in obtaining self-similar flow dynamics under preferred level of detail, which can be represented by initial and boundary value problems under specific assumptions.
Reynolds Averaged Navier-Stokes (RANS) equation solutions of wind turbine wakes
Energy Technology Data Exchange (ETDEWEB)
Ludwig, Daniel Evandro; Horn, Diego Anderson; Petry, Adriane Prisco [Thermal and Energy Study Group, Mechanical Engeneering Department, Federal University of Rio Grande do Sul, Porto Alegre (Brazil)], E-mail: adrianep@mecanica.ufrgs.br
2010-07-01
This paper aims to evaluate the influence of three different turbulence models in the study of a wind turbine wake. Numerical Simulation is used as working tool to characterize the flow through the wind turbines, it is used the numeric simulation. The numerical analysis is based on the finite volume method and the Reynolds Averaged Navier-Stokes (RANS) equations. Three turbulence models are used to represent the total effects of turbulence in the flow: the two equations k-classical and the RNG k- models, based on the turbulent viscosity; and the Shear Stress Transport (SST) model, based on the transport of the Reynolds tensor. The results of the 'u' velocity profiles are compared to experimental data from Vermeer (2003) at distances equivalent to 2, 4, 6, 8, 10 and 16 diameters downstream from the turbine. Results shows that the SST model gives better results until 6 diameters, beyond this distance there is no significant differences between the compared models. (author)
CRPropa 3.1—a low energy extension based on stochastic differential equations
Energy Technology Data Exchange (ETDEWEB)
Merten, Lukas; Tjus, Julia Becker; Eichmann, Björn [Theoretische Physik IV: Plasma-Astroteilchenphysik, Ruhr-Universität Bochum, Universitätsstrasse 150, 44801 Bochum (Germany); Fichtner, Horst [Theoretische Physik IV: Weltraum- und Astrophysik, Ruhr-Universität Bochum, Universitätsstrasse 150, 44801 Bochum (Germany); Sigl, Günter, E-mail: lukas.merten@rub.de, E-mail: julia.tjus@rub.de, E-mail: hf@tp4.rub.de, E-mail: eiche@tp4.rub.de, E-mail: guenter.sigl@desy.de [II Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg (Germany)
2017-06-01
The propagation of charged cosmic rays through the Galactic environment influences all aspects of the observation at Earth. Energy spectrum, composition and arrival directions are changed due to deflections in magnetic fields and interactions with the interstellar medium. Today the transport is simulated with different simulation methods either based on the solution of a transport equation (multi-particle picture) or a solution of an equation of motion (single-particle picture). We developed a new module for the publicly available propagation software CRPropa 3.1, where we implemented an algorithm to solve the transport equation using stochastic differential equations. This technique allows us to use a diffusion tensor which is anisotropic with respect to an arbitrary magnetic background field. The source code of CRPropa is written in C++ with python steering via SWIG which makes it easy to use and computationally fast. In this paper, we present the new low-energy propagation code together with validation procedures that are developed to proof the accuracy of the new implementation. Furthermore, we show first examples of the cosmic ray density evolution, which depends strongly on the ratio of the parallel κ{sub ∥} and perpendicular κ{sub ⊥} diffusion coefficients. This dependency is systematically examined as well the influence of the particle rigidity on the diffusion process.
CRPropa 3.1—a low energy extension based on stochastic differential equations
International Nuclear Information System (INIS)
Merten, Lukas; Tjus, Julia Becker; Eichmann, Björn; Fichtner, Horst; Sigl, Günter
2017-01-01
The propagation of charged cosmic rays through the Galactic environment influences all aspects of the observation at Earth. Energy spectrum, composition and arrival directions are changed due to deflections in magnetic fields and interactions with the interstellar medium. Today the transport is simulated with different simulation methods either based on the solution of a transport equation (multi-particle picture) or a solution of an equation of motion (single-particle picture). We developed a new module for the publicly available propagation software CRPropa 3.1, where we implemented an algorithm to solve the transport equation using stochastic differential equations. This technique allows us to use a diffusion tensor which is anisotropic with respect to an arbitrary magnetic background field. The source code of CRPropa is written in C++ with python steering via SWIG which makes it easy to use and computationally fast. In this paper, we present the new low-energy propagation code together with validation procedures that are developed to proof the accuracy of the new implementation. Furthermore, we show first examples of the cosmic ray density evolution, which depends strongly on the ratio of the parallel κ ∥ and perpendicular κ ⊥ diffusion coefficients. This dependency is systematically examined as well the influence of the particle rigidity on the diffusion process.
CRPropa 3.1—a low energy extension based on stochastic differential equations
Merten, Lukas; Becker Tjus, Julia; Fichtner, Horst; Eichmann, Björn; Sigl, Günter
2017-06-01
The propagation of charged cosmic rays through the Galactic environment influences all aspects of the observation at Earth. Energy spectrum, composition and arrival directions are changed due to deflections in magnetic fields and interactions with the interstellar medium. Today the transport is simulated with different simulation methods either based on the solution of a transport equation (multi-particle picture) or a solution of an equation of motion (single-particle picture). We developed a new module for the publicly available propagation software CRPropa 3.1, where we implemented an algorithm to solve the transport equation using stochastic differential equations. This technique allows us to use a diffusion tensor which is anisotropic with respect to an arbitrary magnetic background field. The source code of CRPropa is written in C++ with python steering via SWIG which makes it easy to use and computationally fast. In this paper, we present the new low-energy propagation code together with validation procedures that are developed to proof the accuracy of the new implementation. Furthermore, we show first examples of the cosmic ray density evolution, which depends strongly on the ratio of the parallel κ∥ and perpendicular κ⊥ diffusion coefficients. This dependency is systematically examined as well the influence of the particle rigidity on the diffusion process.
A new nonlinear turbulence model based on Partially-Averaged Navier-Stokes Equations
International Nuclear Information System (INIS)
Liu, J T; Wu, Y L; Cai, C; Liu, S H; Wang, L Q
2013-01-01
Partially-averaged Navier-Stokes (PANS) Model was recognized as a Reynolds-averaged Navier-Stokes (RANS) to direct numerical simulation (DNS) bridging method. PANS model was purported for any filter width-from RANS to DNS. PANS method also shared some similarities with the currently popular URANS (unsteady RANS) method. In this paper, a new PANS model was proposed, which was based on RNG k-ε turbulence model. The Standard and RNG k-ε turbulence model were both isotropic models, as well as PANS models. The sheer stress in those PANS models was solved by linear equation. The linear hypothesis was not accurate in the simulation of complex flow, such as stall phenomenon. The sheer stress here was solved by nonlinear method proposed by Ehrhard. Then, the nonlinear PANS model was set up. The pressure coefficient of the suction side of the NACA0015 hydrofoil was predicted. The result of pressure coefficient agrees well with experimental result, which proves that the nonlinear PANS model can capture the high pressure gradient flow. A low specific centrifugal pump was used to verify the capacity of the nonlinear PANS model. The comparison between the simulation results of the centrifugal pump and Particle Image Velocimetry (PIV) results proves that the nonlinear PANS model can be used in the prediction of complex flow field
Model identification using stochastic differential equation grey-box models in diabetes.
Duun-Henriksen, Anne Katrine; Schmidt, Signe; Røge, Rikke Meldgaard; Møller, Jonas Bech; Nørgaard, Kirsten; Jørgensen, John Bagterp; Madsen, Henrik
2013-03-01
The acceptance of virtual preclinical testing of control algorithms is growing and thus also the need for robust and reliable models. Models based on ordinary differential equations (ODEs) can rarely be validated with standard statistical tools. Stochastic differential equations (SDEs) offer the possibility of building models that can be validated statistically and that are capable of predicting not only a realistic trajectory, but also the uncertainty of the prediction. In an SDE, the prediction error is split into two noise terms. This separation ensures that the errors are uncorrelated and provides the possibility to pinpoint model deficiencies. An identifiable model of the glucoregulatory system in a type 1 diabetes mellitus (T1DM) patient is used as the basis for development of a stochastic-differential-equation-based grey-box model (SDE-GB). The parameters are estimated on clinical data from four T1DM patients. The optimal SDE-GB is determined from likelihood-ratio tests. Finally, parameter tracking is used to track the variation in the "time to peak of meal response" parameter. We found that the transformation of the ODE model into an SDE-GB resulted in a significant improvement in the prediction and uncorrelated errors. Tracking of the "peak time of meal absorption" parameter showed that the absorption rate varied according to meal type. This study shows the potential of using SDE-GBs in diabetes modeling. Improved model predictions were obtained due to the separation of the prediction error. SDE-GBs offer a solid framework for using statistical tools for model validation and model development. © 2013 Diabetes Technology Society.
Leander, Jacob; Almquist, Joachim; Ahlström, Christine; Gabrielsson, Johan; Jirstrand, Mats
2015-05-01
Inclusion of stochastic differential equations in mixed effects models provides means to quantify and distinguish three sources of variability in data. In addition to the two commonly encountered sources, measurement error and interindividual variability, we also consider uncertainty in the dynamical model itself. To this end, we extend the ordinary differential equation setting used in nonlinear mixed effects models to include stochastic differential equations. The approximate population likelihood is derived using the first-order conditional estimation with interaction method and extended Kalman filtering. To illustrate the application of the stochastic differential mixed effects model, two pharmacokinetic models are considered. First, we use a stochastic one-compartmental model with first-order input and nonlinear elimination to generate synthetic data in a simulated study. We show that by using the proposed method, the three sources of variability can be successfully separated. If the stochastic part is neglected, the parameter estimates become biased, and the measurement error variance is significantly overestimated. Second, we consider an extension to a stochastic pharmacokinetic model in a preclinical study of nicotinic acid kinetics in obese Zucker rats. The parameter estimates are compared between a deterministic and a stochastic NiAc disposition model, respectively. Discrepancies between model predictions and observations, previously described as measurement noise only, are now separated into a comparatively lower level of measurement noise and a significant uncertainty in model dynamics. These examples demonstrate that stochastic differential mixed effects models are useful tools for identifying incomplete or inaccurate model dynamics and for reducing potential bias in parameter estimates due to such model deficiencies.
Efficient Multilevel and Multi-index Sampling Methods in Stochastic Differential Equations
Haji-Ali, Abdul Lateef
2016-05-22
of this thesis is the novel Multi-index Monte Carlo (MIMC) method which is an extension of MLMC in high dimensional problems with significant computational savings. Under reasonable assumptions on the weak and variance convergence, which are related to the mixed regularity of the underlying problem and the discretization method, the order of the computational complexity of MIMC is, at worst up to a logarithmic factor, independent of the dimensionality of the underlying parametric equation. We also apply the same multi-index methodology to another sampling method, namely the Stochastic Collocation method. Hence, the novel Multi-index Stochastic Collocation method is proposed and is shown to be more efficient in problems with sufficient mixed regularity than our novel MIMC method and other standard methods. Finally, MIMC is applied to approximate quantities of interest of stochastic particle systems in the mean-field when the number of particles tends to infinity. To approximate these quantities of interest up to an error tolerance, TOL, MIMC has a computational complexity of O(TOL-2log(TOL)2). This complexity is achieved by building a hierarchy based on two discretization parameters: the number of time steps in an Milstein scheme and the number of particles in the particle system. Moreover, we use a partitioning estimator to increase the correlation between two stochastic particle systems with different sizes. In comparison, the optimal computational complexity of MLMC in this case is O(TOL-3) and the computational complexity of Monte Carlo is O(TOL-4).
Analyzing a stochastic time series obeying a second-order differential equation.
Lehle, B; Peinke, J
2015-06-01
The stochastic properties of a Langevin-type Markov process can be extracted from a given time series by a Markov analysis. Also processes that obey a stochastically forced second-order differential equation can be analyzed this way by employing a particular embedding approach: To obtain a Markovian process in 2N dimensions from a non-Markovian signal in N dimensions, the system is described in a phase space that is extended by the temporal derivative of the signal. For a discrete time series, however, this derivative can only be calculated by a differencing scheme, which introduces an error. If the effects of this error are not accounted for, this leads to systematic errors in the estimation of the drift and diffusion functions of the process. In this paper we will analyze these errors and we will propose an approach that correctly accounts for them. This approach allows an accurate parameter estimation and, additionally, is able to cope with weak measurement noise, which may be superimposed to a given time series.
Stochastic Differential Equation Models for the Price of European CO2 Emissions Allowances
Directory of Open Access Journals (Sweden)
Wugan Cai
2017-02-01
Full Text Available Understanding the stochastic nature of emissions allowances is crucial for risk management in emissions trading markets. In this study, we discuss the emissions allowances spot price within the European Union Emissions Trading Scheme: Powernext and European Climate Exchange. To compare the fitness of five stochastic differential equations (SDEs to the European Union allowances spot price, we apply regression theory to obtain the point and interval estimations for the parameters of the SDEs. An empirical evaluation demonstrates that the mean reverting square root process (MRSRP has the best fitness of five SDEs to forecast the spot price. To reduce the degree of smog, we develop a new trading scheme in which firms have to hand many more allowances to the government when they emit one unit of air pollution on heavy pollution days, versus one allowance on clean days. Thus, we set up the SDE MRSRP model with Markovian switching to analyse the evolution of the spot price in such a scheme. The analysis shows that the allowances spot price will not jump too much in the new scheme. The findings of this study could contribute to developing a new type of emissions trading.
Sanz, Luis; Alonso, Juan Antonio
2017-12-01
In this work we develop approximate aggregation techniques in the context of slow-fast linear population models governed by stochastic differential equations and apply the results to the treatment of populations with spatial heterogeneity. Approximate aggregation techniques allow one to transform a complex system involving many coupled variables and in which there are processes with different time scales, by a simpler reduced model with a fewer number of 'global' variables, in such a way that the dynamics of the former can be approximated by that of the latter. In our model we contemplate a linear fast deterministic process together with a linear slow process in which the parameters are affected by additive noise, and give conditions for the solutions corresponding to positive initial conditions to remain positive for all times. By letting the fast process reach equilibrium we build a reduced system with a lesser number of variables, and provide results relating the asymptotic behaviour of the first- and second-order moments of the population vector for the original and the reduced system. The general technique is illustrated by analysing a multiregional stochastic system in which dispersal is deterministic and the rate growth of the populations in each patch is affected by additive noise.
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data
Babuška, Ivo; Nobile, Fabio; Tempone, Raul
2010-01-01
This work proposes and analyzes a stochastic collocation method for solving elliptic partial differential equations with random coefficients and forcing terms. These input data are assumed to depend on a finite number of random variables. The method consists of a Galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space, and naturally leads to the solution of uncoupled deterministic problems as in the Monte Carlo approach. It treats easily a wide range of situations, such as input data that depend nonlinearly on the random variables, diffusivity coefficients with unbounded second moments, and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate exponential convergence of the “probability error” with respect to the number of Gauss points in each direction of the probability space, under some regularity assumptions on the random input data. Numerical examples show the effectiveness of the method. Finally, we include a section with developments posterior to the original publication of this work. There we review sparse grid stochastic collocation methods, which are effective collocation strategies for problems that depend on a moderately large number of random variables.
A stochastic differential equation model for the foraging behavior of fish schools.
Tạ, Tôn Việt; Nguyen, Linh Thi Hoai
2018-03-15
Constructing models of living organisms locating food sources has important implications for understanding animal behavior and for the development of distribution technologies. This paper presents a novel simple model of stochastic differential equations for the foraging behavior of fish schools in a space including obstacles. The model is studied numerically. Three configurations of space with various food locations are considered. In the first configuration, fish swim in free but limited space. All individuals can find food with large probability while keeping their school structure. In the second and third configurations, they move in limited space with one and two obstacles, respectively. Our results reveal that the probability of foraging success is highest in the first configuration, and smallest in the third one. Furthermore, when school size increases up to an optimal value, the probability of foraging success tends to increase. When it exceeds an optimal value, the probability tends to decrease. The results agree with experimental observations.
Hall, Eric Joseph
2016-12-08
We derive computable error estimates for finite element approximations of linear elliptic partial differential equations with rough stochastic coefficients. In this setting, the exact solutions contain high frequency content that standard a posteriori error estimates fail to capture. We propose goal-oriented estimates, based on local error indicators, for the pathwise Galerkin and expected quadrature errors committed in standard, continuous, piecewise linear finite element approximations. Derived using easily validated assumptions, these novel estimates can be computed at a relatively low cost and have applications to subsurface flow problems in geophysics where the conductivities are assumed to have lognormal distributions with low regularity. Our theory is supported by numerical experiments on test problems in one and two dimensions.
Effects of QCD equation of state on the stochastic gravitational wave background
Energy Technology Data Exchange (ETDEWEB)
Anand, Sampurn; Mohanty, Subhendra [Physical Research Laboratory, Ahmedabad 380009 (India); Dey, Ujjal Kumar, E-mail: sampurn@prl.res.in, E-mail: ujjal@cts.iitkgp.ernet.in, E-mail: mohanty@prl.res.in [Centre for Theoretical Studies, Indian Institute of Technology, Kharagpur 721302 (India)
2017-03-01
Cosmological phase transitions can be a source of Stochastic Gravitational Wave (SGW) background. Apart from the dynamics of the phase transition, the characteristic frequency and the fractional energy density Ω{sub gw} of the SGW depends upon the temperature of the transition. In this article, we compute the SGW spectrum in the light of QCD equation of state provided by the lattice results. We find that the inclusion of trace anomaly from lattice QCD, enhances the SGW signal generated during QCD phase transition by ∼ 50% and the peak frequency of the QCD era SGW are shifted higher by ∼ 25% as compared to the earlier estimates without trace anomaly. This result is extremely significant for testing the phase transition dynamics near QCD epoch.
International Nuclear Information System (INIS)
Di Nunno, Giulia; Khedher, Asma; Vanmaele, Michèle
2015-01-01
We consider a backward stochastic differential equation with jumps (BSDEJ) which is driven by a Brownian motion and a Poisson random measure. We present two candidate-approximations to this BSDEJ and we prove that the solution of each candidate-approximation converges to the solution of the original BSDEJ in a space which we specify. We use this result to investigate in further detail the consequences of the choice of the model to (partial) hedging in incomplete markets in finance. As an application, we consider models in which the small variations in the price dynamics are modeled with a Poisson random measure with infinite activity and models in which these small variations are modeled with a Brownian motion or are cut off. Using the convergence results on BSDEJs, we show that quadratic hedging strategies are robust towards the approximation of the market prices and we derive an estimation of the model risk
Model Identification Using Stochastic Differential Equation Grey-Box Models in Diabetes
DEFF Research Database (Denmark)
Duun-Henriksen, Anne Katrine; Schmidt, Signe; Røge, Rikke Meldgaard
2013-01-01
are uncorrelated and provides the possibility to pinpoint model deficiencies. METHODS: An identifiable model of the glucoregulatory system in a type 1 diabetes mellitus (T1DM) patient is used as the basis for development of a stochastic-differential-equation-based grey-box model (SDE-GB). The parameters...... in a significant improvement in the prediction and uncorrelated errors. Tracking of the "peak time of meal absorption" parameter showed that the absorption rate varied according to meal type. CONCLUSION: This study shows the potential of using SDE-GBs in diabetes modeling. Improved model predictions were obtained...... are estimated on clinical data from four T1DM patients. The optimal SDE-GB is determined from likelihood-ratio tests. Finally, parameter tracking is used to track the variation in the "time to peak of meal response" parameter. RESULTS: We found that the transformation of the ODE model into an SDE-GB resulted...
Excitation transfer pathways in excitonic aggregates revealed by the stochastic Schrödinger equation
Energy Technology Data Exchange (ETDEWEB)
Abramavicius, Vytautas, E-mail: vytautas.ab@gmail.com; Abramavicius, Darius, E-mail: darius.abramavicius@ff.vu.lt [Faculty of Physics, Department of Theoretical Physics, Vilnius University, Saulėtekio 9, LT-10222 Vilnius (Lithuania)
2014-02-14
We derive the stochastic Schrödinger equation for the system wave vector and use it to describe the excitation energy transfer dynamics in molecular aggregates. We suggest a quantum-measurement based method of estimating the excitation transfer time. Adequacy of the proposed approach is demonstrated by performing calculations on a model system. The theory is then applied to study the excitation transfer dynamics in a photosynthetic pigment-protein Fenna-Matthews-Olson (FMO) aggregate using both the Debye spectral density and the spectral density obtained from earlier molecular dynamics simulations containing strong vibrational high-frequency modes. The obtained results show that the excitation transfer times in the FMO system are affected by the presence of the vibrational modes; however, the transfer pathways remain the same.
Gerencsér, Máté; Jentzen, Arnulf; Salimova, Diyora
2017-11-01
In a recent article (Jentzen et al. 2016 Commun. Math. Sci. 14 , 1477-1500 (doi:10.4310/CMS.2016.v14.n6.a1)), it has been established that, for every arbitrarily slow convergence speed and every natural number d ∈{4,5,…}, there exist d -dimensional stochastic differential equations with infinitely often differentiable and globally bounded coefficients such that no approximation method based on finitely many observations of the driving Brownian motion can converge in absolute mean to the solution faster than the given speed of convergence. In this paper, we strengthen the above result by proving that this slow convergence phenomenon also arises in two ( d =2) and three ( d =3) space dimensions.
Zhang, Ling
2017-01-01
The main purpose of this paper is to investigate the strong convergence and exponential stability in mean square of the exponential Euler method to semi-linear stochastic delay differential equations (SLSDDEs). It is proved that the exponential Euler approximation solution converges to the analytic solution with the strong order [Formula: see text] to SLSDDEs. On the one hand, the classical stability theorem to SLSDDEs is given by the Lyapunov functions. However, in this paper we study the exponential stability in mean square of the exact solution to SLSDDEs by using the definition of logarithmic norm. On the other hand, the implicit Euler scheme to SLSDDEs is known to be exponentially stable in mean square for any step size. However, in this article we propose an explicit method to show that the exponential Euler method to SLSDDEs is proved to share the same stability for any step size by the property of logarithmic norm.
Regularity of random attractors for fractional stochastic reaction-diffusion equations on Rn
Gu, Anhui; Li, Dingshi; Wang, Bixiang; Yang, Han
2018-06-01
We investigate the regularity of random attractors for the non-autonomous non-local fractional stochastic reaction-diffusion equations in Hs (Rn) with s ∈ (0 , 1). We prove the existence and uniqueness of the tempered random attractor that is compact in Hs (Rn) and attracts all tempered random subsets of L2 (Rn) with respect to the norm of Hs (Rn). The main difficulty is to show the pullback asymptotic compactness of solutions in Hs (Rn) due to the noncompactness of Sobolev embeddings on unbounded domains and the almost sure nondifferentiability of the sample paths of the Wiener process. We establish such compactness by the ideas of uniform tail-estimates and the spectral decomposition of solutions in bounded domains.
A stochastic differential equation model for the foraging behavior of fish schools
Tạ, Tôn ệt, Vi; Hoai Nguyen, Linh Thi
2018-05-01
Constructing models of living organisms locating food sources has important implications for understanding animal behavior and for the development of distribution technologies. This paper presents a novel simple model of stochastic differential equations for the foraging behavior of fish schools in a space including obstacles. The model is studied numerically. Three configurations of space with various food locations are considered. In the first configuration, fish swim in free but limited space. All individuals can find food with large probability while keeping their school structure. In the second and third configurations, they move in limited space with one and two obstacles, respectively. Our results reveal that the probability of foraging success is highest in the first configuration, and smallest in the third one. Furthermore, when school size increases up to an optimal value, the probability of foraging success tends to increase. When it exceeds an optimal value, the probability tends to decrease. The results agree with experimental observations.
Directory of Open Access Journals (Sweden)
Ling Zhang
2017-10-01
Full Text Available Abstract The main purpose of this paper is to investigate the strong convergence and exponential stability in mean square of the exponential Euler method to semi-linear stochastic delay differential equations (SLSDDEs. It is proved that the exponential Euler approximation solution converges to the analytic solution with the strong order 1 2 $\\frac{1}{2}$ to SLSDDEs. On the one hand, the classical stability theorem to SLSDDEs is given by the Lyapunov functions. However, in this paper we study the exponential stability in mean square of the exact solution to SLSDDEs by using the definition of logarithmic norm. On the other hand, the implicit Euler scheme to SLSDDEs is known to be exponentially stable in mean square for any step size. However, in this article we propose an explicit method to show that the exponential Euler method to SLSDDEs is proved to share the same stability for any step size by the property of logarithmic norm.
Energy Technology Data Exchange (ETDEWEB)
Di Nunno, Giulia, E-mail: giulian@math.uio.no [University of Oslo, Center of Mathematics for Applications (Norway); Khedher, Asma, E-mail: asma.khedher@tum.de [Technische Universität München, Chair of Mathematical Finance (Germany); Vanmaele, Michèle, E-mail: michele.vanmaele@ugent.be [Ghent University, Department of Applied Mathematics, Computer Science and Statistics (Belgium)
2015-12-15
We consider a backward stochastic differential equation with jumps (BSDEJ) which is driven by a Brownian motion and a Poisson random measure. We present two candidate-approximations to this BSDEJ and we prove that the solution of each candidate-approximation converges to the solution of the original BSDEJ in a space which we specify. We use this result to investigate in further detail the consequences of the choice of the model to (partial) hedging in incomplete markets in finance. As an application, we consider models in which the small variations in the price dynamics are modeled with a Poisson random measure with infinite activity and models in which these small variations are modeled with a Brownian motion or are cut off. Using the convergence results on BSDEJs, we show that quadratic hedging strategies are robust towards the approximation of the market prices and we derive an estimation of the model risk.
Kinetics of subdiffusion-assisted reactions: non-Markovian stochastic Liouville equation approach
International Nuclear Information System (INIS)
Shushin, A I
2005-01-01
Anomalous specific features of the kinetics of subdiffusion-assisted bimolecular reactions (time-dependence, dependence on parameters of systems, etc) are analysed in detail with the use of the non-Markovian stochastic Liouville equation (SLE), which has been recently derived within the continuous-time random-walk (CTRW) approach. In the CTRW approach, subdiffusive motion of particles is modelled by jumps whose onset probability distribution function is of a long-tailed form. The non-Markovian SLE allows for rigorous describing of some peculiarities of these reactions; for example, very slow long-time behaviour of the kinetics, non-analytical dependence of the reaction rate on the reactivity of particles, strong manifestation of fluctuation kinetics showing itself in very slowly decreasing behaviour of the kinetics at very long times, etc
Xie, Bin
2018-01-01
In this paper, the main topic is to investigate the intermittent property of the one-dimensional stochastic heat equation driven by an inhomogeneous Brownian sheet, which is a noise deduced from the study of the catalytic super-Brownian motion. Under some proper conditions on the catalytic measure of the inhomogeneous Brownian sheet, we show that the solution is weakly full intermittent based on the estimates of moments of the solution. In particular, it is proved that the second moment of the solution grows at the exponential rate. The novelty is that the catalytic measure relative to the inhomogeneous noise is not required to be absolutely continuous with respect to the Lebesgue measure on R.
Huang, Guanghui; Wan, Jianping; Chen, Hui
2013-02-01
Nonlinear stochastic differential equation models with unobservable state variables are now widely used in analysis of PK/PD data. Unobservable state variables are usually estimated with extended Kalman filter (EKF), and the unknown pharmacokinetic parameters are usually estimated by maximum likelihood estimator. However, EKF is inadequate for nonlinear PK/PD models, and MLE is known to be biased downwards. A density-based Monte Carlo filter (DMF) is proposed to estimate the unobservable state variables, and a simulation-based M estimator is proposed to estimate the unknown parameters in this paper, where a genetic algorithm is designed to search the optimal values of pharmacokinetic parameters. The performances of EKF and DMF are compared through simulations for discrete time and continuous time systems respectively, and it is found that the results based on DMF are more accurate than those given by EKF with respect to mean absolute error. Copyright © 2012 Elsevier Ltd. All rights reserved.
Probabilistic Forecasts of Wind Power Generation by Stochastic Differential Equation Models
DEFF Research Database (Denmark)
Møller, Jan Kloppenborg; Zugno, Marco; Madsen, Henrik
2016-01-01
The increasing penetration of wind power has resulted in larger shares of volatile sources of supply in power systems worldwide. In order to operate such systems efficiently, methods for reliable probabilistic forecasts of future wind power production are essential. It is well known...... that the conditional density of wind power production is highly dependent on the level of predicted wind power and prediction horizon. This paper describes a new approach for wind power forecasting based on logistic-type stochastic differential equations (SDEs). The SDE formulation allows us to calculate both state......-dependent conditional uncertainties as well as correlation structures. Model estimation is performed by maximizing the likelihood of a multidimensional random vector while accounting for the correlation structure defined by the SDE formulation. We use non-parametric modelling to explore conditional correlation...
International Nuclear Information System (INIS)
Phan Thanh An; Phan Le Na; Ngo Quoc Chung
2004-05-01
We describe a practical implementation for finding parametric domain for asymptotic stability with probability one of zero solution of linear Ito stochastic differential equations based on Korenevskij and Mitropolskij's sufficient condition and our sufficient conditions. Numerical results show that all of these sufficient conditions are crucial in the implementation. (author)
DEFF Research Database (Denmark)
Simonsen, Maria; Schiøler, Henrik; Leth, John-Josef
2014-01-01
The Euler-Maruyama method is applied to a simple stochastic differential equation (SDE) with discontinuous drift. Convergence aspects are investigated in the case, where the Euler-Maruyama method is simulated in dyadic points. A strong rate of convergence is presented for the numerical simulations...
DEFF Research Database (Denmark)
Overgaard, Rune Viig; Holford, Nick; Rytved, K. A.
2007-01-01
Purpose To describe the pharmacodynamic effects of recombinant human interleukin-21 (IL-21) on core body temperature in cynomolgus monkeys using basic mechanisms of heat regulation. A major effort was devoted to compare the use of ordinary differential equations (ODEs) with stochastic differentia...
Lipan, Ovidiu; Ferwerda, Cameron
2018-02-01
The deterministic Hill function depends only on the average values of molecule numbers. To account for the fluctuations in the molecule numbers, the argument of the Hill function needs to contain the means, the standard deviations, and the correlations. Here we present a method that allows for stochastic Hill functions to be constructed from the dynamical evolution of stochastic biocircuits with specific topologies. These stochastic Hill functions are presented in a closed analytical form so that they can be easily incorporated in models for large genetic regulatory networks. Using a repressive biocircuit as an example, we show by Monte Carlo simulations that the traditional deterministic Hill function inaccurately predicts time of repression by an order of two magnitudes. However, the stochastic Hill function was able to capture the fluctuations and thus accurately predicted the time of repression.
DEFF Research Database (Denmark)
method allows us to develop a new expression for the growth rate. The method is based on the stochastic continuous-discrete time state-space model, with a continuous-time state equation (a stochastic differential equation, SDE) combined with a discrete-time measurement equation. In our study the SDE...... described by Kristensen et. al [2]. The resulting time series allows us graphically to inspect the functional dependence of the growth rate on the substrate content. From the method described above we find three new plausible expressions for μ(S). Therefore we apply the likelihood-ratio test to compare...... for the Monod expression. Thus, the method was applied to successfully determine a significant better expression for the substrate dependent growth expression, and we find the method generally applicable for model development. References [1] Kristensen NR, Madsen H, Jørgensen, SB (2004) A method for systematic...
International Nuclear Information System (INIS)
Wang Huaiyu; Long Yao; Chen Nanxian
2006-01-01
In this paper, the solution of Chebyshev equation with its argument being greater than 1 is obtained. The initial value of the derivative of the solution is the expression of magnetization, which is valid for any spin quantum number S. The Chebyshev equation is transformed from an ordinary differential equation obtained when we dealt with Heisenberg model, in order to calculate all three components of magnetization, by many-body Green's function under random phase approximation. The Chebyshev functions with argument being greater than 1 are discussed. This paper shows that the Chebyshev polynomials with their argument being greater than 1 have their physical application.
International Nuclear Information System (INIS)
Ray A. Berry
2005-01-01
At the INL researchers and engineers routinely encounter multiphase, multi-component, and/or multi-material flows. Some examples include: Reactor coolant flows Molten corium flows Dynamic compaction of metal powders Spray forming and thermal plasma spraying Plasma quench reactor Subsurface flows, particularly in the vadose zone Internal flows within fuel cells Black liquor atomization and combustion Wheat-chaff classification in combine harvesters Generation IV pebble bed, high temperature gas reactor The complexity of these flows dictates that they be examined in an averaged sense. Typically one would begin with known (or at least postulated) microscopic flow relations that hold on the ''small'' scale. These include continuum level conservation of mass, balance of species mass and momentum, conservation of energy, and a statement of the second law of thermodynamics often in the form of an entropy inequality (such as the Clausius-Duhem inequality). The averaged or macroscopic conservation equations and entropy inequalities are then obtained from the microscopic equations through suitable averaging procedures. At this stage a stronger form of the second law may also be postulated for the mixture of phases or materials. To render the evolutionary material flow balance system unique, constitutive equations and phase or material interaction relations are introduced from experimental observation, or by postulation, through strict enforcement of the constraints or restrictions resulting from the averaged entropy inequalities. These averaged equations form the governing equation system for the dynamic evolution of these mixture flows. Most commonly, the averaging technique utilized is either volume or time averaging or a combination of the two. The flow restrictions required for volume and time averaging to be valid can be severe, and violations of these restrictions are often found. A more general, less restrictive (and far less commonly used) type of averaging known as
On the XFEL Schrödinger Equation: Highly Oscillatory Magnetic Potentials and Time Averaging
Antonelli, Paolo; Athanassoulis, Agisillaos; Hajaiej, Hichem; Markowich, Peter A.
2014-01-01
We analyse a nonlinear Schrödinger equation for the time-evolution of the wave function of an electron beam, interacting selfconsistently through a Hartree-Fock nonlinearity and through the repulsive Coulomb interaction of an atomic nucleus
Research on nonlinear stochastic dynamical price model
International Nuclear Information System (INIS)
Li Jiaorui; Xu Wei; Xie Wenxian; Ren Zhengzheng
2008-01-01
In consideration of many uncertain factors existing in economic system, nonlinear stochastic dynamical price model which is subjected to Gaussian white noise excitation is proposed based on deterministic model. One-dimensional averaged Ito stochastic differential equation for the model is derived by using the stochastic averaging method, and applied to investigate the stability of the trivial solution and the first-passage failure of the stochastic price model. The stochastic price model and the methods presented in this paper are verified by numerical studies
Interpretation of non-Markovian stochastic Schroedinger equations as a hidden-variable theory
International Nuclear Information System (INIS)
Gambetta, Jay; Wiseman, H.M.
2003-01-01
Do diffusive non-Markovian stochastic Schroedinger equations (SSEs) for open quantum systems have a physical interpretation? In a recent paper [Phys. Rev. A 66, 012108 (2002)] we investigated this question using the orthodox interpretation of quantum mechanics. We found that the solution of a non-Markovian SSE represents the state the system would be in at that time if a measurement was performed on the environment at that time, and yielded a particular result. However, the linking of solutions at different times to make a trajectory is, we concluded, a fiction. In this paper we investigate this question using the modal (hidden variable) interpretation of quantum mechanics. We find that the noise function z(t) appearing in the non-Markovian SSE can be interpreted as a hidden variable for the environment. That is, some chosen property (beable) of the environment has a definite value z(t) even in the absence of measurement on the environment. The non-Markovian SSE gives the evolution of the state of the system 'conditioned' on this environment hidden variable. We present the theory for diffusive non-Markovian SSEs that have as their Markovian limit SSEs corresponding to homodyne and heterodyne detection, as well as one which has no Markovian limit
Stochastic differential equation (SDE) model of opening gold share price of bursa saham malaysia
Hussin, F. N.; Rahman, H. A.; Bahar, A.
2017-09-01
Black and Scholes option pricing model is one of the most recognized stochastic differential equation model in mathematical finance. Two parameter estimation methods have been utilized for the Geometric Brownian model (GBM); historical and discrete method. The historical method is a statistical method which uses the property of independence and normality logarithmic return, giving out the simplest parameter estimation. Meanwhile, discrete method considers the function of density of transition from the process of diffusion normal log which has been derived from maximum likelihood method. These two methods are used to find the parameter estimates samples of Malaysians Gold Share Price data such as: Financial Times and Stock Exchange (FTSE) Bursa Malaysia Emas, and Financial Times and Stock Exchange (FTSE) Bursa Malaysia Emas Shariah. Modelling of gold share price is essential since fluctuation of gold affects worldwide economy nowadays, including Malaysia. It is found that discrete method gives the best parameter estimates than historical method due to the smallest Root Mean Square Error (RMSE) value.
Weak-noise limit of a piecewise-smooth stochastic differential equation.
Chen, Yaming; Baule, Adrian; Touchette, Hugo; Just, Wolfram
2013-11-01
We investigate the validity and accuracy of weak-noise (saddle-point or instanton) approximations for piecewise-smooth stochastic differential equations (SDEs), taking as an illustrative example a piecewise-constant SDE, which serves as a simple model of Brownian motion with solid friction. For this model, we show that the weak-noise approximation of the path integral correctly reproduces the known propagator of the SDE at lowest order in the noise power, as well as the main features of the exact propagator with higher-order corrections, provided the singularity of the path integral associated with the nonsmooth SDE is treated with some heuristics. We also show that, as in the case of smooth SDEs, the deterministic paths of the noiseless system correctly describe the behavior of the nonsmooth SDE in the low-noise limit. Finally, we consider a smooth regularization of the piecewise-constant SDE and study to what extent this regularization can rectify some of the problems encountered when dealing with discontinuous drifts and singularities in SDEs.
A stochastic collocation method for the second order wave equation with a discontinuous random speed
Motamed, Mohammad
2012-08-31
In this paper we propose and analyze a stochastic collocation method for solving the second order wave equation with a random wave speed and subjected to deterministic boundary and initial conditions. The speed is piecewise smooth in the physical space and depends on a finite number of random variables. The numerical scheme consists of a finite difference or finite element method in the physical space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space. This approach leads to the solution of uncoupled deterministic problems as in the Monte Carlo method. We consider both full and sparse tensor product spaces of orthogonal polynomials. We provide a rigorous convergence analysis and demonstrate different types of convergence of the probability error with respect to the number of collocation points for full and sparse tensor product spaces and under some regularity assumptions on the data. In particular, we show that, unlike in elliptic and parabolic problems, the solution to hyperbolic problems is not in general analytic with respect to the random variables. Therefore, the rate of convergence may only be algebraic. An exponential/fast rate of convergence is still possible for some quantities of interest and for the wave solution with particular types of data. We present numerical examples, which confirm the analysis and show that the collocation method is a valid alternative to the more traditional Monte Carlo method for this class of problems. © 2012 Springer-Verlag.
THE GROSSER ALETSCHGLETSCHER DYNAMICS: FROM A “MINIMAL MODEL” TO A STOCHASTIC EQUATION
Directory of Open Access Journals (Sweden)
Alexander V. Kislov
2016-01-01
Full Text Available Mountain glaciers manifest oscillations at different time-scales. Apart from synchronous reaction to lasting changes, there is asynchronism between climatic forcing and observed anomalies of the glaciers. Based on general theories on the laws of temporal dynamics relating to massive inertial objects, the observed interannual changes of glacier length could result from the accumulation of small anomalies in the heat/water fluxes. Despite the fact that the original model of the dynamics of mountain glaciers is deterministically based on the physical law of conservation of water mass, the model of length change is interpreted as stochastic; from this perspective, it is the Langevin equation that incorporates the action of temperature anomalies and precipitation like random white noise. The process is analogous to Brownian motion. Under these conditions, the Grosser Aletschgletscher (selected as an example is represented by a system undergoing a random walk. It was shown that the possible range of variability covers the observed interval of length fluctuations.
Flohr, J R; Dritz, S S; Tokach, M D; Woodworth, J C; DeRouchey, J M; Goodband, R D
2018-05-01
Floor space allowance for pigs has substantial effects on pig growth and welfare. Data from 30 papers examining the influence of floor space allowance on the growth of finishing pigs was used in a meta-analysis to develop alternative prediction equations for average daily gain (ADG), average daily feed intake (ADFI) and gain : feed ratio (G : F). Treatment means were compiled in a database that contained 30 papers for ADG and 28 papers for ADFI and G : F. The predictor variables evaluated were floor space (m2/pig), k (floor space/final BW0.67), Initial BW, Final BW, feed space (pigs per feeder hole), water space (pigs per waterer), group size (pigs per pen), gender, floor type and study length (d). Multivariable general linear mixed model regression equations were used. Floor space treatments within each experiment were the observational and experimental unit. The optimum equations to predict ADG, ADFI and G : F were: ADG, g=337.57+(16 468×k)-(237 350×k 2)-(3.1209×initial BW (kg))+(2.569×final BW (kg))+(71.6918×k×initial BW (kg)); ADFI, g=833.41+(24 785×k)-(388 998×k 2)-(3.0027×initial BW (kg))+(11.246×final BW (kg))+(187.61×k×initial BW (kg)); G : F=predicted ADG/predicted ADFI. Overall, the meta-analysis indicates that BW is an important predictor of ADG and ADFI even after computing the constant coefficient k, which utilizes final BW in its calculation. This suggests including initial and final BW improves the prediction over using k as a predictor alone. In addition, the analysis also indicated that G : F of finishing pigs is influenced by floor space allowance, whereas individual studies have concluded variable results.
International Nuclear Information System (INIS)
Cockburn, S. P.; Gallucci, D.; Proukakis, N. P.
2011-01-01
The stochastic Gross-Pitaevskii equation is shown to be an excellent model for quasi-one-dimensional Bose gas experiments, accurately reproducing the in situ density profiles recently obtained in the experiments of Trebbia et al.[Phys. Rev. Lett. 97, 250403 (2006)] and van Amerongen et al.[Phys. Rev. Lett. 100, 090402 (2008)] and the density fluctuation data reported by Armijo et al.[Phys. Rev. Lett. 105, 230402 (2010)]. To facilitate such agreement, we propose and implement a quasi-one-dimensional extension to the one-dimensional stochastic Gross-Pitaevskii equation for the low-energy, axial modes, while atoms in excited transverse modes are treated as independent ideal Bose gases.
DEFF Research Database (Denmark)
Møller, Jonas Bech; Overgaard, R.V.; Madsen, Henrik
2010-01-01
Several articles have investigated stochastic differential equations (SDEs) in PK/PD models, but few have quantitatively investigated the benefits to predictive performance of models based on real data. Estimation of first phase insulin secretion which reflects beta-cell function using models of ...... obtained from the glucose tolerance tests. Since, the estimation time of extended models was not heavily increased compared to basic models, the applied method is concluded to have high relevance not only in theory but also in practice....
Crisan, Dan
2011-01-01
"Stochastic Analysis" aims to provide mathematical tools to describe and model high dimensional random systems. Such tools arise in the study of Stochastic Differential Equations and Stochastic Partial Differential Equations, Infinite Dimensional Stochastic Geometry, Random Media and Interacting Particle Systems, Super-processes, Stochastic Filtering, Mathematical Finance, etc. Stochastic Analysis has emerged as a core area of late 20th century Mathematics and is currently undergoing a rapid scientific development. The special volume "Stochastic Analysis 2010" provides a sa
Debussche, A.; Glatt-Holtz, N.; Temam, R.; Ziane, M.
2012-07-01
The primitive equations (PEs) are a basic model in the study of large scale oceanic and atmospheric dynamics. These systems form the analytical core of the most advanced general circulation models. For this reason and due to their challenging nonlinear and anisotropic structure, the PEs have recently received considerable attention from the mathematical community. On the other hand, in view of the complex multi-scale nature of the earth's climate system, many uncertainties appear that should be accounted for in the basic dynamical models of atmospheric and oceanic processes. In the climate community stochastic methods have come into extensive use in this connection. For this reason there has appeared a need to further develop the foundations of nonlinear stochastic partial differential equations in connection with the PEs and more generally. In this work we study a stochastic version of the PEs. We establish the global existence and uniqueness of strong, pathwise solutions for these equations in dimension 3 for the case of a nonlinear multiplicative noise. The proof makes use of anisotropic estimates, L^{p}_{t}L^{q}_{x} estimates on the pressure and stopping time arguments.
International Nuclear Information System (INIS)
Debussche, A; Glatt-Holtz, N; Temam, R; Ziane, M
2012-01-01
The primitive equations (PEs) are a basic model in the study of large scale oceanic and atmospheric dynamics. These systems form the analytical core of the most advanced general circulation models. For this reason and due to their challenging nonlinear and anisotropic structure, the PEs have recently received considerable attention from the mathematical community. On the other hand, in view of the complex multi-scale nature of the earth's climate system, many uncertainties appear that should be accounted for in the basic dynamical models of atmospheric and oceanic processes. In the climate community stochastic methods have come into extensive use in this connection. For this reason there has appeared a need to further develop the foundations of nonlinear stochastic partial differential equations in connection with the PEs and more generally. In this work we study a stochastic version of the PEs. We establish the global existence and uniqueness of strong, pathwise solutions for these equations in dimension 3 for the case of a nonlinear multiplicative noise. The proof makes use of anisotropic estimates, L p t L q x estimates on the pressure and stopping time arguments
Stochastic processes and applications diffusion processes, the Fokker-Planck and Langevin equations
Pavliotis, Grigorios A
2014-01-01
This book presents various results and techniques from the theory of stochastic processes that are useful in the study of stochastic problems in the natural sciences. The main focus is analytical methods, although numerical methods and statistical inference methodologies for studying diffusion processes are also presented. The goal is the development of techniques that are applicable to a wide variety of stochastic models that appear in physics, chemistry and other natural sciences. Applications such as stochastic resonance, Brownian motion in periodic potentials and Brownian motors are studied and the connection between diffusion processes and time-dependent statistical mechanics is elucidated. The book contains a large number of illustrations, examples, and exercises. It will be useful for graduate-level courses on stochastic processes for students in applied mathematics, physics and engineering. Many of the topics covered in this book (reversible diffusions, convergence to eq...
Energy Technology Data Exchange (ETDEWEB)
Suescun D, D.; Oviedo T, M., E-mail: daniel.suescun@usco.edu.co [Universidad Surcolombiana, Av. Pastrana Borrero - Carrera 1, Neiva, Huila (Colombia)
2017-09-15
In this paper, a numerical study of stochastic differential equations that describe the kinetics in a nuclear reactor is presented. These equations, known as the stochastic equations of punctual kinetics they model temporal variations in neutron population density and concentrations of deferred neutron precursors. Because these equations are probabilistic in nature (since random oscillations in the neutrons and population of precursors were considered to be approximately normally distributed, and these equations also possess strong coupling and stiffness properties) the proposed method for the numerical simulations is the Euler-Maruyama scheme that provides very good approximations for calculating the neutron population and concentrations of deferred neutron precursors. The method proposed for this work was computationally tested for different seeds, initial conditions, experimental data and forms of reactivity for a group of precursors and then for six groups of deferred neutron precursors at each time step with 5000 Brownian movements per seed. In a paper reported in the literature, the Euler-Maruyama method was proposed, but there are many doubts about the reported values, in addition to not reporting the seed used, so in this work is expected to rectify the reported values. After taking the average of the different seeds used to generate the pseudo-random numbers the results provided by the Euler-Maruyama scheme will be compared in mean and standard deviation with other methods reported in the literature and results of the deterministic model of the equations of the punctual kinetics. This comparison confirms in particular that the Euler-Maruyama scheme is an efficient method to solve the equations of stochastic point kinetics but different from the values found and reported by another author. The Euler-Maruyama method is simple and easy to implement, provides acceptable results for neutron population density and concentration of deferred neutron precursors and
García, Isaac A.; Llibre, Jaume; Maza, Susanna
2018-06-01
In this work we consider real analytic functions , where , Ω is a bounded open subset of , is an interval containing the origin, are parameters, and ε is a small parameter. We study the branching of the zero-set of at multiple points when the parameter ε varies. We apply the obtained results to improve the classical averaging theory for computing T-periodic solutions of λ-families of analytic T-periodic ordinary differential equations defined on , using the displacement functions defined by these equations. We call the coefficients in the Taylor expansion of in powers of ε the averaged functions. The main contribution consists in analyzing the role that have the multiple zeros of the first non-zero averaged function. The outcome is that these multiple zeros can be of two different classes depending on whether the zeros belong or not to the analytic set defined by the real variety associated to the ideal generated by the averaged functions in the Noetheriang ring of all the real analytic functions at . We bound the maximum number of branches of isolated zeros that can bifurcate from each multiple zero z 0. Sometimes these bounds depend on the cardinalities of minimal bases of the former ideal. Several examples illustrate our results and they are compared with the classical theory, branching theory and also under the light of singularity theory of smooth maps. The examples range from polynomial vector fields to Abel differential equations and perturbed linear centers.
International Nuclear Information System (INIS)
Laurence, D.
1997-01-01
The k-ε model and Reynolds stress transport model are set out in a few words. Limitations of models are shown, particularly for turbulence generation in the turbulent viscosity context, and, more generally, the uncertainties and miscellaneous changes made to the dissipation equation. The performances of models are then compared, using results of the three latest ERCOFTA/IAHR workshops. It is shown that algebraic constraints which can be derived exactly by assuming asymptotic limits (rapid distortion, homogeneous shear at infinite time, 2D turbulence) have inhibited a better tuning of the models for real life flow where these limits are not encountered. A more pragmatic approach could be taken by allowing the constants to be functions of invariant parameters. But these functions, making the models non-linear, can lead to bifurcations or instability. One essential parameter is the distance to the wall, which recent models have tried to eliminate, although this parameter appears indirectly through the Poisson equation for the fluctuating pressure. A possible indirect model is the elliptic relaxation. Progress was recently achieved in near-wall low Re modelling, but these advances do not always result in benefits to industry since only the 'wall function' approaches can be used in the high Re, 3D flows that we need to study. With the knowledge gained from near-wall modelling, it might be profitable to revisit the 'wall functions' devised 20 years ago. (author)
Average value of the shape and direction factor in the equation of refractive index
Zhang, Tao
2017-10-01
The theoretical calculation of the refractive indices is of great significance for the developments of new optical materials. The calculation method of refractive index, which was deduced from the electron-cloud-conductor model, contains the shape and direction factor 〈g〉. 〈g〉 affects the electromagnetic-induction energy absorbed by the electron clouds, thereby influencing the refractive indices. It is not yet known how to calculate 〈g〉 value of non-spherical electron clouds. In this paper, 〈g〉 value is derived by imaginatively dividing the electron cloud into numerous little volume elements and then regrouping them. This paper proves that 〈g〉 = 2/3 when molecules’ spatial orientations distribute randomly. The calculations of the refractive indices of several substances validate this equation. This result will help to promote the application of the calculation method of refractive index.
Probabilistic Forecast of Wind Power Generation by Stochastic Differential Equation Models
Elkantassi, Soumaya
2017-01-01
Reliable forecasting of wind power generation is crucial to optimal control of costs in generation of electricity with respect to the electricity demand. Here, we propose and analyze stochastic wind power forecast models described by parametrized
Directory of Open Access Journals (Sweden)
Li-ren Yu
2012-03-01
Full Text Available This paper describes a numerical simulation in the Amazon water system, aiming to develop a quasi-three-dimensional numerical tool for refined modeling of turbulent flow and passive transport of mass in natural waters. Three depth-averaged two-equation turbulence closure models, k˜−ε˜,k˜−w˜, and k˜−ω˜ , were used to close the non-simplified quasi-three dimensional hydrodynamic fundamental governing equations. The discretized equations were solved with the advanced multi-grid iterative method using non-orthogonal body-fitted coarse and fine grids with collocated variable arrangement. Except for steady flow computation, the processes of contaminant inpouring and plume development at the beginning of discharge, caused by a side-discharge of a tributary, have also been numerically investigated. The three depth-averaged two-equation closure models are all suitable for modeling strong mixing turbulence. The newly established turbulence models such as the k˜−ω˜ model, with a higher order of magnitude of the turbulence parameter, provide a possibility for improving computational precision.
Deng, Chenhui; Plan, Elodie L; Karlsson, Mats O
2016-06-01
Parameter variation in pharmacometric analysis studies can be characterized as within subject parameter variability (WSV) in pharmacometric models. WSV has previously been successfully modeled using inter-occasion variability (IOV), but also stochastic differential equations (SDEs). In this study, two approaches, dynamic inter-occasion variability (dIOV) and adapted stochastic differential equations, were proposed to investigate WSV in pharmacometric count data analysis. These approaches were applied to published count models for seizure counts and Likert pain scores. Both approaches improved the model fits significantly. In addition, stochastic simulation and estimation were used to explore further the capability of the two approaches to diagnose and improve models where existing WSV is not recognized. The results of simulations confirmed the gain in introducing WSV as dIOV and SDEs when parameters vary randomly over time. Further, the approaches were also informative as diagnostics of model misspecification, when parameters changed systematically over time but this was not recognized in the structural model. The proposed approaches in this study offer strategies to characterize WSV and are not restricted to count data.
Comparative study of dense plasma state equations obtained from different models of average-atom
International Nuclear Information System (INIS)
Fromy, Patrice
1991-01-01
This research thesis addresses the influence of temperature and density effects on magnitudes such as pressure, energy, ionisation, and on energy levels of a body described according to the approximation of an electrically neutral isolated atomic sphere. Starting from the general formalism of the functional density, with some approximations, the author deduces the Thomas-Fermi, Thomas-Fermi-Dirac, and Thomas-Fermi-Dirac-Weizsaecker models, and an average-atom approximated quantum model. For each of these models, the author presents an explicit method of resolution, as well as the determination of different magnitudes taken into account in this study. For the different studied magnitudes, the author highlights effects due to the influence of temperature and of density, as well as variations due to the different models [fr
Rouz, Omid Farkhondeh; Ahmadian, Davood; Milev, Mariyan
2017-12-01
This paper establishes exponential mean square stability of two classes of theta Milstein methods, namely split-step theta Milstein (SSTM) method and stochastic theta Milstein (STM) method, for stochastic differential delay equations (SDDEs). We consider the SDDEs problem under a coupled monotone condition on drift and diffusion coefficients, as well as a necessary linear growth condition on the last term of theta Milstein method. It is proved that the SSTM method with θ ∈ [0, ½] can recover the exponential mean square stability of the exact solution with some restrictive conditions on stepsize, but for θ ∈ (½, 1], we proved that the stability results hold for any stepsize. Then, based on the stability results of SSTM method, we examine the exponential mean square stability of the STM method and obtain the similar stability results to that of the SSTM method. In the numerical section the figures show thevalidity of our claims.
Energy Technology Data Exchange (ETDEWEB)
Jin, Shi, E-mail: sjin@wisc.edu [Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706 (United States); Institute of Natural Sciences, Department of Mathematics, MOE-LSEC and SHL-MAC, Shanghai Jiao Tong University, Shanghai 200240 (China); Lu, Hanqing, E-mail: hanqing@math.wisc.edu [Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706 (United States)
2017-04-01
In this paper, we develop an Asymptotic-Preserving (AP) stochastic Galerkin scheme for the radiative heat transfer equations with random inputs and diffusive scalings. In this problem the random inputs arise due to uncertainties in cross section, initial data or boundary data. We use the generalized polynomial chaos based stochastic Galerkin (gPC-SG) method, which is combined with the micro–macro decomposition based deterministic AP framework in order to handle efficiently the diffusive regime. For linearized problem we prove the regularity of the solution in the random space and consequently the spectral accuracy of the gPC-SG method. We also prove the uniform (in the mean free path) linear stability for the space-time discretizations. Several numerical tests are presented to show the efficiency and accuracy of proposed scheme, especially in the diffusive regime.
Preston, L. A.
2017-12-01
Marine hydrokinetic (MHK) devices offer a clean, renewable alternative energy source for the future. Responsible utilization of MHK devices, however, requires that the effects of acoustic noise produced by these devices on marine life and marine-related human activities be well understood. Paracousti is a 3-D full waveform acoustic modeling suite that can accurately propagate MHK noise signals in the complex bathymetry found in the near-shore to open ocean environment and considers real properties of the seabed, water column, and air-surface interface. However, this is a deterministic simulation that assumes the environment and source are exactly known. In reality, environmental and source characteristics are often only known in a statistical sense. Thus, to fully characterize the expected noise levels within the marine environment, this uncertainty in environmental and source factors should be incorporated into the acoustic simulations. One method is to use Monte Carlo (MC) techniques where simulation results from a large number of deterministic solutions are aggregated to provide statistical properties of the output signal. However, MC methods can be computationally prohibitive since they can require tens of thousands or more simulations to build up an accurate representation of those statistical properties. An alternative method, using the technique of stochastic partial differential equations (SPDE), allows computation of the statistical properties of output signals at a small fraction of the computational cost of MC. We are developing a SPDE solver for the 3-D acoustic wave propagation problem called Paracousti-UQ to help regulators and operators assess the statistical properties of environmental noise produced by MHK devices. In this presentation, we present the SPDE method and compare statistical distributions of simulated acoustic signals in simple models to MC simulations to show the accuracy and efficiency of the SPDE method. Sandia National Laboratories
Energy Technology Data Exchange (ETDEWEB)
Hahn, Y.K., E-mail: ykhahn22@verizon.net
2014-12-15
The self-consistent field theory of collisions is formulated, incorporating the unique dynamics generated by the self-averaged potentials. The bound state Hartree–Fock approach is extended for the first time to scattering states, by properly resolving the principal difficulties of non-integrable continuum orbitals and imposing complex asymptotic conditions. The recently developed asymptotic source theory provides the natural theoretical basis, as the asymptotic conditions are completely transferred to the source terms and the new scattering function is made fullyintegrable. The scattering solutions can then be directly expressed in terms of bound state HF configurations, establishing the relationship between the bound and scattering state solutions. Alternatively, the integrable spin orbitals are generated by constructing the individual orbital equations that contain asymptotic sources and self-averaged potentials. However, the orbital energies are not determined by the equations, and a special channel energy fixing procedure is developed to secure the solutions. It is also shown that the variational construction of the orbital equations has intrinsic ambiguities that are generally associated with the self-consistent approach. On the other hand, when a small subset of open channels is included in the source term, the solutions are only partiallyintegrable, but the individual open channels can then be treated more simply by properly selecting the orbital energies. The configuration mixing and channel coupling are then necessary to complete the solution. The new theory improves the earlier continuum HF model. - Highlights: • First extension of HF to scattering states, with proper asymptotic conditions. • Orbital equations with asymptotic sources and integrable orbital solutions. • Construction of self-averaged potentials, and orbital energy fixing. • Channel coupling and configuration mixing, involving the new orbitals. • Critical evaluation of the
International Nuclear Information System (INIS)
Zhang Guangjun; Xu Jianxue; Wang Jue; Yue Zhifeng; Zou Hailin
2009-01-01
In this paper stochastic resonance induced by the novel random transitions of two-dimensional weak damping bistable Duffing oscillator is analyzed by moment method. This kind of novel transition refers to the one among three potential well on two sides of bifurcation point of original system at the presence of internal noise. Several conclusions are drawn. First, the semi-analytical result of stochastic resonance induced by the novel random transitions of two-dimensional weak damping bistable Duffing oscillator can be obtained, and the semi-analytical result is qualitatively compatible with the one of Monte Carlo simulation. Second, a bifurcation of double-branch fixed point curves occurs in the moment equations with noise intensity as their bifurcation parameter. Third, the bifurcation of moment equations corresponds to stochastic resonance of original system. Finally, the mechanism of stochastic resonance is presented from another viewpoint through analyzing the energy transfer induced by the bifurcation of moment equation.
Caglar, Mehmet Umut; Pal, Ranadip
2011-03-01
Central dogma of molecular biology states that ``information cannot be transferred back from protein to either protein or nucleic acid''. However, this assumption is not exactly correct in most of the cases. There are a lot of feedback loops and interactions between different levels of systems. These types of interactions are hard to analyze due to the lack of cell level data and probabilistic - nonlinear nature of interactions. Several models widely used to analyze and simulate these types of nonlinear interactions. Stochastic Master Equation (SME) models give probabilistic nature of the interactions in a detailed manner, with a high calculation cost. On the other hand Probabilistic Boolean Network (PBN) models give a coarse scale picture of the stochastic processes, with a less calculation cost. Differential Equation (DE) models give the time evolution of mean values of processes in a highly cost effective way. The understanding of the relations between the predictions of these models is important to understand the reliability of the simulations of genetic regulatory networks. In this work the success of the mapping between SME, PBN and DE models is analyzed and the accuracy and affectivity of the control policies generated by using PBN and DE models is compared.
Numerical solutions of stochastic Lotka-Volterra equations via operational matrices
Directory of Open Access Journals (Sweden)
F. Hosseini Shekarabi
2016-03-01
Full Text Available In this paper, an efficient and convenient method for numerical solutions of stochastic Lotka-Volterra dynamical system is proposed. Here, we consider block pulse functions and their operational matrices of integration. Illustrative example is included to demonstrate the procedure and accuracy of the operational matrices based on block pulse functions.
Avanzini, Francesco; Moro, Giorgio J
2018-03-15
The quantum molecular trajectory is the deterministic trajectory, arising from the Bohm theory, that describes the instantaneous positions of the nuclei of molecules by assuring the agreement with the predictions of quantum mechanics. Therefore, it provides the suitable framework for representing the geometry and the motions of molecules without neglecting their quantum nature. However, the quantum molecular trajectory is extremely demanding from the computational point of view, and this strongly limits its applications. To overcome such a drawback, we derive a stochastic representation of the quantum molecular trajectory, through projection operator techniques, for the degrees of freedom of an open quantum system. The resulting Fokker-Planck operator is parametrically dependent upon the reduced density matrix of the open system. Because of the pilot role played by the reduced density matrix, this stochastic approach is able to represent accurately the main features of the open system motions both at equilibrium and out of equilibrium with the environment. To verify this procedure, the predictions of the stochastic and deterministic representation are compared for a model system of six interacting harmonic oscillators, where one oscillator is taken as the open quantum system of interest. The undeniable advantage of the stochastic approach is that of providing a simplified and self-contained representation of the dynamics of the open system coordinates. Furthermore, it can be employed to study the out of equilibrium dynamics and the relaxation of quantum molecular motions during photoinduced processes, like photoinduced conformational changes and proton transfers.
International Nuclear Information System (INIS)
El-Tawil, M A; Al-Jihany, A S
2008-01-01
In this paper, nonlinear oscillators under quadratic nonlinearity with stochastic inputs are considered. Different methods are used to obtain first order approximations, namely, the WHEP technique, the perturbation method, the Pickard approximations, the Adomian decompositions and the homotopy perturbation method (HPM). Some statistical moments are computed for the different methods using mathematica 5. Comparisons are illustrated through figures for different case-studies
Directory of Open Access Journals (Sweden)
2006-01-01
Full Text Available We show the existence of a solution for the double-barrier reflected BSDE when the barriers are completely separate and the generator is continuous with quadratic growth. As an application, we solve the risk-sensitive mixed zero-sum stochastic differential game. In addition we deal with recallable options under Knightian uncertainty.
Stochastic theory of fatigue corrosion
Hu, Haiyun
1999-10-01
A stochastic theory of corrosion has been constructed. The stochastic equations are described giving the transportation corrosion rate and fluctuation corrosion coefficient. In addition the pit diameter distribution function, the average pit diameter and the most probable pit diameter including other related empirical formula have been derived. In order to clarify the effect of stress range on the initiation and growth behaviour of pitting corrosion, round smooth specimen were tested under cyclic loading in 3.5% NaCl solution.
Fractional Stochastic Field Theory
Honkonen, Juha
2018-02-01
Models describing evolution of physical, chemical, biological, social and financial processes are often formulated as differential equations with the understanding that they are large-scale equations for averages of quantities describing intrinsically random processes. Explicit account of randomness may lead to significant changes in the asymptotic behaviour (anomalous scaling) in such models especially in low spatial dimensions, which in many cases may be captured with the use of the renormalization group. Anomalous scaling and memory effects may also be introduced with the use of fractional derivatives and fractional noise. Construction of renormalized stochastic field theory with fractional derivatives and fractional noise in the underlying stochastic differential equations and master equations and the interplay between fluctuation-induced and built-in anomalous scaling behaviour is reviewed and discussed.
International Nuclear Information System (INIS)
Doering, C.R.
1985-01-01
Applications of nonlinear parabolic stochastic differential equations with additive colored noise in equilibrium and nonequilibrium statistical mechanics and quantum field theory are developed in detail, providing a new unified mathematical approach to many problems. The existence and uniqueness of solutions to these equations is established, and some of the properties of the solutions are investigated. In particular, asymptotic expansions for the correlation functions of the solutions are introduced and compared to rigorous nonperturbative bounds on the moments. It is found that the perturbative analysis is in qualitative disagreement with the exact result in models corresponding to cut-off self-interacting nonperturbatively renormalizable scalar quantum field theories. For these theories the nonlinearities cannot be considered as perturbations of the linearized theory
International Nuclear Information System (INIS)
Mieck, B.
2007-01-01
We consider bosonic atoms with a repulsive contact interaction in a trap potential for a Bose-Einstein condensation (BEC) and additionally include a random potential. The ensemble averages for two models of static (I) and dynamic (II) disorder are performed and investigated in parallel. The bosonic many body systems of the two disorder models are represented by coherent state path integrals on the Keldysh time contour which allow exact ensemble averages for zero and finite temperatures. These ensemble averages of coherent state path integrals therefore present alternatives to replica field theories or super-symmetric averaging techniques. Hubbard-Stratonovich transformations (HST) lead to two corresponding self-energies for the hermitian repulsive interaction and for the non-hermitian disorder-interaction. The self-energy of the repulsive interaction is absorbed by a shift into the disorder-self-energy which comprises as an element of a larger symplectic Lie algebra sp(4M) the self-energy of the repulsive interaction as a subalgebra (which is equivalent to the direct product of M x sp(2); 'M' is the number of discrete time intervals of the disorder-self-energy in the generating function). After removal of the remaining Gaussian integral for the self-energy of the repulsive interaction, the first order variations of the coherent state path integrals result in the exact mean field or saddle point equations, solely depending on the disorder-self-energy matrix. These equations can be solved by continued fractions and are reminiscent to the 'Nambu-Gorkov' Green function formalism in superconductivity because anomalous terms or pair condensates of the bosonic atoms are also included into the selfenergies. The derived mean field equations of the models with static (I) and dynamic (II) disorder are particularly applicable for BEC in d=3 spatial dimensions because of the singularity of the density of states at vanishing wavevector. However, one usually starts out from
Chaynikov, S.; Porta, G.; Riva, M.; Guadagnini, A.
2012-04-01
We focus on a theoretical analysis of nonreactive solute transport in porous media through the volume averaging technique. Darcy-scale transport models based on continuum formulations typically include large scale dispersive processes which are embedded in a pore-scale advection diffusion equation through a Fickian analogy. This formulation has been extensively questioned in the literature due to its inability to depict observed solute breakthrough curves in diverse settings, ranging from the laboratory to the field scales. The heterogeneity of the pore-scale velocity field is one of the key sources of uncertainties giving rise to anomalous (non-Fickian) dispersion in macro-scale porous systems. Some of the models which are employed to interpret observed non-Fickian solute behavior make use of a continuum formulation of the porous system which assumes a two-region description and includes a bimodal velocity distribution. A first class of these models comprises the so-called ''mobile-immobile'' conceptualization, where convective and dispersive transport mechanisms are considered to dominate within a high velocity region (mobile zone), while convective effects are neglected in a low velocity region (immobile zone). The mass exchange between these two regions is assumed to be controlled by a diffusive process and is macroscopically described by a first-order kinetic. An extension of these ideas is the two equation ''mobile-mobile'' model, where both transport mechanisms are taken into account in each region and a first-order mass exchange between regions is employed. Here, we provide an analytical derivation of two region "mobile-mobile" meso-scale models through a rigorous upscaling of the pore-scale advection diffusion equation. Among the available upscaling methodologies, we employ the Volume Averaging technique. In this approach, the heterogeneous porous medium is supposed to be pseudo-periodic, and can be represented through a (spatially) periodic unit cell
Detailed balance principle and finite-difference stochastic equation in a field theory
International Nuclear Information System (INIS)
Kozhamkulov, T.A.
1986-01-01
A finite-difference equation, which is a generalization of the Langevin equation in field theory, has been obtained basing upon the principle of detailed balance for the Markov chain. Advantages of the present approach as compared with the conventional Parisi-Wu method are shown for examples of an exactly solvable problem of zero-dimensional quantum theory and a simple numerical simulation
Principle of detailed balance and the finite-difference stochastic equation in field theory
International Nuclear Information System (INIS)
Kozhamkulov, T.A.
1986-01-01
The principle of detailed balance for the Markov chain is used to obtain a finite-difference equation which generalizes the Langevin equation in field theory. The advantages of using this approach compared to the conventional Parisi-Wu method are demonstrated for the examples of an exactly solvable problem in zero-dimensional quantum theory and a simple numerical simulation
International Nuclear Information System (INIS)
Keanini, R.G.
2011-01-01
Research highlights: → Systematic approach for physically probing nonlinear and random evolution problems. → Evolution of vortex sheets corresponds to evolution of an Ornstein-Uhlenbeck process. → Organization of near-molecular scale vorticity mediated by hydrodynamic modes. → Framework allows calculation of vorticity evolution within random strain fields. - Abstract: A framework which combines Green's function (GF) methods and techniques from the theory of stochastic processes is proposed for tackling nonlinear evolution problems. The framework, established by a series of easy-to-derive equivalences between Green's function and stochastic representative solutions of linear drift-diffusion problems, provides a flexible structure within which nonlinear evolution problems can be analyzed and physically probed. As a preliminary test bed, two canonical, nonlinear evolution problems - Burgers' equation and the nonlinear Schroedinger's equation - are first treated. In the first case, the framework provides a rigorous, probabilistic derivation of the well known Cole-Hopf ansatz. Likewise, in the second, the machinery allows systematic recovery of a known soliton solution. The framework is then applied to a fairly extensive exploration of physical features underlying evolution of randomly stretched and advected Burger's vortex sheets. Here, the governing vorticity equation corresponds to the Fokker-Planck equation of an Ornstein-Uhlenbeck process, a correspondence that motivates an investigation of sub-sheet vorticity evolution and organization. Under the assumption that weak hydrodynamic fluctuations organize disordered, near-molecular-scale, sub-sheet vorticity, it is shown that these modes consist of two weakly damped counter-propagating cross-sheet acoustic modes, a diffusive cross-sheet shear mode, and a diffusive cross-sheet entropy mode. Once a consistent picture of in-sheet vorticity evolution is established, a number of analytical results, describing the
The average kinetic energy of the heavy quark in Λb in the Bethe-Salpeter equation approach
International Nuclear Information System (INIS)
Guo, X.-H.; Wu, H.-K.
2007-01-01
In the previous paper, based on the SU(2) f xSU(2) s heavy quark symmetries of the QCD Lagrangian in the heavy quark limit, the Bethe-Salpeter equation for the heavy baryon Λ b was established with the picture that Λ b is composed of a heavy quark and a scalar light diquark. In the present work, we apply this model to calculate μ π 2 for Λ b , the average kinetic energy of the heavy quark inside Λ b . This quantity is particularly interesting since it can be measured in experiments and since it contributes to the inclusive semileptonic decays of Λ b when contributions from higher order terms in 1/M b expansions are taken into account and consequently influences the determination of the Cabibbo-Kobayashi-Maskawa matrix elements V ub and V cb . We find that μ π 2 for Λ b is 0.25GeV 2 ∼0.95GeV 2 , depending on the parameters in the model including the light diquark mass and the interaction strength between the heavy quark and the light diquark in the kernel of the BS equation. We also find that this result is consistent with the value of μ π 2 for Λ b which is derived from the experimental value of μ π 2 for the B meson with the aid of the heavy quark effective theory
Dolgov, Sergey
2015-11-03
We apply the tensor train (TT) decomposition to construct the tensor product polynomial chaos expansion (PCE) of a random field, to solve the stochastic elliptic diffusion PDE with the stochastic Galerkin discretization, and to compute some quantities of interest (mean, variance, and exceedance probabilities). We assume that the random diffusion coefficient is given as a smooth transformation of a Gaussian random field. In this case, the PCE is delivered by a complicated formula, which lacks an analytic TT representation. To construct its TT approximation numerically, we develop the new block TT cross algorithm, a method that computes the whole TT decomposition from a few evaluations of the PCE formula. The new method is conceptually similar to the adaptive cross approximation in the TT format but is more efficient when several tensors must be stored in the same TT representation, which is the case for the PCE. In addition, we demonstrate how to assemble the stochastic Galerkin matrix and to compute the solution of the elliptic equation and its postprocessing, staying in the TT format. We compare our technique with the traditional sparse polynomial chaos and the Monte Carlo approaches. In the tensor product polynomial chaos, the polynomial degree is bounded for each random variable independently. This provides higher accuracy than the sparse polynomial set or the Monte Carlo method, but the cardinality of the tensor product set grows exponentially with the number of random variables. However, when the PCE coefficients are implicitly approximated in the TT format, the computations with the full tensor product polynomial set become possible. In the numerical experiments, we confirm that the new methodology is competitive in a wide range of parameters, especially where high accuracy and high polynomial degrees are required.
Random Attractors for the Stochastic Navier-Stokes Equations on the 2D Unit Sphere
Brzeźniak, Z.; Goldys, B.; Le Gia, Q. T.
2018-03-01
In this paper we prove the existence of random attractors for the Navier-Stokes equations on 2 dimensional sphere under random forcing irregular in space and time. We also deduce the existence of an invariant measure.
Directory of Open Access Journals (Sweden)
Weifeng Wang
2014-01-01
Full Text Available We study an optimal control problem governed by a semilinear parabolic equation, whose control variable is contained only in the boundary condition. An existence theorem for the optimal control is obtained.
Johnson, Paul; Howell, Sydney; Duck, Peter
2017-08-13
A mixed financial/physical partial differential equation (PDE) can optimize the joint earnings of a single wind power generator (WPG) and a generic energy storage device (ESD). Physically, the PDE includes constraints on the ESD's capacity, efficiency and maximum speeds of charge and discharge. There is a mean-reverting daily stochastic cycle for WPG power output. Physically, energy can only be produced or delivered at finite rates. All suppliers must commit hourly to a finite rate of delivery C , which is a continuous control variable that is changed hourly. Financially, we assume heavy 'system balancing' penalties in continuous time, for deviations of output rate from the commitment C Also, the electricity spot price follows a mean-reverting stochastic cycle with a strong evening peak, when system balancing penalties also peak. Hence the economic goal of the WPG plus ESD, at each decision point, is to maximize expected net present value (NPV) of all earnings (arbitrage) minus the NPV of all expected system balancing penalties, along all financially/physically feasible future paths through state space. Given the capital costs for the various combinations of the physical parameters, the design and operating rules for a WPG plus ESD in a finite market may be jointly optimizable.This article is part of the themed issue 'Energy management: flexibility, risk and optimization'. © 2017 The Author(s).
International Nuclear Information System (INIS)
Kist, Tarso B.L.; Orszag, M.; Davidovich, L.
1997-01-01
The dynamics of open system is frequently modeled in terms of a small system S coupled to a reservoir R, the last having an infinitely larger number of degree of freedom than S. Usually the dynamics of the S variables may be of interest, which can be studied using either Langevin equations, or master equations, or yet the path integral formulation. Useful alternatives for the master equation method are the Monte Carlo Wave-function method (MCWF), and Stochastic Schroedinger Equations (SSE's). The methods MCWF and SSE's recently experienced a fast development both in their theoretical background and applications to the study of the dissipative quantum systems dynamics in quantum optics. Even though these alternatives can be shown to be formally equivalent to the master equation approach, they are often regarded as mathematical tricks, with no relation to a concrete physical evolution of the system. The advantage of using them is that one has to deal with state vectors, instead of density matrices, thus reducing the total amount of matrix elements to be calculated. In this work, we consider the possibility of giving a physical interpretation to these methods, in terms of continuous measurements made on the evolving system. We show that physical realizations of the two methods are indeed possible, for a mode of the electromagnetic field in a cavity interacting with a continuum of modes corresponding to the field outside the cavity. Two schemes are proposed, consisting of a mode of the electromagnetic field interacting with a beam of Rydberg two-level atoms. In these schemes, the field mode plays the role of a small system and the atomic beam plays the role of a reservoir (infinitely larger number of degrees of freedom at finite temperature, the interaction between them being given by the Jaynes-Cummings model
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Basharov, A. M., E-mail: basharov@gmail.com [National Research Centre ' Kurchatov Institute,' (Russian Federation)
2012-09-15
It is shown that the effective Hamiltonian representation, as it is formulated in author's papers, serves as a basis for distinguishing, in a broadband environment of an open quantum system, independent noise sources that determine, in terms of the stationary quantum Wiener and Poisson processes in the Markov approximation, the effective Hamiltonian and the equation for the evolution operator of the open system and its environment. General stochastic differential equations of generalized Langevin (non-Wiener) type for the evolution operator and the kinetic equation for the density matrix of an open system are obtained, which allow one to analyze the dynamics of a wide class of localized open systems in the Markov approximation. The main distinctive features of the dynamics of open quantum systems described in this way are the stabilization of excited states with respect to collective processes and an additional frequency shift of the spectrum of the open system. As an illustration of the general approach developed, the photon dynamics in a single-mode cavity without losses on the mirrors is considered, which contains identical intracavity atoms coupled to the external vacuum electromagnetic field. For some atomic densities, the photons of the cavity mode are 'locked' inside the cavity, thus exhibiting a new phenomenon of radiation trapping and non-Wiener dynamics.
International Nuclear Information System (INIS)
Basharov, A. M.
2012-01-01
It is shown that the effective Hamiltonian representation, as it is formulated in author’s papers, serves as a basis for distinguishing, in a broadband environment of an open quantum system, independent noise sources that determine, in terms of the stationary quantum Wiener and Poisson processes in the Markov approximation, the effective Hamiltonian and the equation for the evolution operator of the open system and its environment. General stochastic differential equations of generalized Langevin (non-Wiener) type for the evolution operator and the kinetic equation for the density matrix of an open system are obtained, which allow one to analyze the dynamics of a wide class of localized open systems in the Markov approximation. The main distinctive features of the dynamics of open quantum systems described in this way are the stabilization of excited states with respect to collective processes and an additional frequency shift of the spectrum of the open system. As an illustration of the general approach developed, the photon dynamics in a single-mode cavity without losses on the mirrors is considered, which contains identical intracavity atoms coupled to the external vacuum electromagnetic field. For some atomic densities, the photons of the cavity mode are “locked” inside the cavity, thus exhibiting a new phenomenon of radiation trapping and non-Wiener dynamics.
Belkina, T. A.; Konyukhova, N. B.; Kurochkin, S. V.
2012-10-01
A singular boundary value problem for a second-order linear integrodifferential equation with Volterra and non-Volterra integral operators is formulated and analyzed. The equation is defined on ℝ+, has a weak singularity at zero and a strong singularity at infinity, and depends on several positive parameters. Under natural constraints on the coefficients of the equation, existence and uniqueness theorems for this problem with given limit boundary conditions at singular points are proved, asymptotic representations of the solution are given, and an algorithm for its numerical determination is described. Numerical computations are performed and their interpretation is given. The problem arises in the study of the survival probability of an insurance company over infinite time (as a function of its initial surplus) in a dynamic insurance model that is a modification of the classical Cramer-Lundberg model with a stochastic process rate of premium under a certain investment strategy in the financial market. A comparative analysis of the results with those produced by the model with deterministic premiums is given.
DEFF Research Database (Denmark)
Lodi, C.; Bacher, Peder; Cipriano, J.
2012-01-01
reduce the ventilation thermal losses of the building by pre-heating the fresh air. Furthermore, by decreasing PV module temperature, the ventilation air heat extraction can simultaneously increase electrical and thermal energy production of the building. A correct prediction of the PV module temperature...... and heat transfer coefficients is fundamental in order to improve the thermo-electrical production.The considered grey-box models are composed of a set of continuous time stochastic differential equations, holding the physical description of the system, combined with a set of discrete time measurement......This paper deals with grey-box modelling of the energy transfer of a double skin Building Integrated Photovoltaic (BIPV) system. Grey-box models are based on a combination of prior physical knowledge and statistics, which enable identification of the unknown parameters in the system and accurate...
Faizullah, Faiz
2016-01-01
The aim of the current paper is to present the path-wise and moment estimates for solutions to stochastic functional differential equations with non-linear growth condition in the framework of G-expectation and G-Brownian motion. Under the nonlinear growth condition, the pth moment estimates for solutions to SFDEs driven by G-Brownian motion are proved. The properties of G-expectations, Hölder's inequality, Bihari's inequality, Gronwall's inequality and Burkholder-Davis-Gundy inequalities are used to develop the above mentioned theory. In addition, the path-wise asymptotic estimates and continuity of pth moment for the solutions to SFDEs in the G-framework, with non-linear growth condition are shown.
Kaulakys, B.; Alaburda, M.; Ruseckas, J.
2016-05-01
A well-known fact in the financial markets is the so-called ‘inverse cubic law’ of the cumulative distributions of the long-range memory fluctuations of market indicators such as a number of events of trades, trading volume and the logarithmic price change. We propose the nonlinear stochastic differential equation (SDE) giving both the power-law behavior of the power spectral density and the long-range dependent inverse cubic law of the cumulative distribution. This is achieved using the suggestion that when the market evolves from calm to violent behavior there is a decrease of the delay time of multiplicative feedback of the system in comparison to the driving noise correlation time. This results in a transition from the Itô to the Stratonovich sense of the SDE and yields a long-range memory process.
A stochastic multiscale method for the elastodynamic wave equation arising from fiber composites
Babuška, Ivo; Motamed, Mohammad; Tempone, Raul
2014-01-01
We present a stochastic multilevel global–local algorithm for computing elastic waves propagating in fiber-reinforced composite materials. Here, the materials properties and the size and location of fibers may be random. The method aims at approximating statistical moments of some given quantities of interest, such as stresses, in regions of relatively small size, e.g. hot spots or zones that are deemed vulnerable to failure. For a fiber-reinforced cross-plied laminate, we introduce three problems (macro, meso, micro) corresponding to the three natural scales, namely the sizes of laminate, ply, and fiber. The algorithm uses the homogenized global solution to construct a good local approximation that captures the microscale features of the real solution. We perform numerical experiments to show the applicability and efficiency of the method.
Passive tracer in a flow corresponding to two-dimensional stochastic Navier–Stokes equations
International Nuclear Information System (INIS)
Komorowski, Tomasz; Peszat, Szymon; Szarek, Tomasz
2013-01-01
In this paper we prove the law of large numbers and central limit theorem for trajectories of a particle carried by a two-dimensional Eulerian velocity field. The field is given by a solution of a stochastic Navier–Stokes system with non-degenerate noise. The spectral gap property, with respect to the Wasserstein metric, for such a system was shown in Hairer and Mattingly (2008 Ann. Probab. 36 2050–91). In this paper we show that a similar property holds for the environment process corresponding to the Lagrangian observations of the velocity. Consequently we conclude the law of large numbers and the central limit theorem for the tracer. The proof of the central limit theorem relies on the martingale approximation of the trajectory process. (paper)
A stochastic multiscale method for the elastodynamic wave equation arising from fiber composites
Babuška, Ivo
2014-03-21
We present a stochastic multilevel global–local algorithm for computing elastic waves propagating in fiber-reinforced composite materials. Here, the materials properties and the size and location of fibers may be random. The method aims at approximating statistical moments of some given quantities of interest, such as stresses, in regions of relatively small size, e.g. hot spots or zones that are deemed vulnerable to failure. For a fiber-reinforced cross-plied laminate, we introduce three problems (macro, meso, micro) corresponding to the three natural scales, namely the sizes of laminate, ply, and fiber. The algorithm uses the homogenized global solution to construct a good local approximation that captures the microscale features of the real solution. We perform numerical experiments to show the applicability and efficiency of the method.
A Stochastic Multiscale Method for the Elastic Wave Equations Arising from Fiber Composites
Babuska, Ivo
2016-01-06
We present a stochastic multilevel global-local algorithm [1] for computing elastic waves propagating in fiber-reinforced polymer composites, where the material properties and the size and distribution of fibers in the polymer matrix may be random. The method aims at approximating statistical moments of some given quantities of interest, such as stresses, in regions of relatively small size, e.g. hot spots or zones that are deemed vulnerable to failure. For a fiber-reinforced cross-plied laminate, we introduce three problems: 1) macro; 2) meso; and 3) micro problems, corresponding to the three natural length scales: 1) the sizes of plate; 2) the tickles of plies; and 3) and the diameter of fibers. The algorithm uses a homogenized global solution to construct a local approximation that captures the microscale features of the problem. We perform numerical experiments to show the applicability and efficiency of the method.
International Nuclear Information System (INIS)
Shiino, Masatoshi; Yamana, Michiko
2004-01-01
We study the statistical mechanical aspects of stochastic analog neural network models for associative memory with correlation type learning. We take three approaches to derive the set of the order parameter equations for investigating statistical properties of retrieval states: the self-consistent signal-to-noise analysis (SCSNA), the Thouless-Anderson-Palmer (TAP) equation, and the replica symmetric calculation. On the basis of the cavity method the SCSNA can be generalized to deal with stochastic networks. We establish the close connection between the TAP equation and the SCSNA to elucidate the relationship between the Onsager reaction term of the TAP equation and the output proportional term of the SCSNA that appear in the expressions for the local fields
Stochastic integration of the Bethe-Salpeter equation for two bound fermions
International Nuclear Information System (INIS)
Salomon, M.
1988-09-01
A non-perturbative method using a Monte Carlo algorithm is used to integrate the Bethe-Salpeter equation in momentum space. Solutions for two scalars and two fermions with an arbitrary coupling constant are calculated for bound states in the ladder approximation. The results are compared with other numerical methods. (Author) (13 refs., 2 figs.)
Rare event simulation for processes generated via stochastic fixed point equations
DEFF Research Database (Denmark)
Collamore, Jeffrey F.; Diao, Guoqing; Vidyashankar, Anand N.
2014-01-01
In a number of applications, particularly in financial and actuarial mathematics, it is of interest to characterize the tail distribution of a random variable V satisfying the distributional equation V=_D f(V), for some random function f. This paper is concerned with computational methods for eva...
Directory of Open Access Journals (Sweden)
Maxim Ioan
2009-05-01
Full Text Available In our paper we build a reccurence from generalized Garman equation and discretization of 3-dimensional domain. From reccurence we build an algorithm for computing values of an option based on time, momentan volatility of support and value of support on a
Random recurrence equations and ruin in a Markov-dependent stochastic economic environment
DEFF Research Database (Denmark)
Collamore, Jeffrey F.
2009-01-01
series models. Our results build upon work of Goldie, who has developed tail asymptotics applicable for independent sequences of random variables subject to a random recurrence equation. In contrast, we adopt a general approach based on the theory of Harris recurrent Markov chains and the associated...
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
In this paper, the stochastic flow of mappings generated by a Feller convolution semigroup on a compact metric space is studied. This kind of flow is the generalization of superprocesses of stochastic flows and stochastic diffeomorphism induced by the strong solutions of stochastic differential equations.
Energy Technology Data Exchange (ETDEWEB)
Decker, J.; Peysson, Y
2004-12-01
A new original code for solving the 3-D relativistic and bounce-averaged electron drift kinetic equation is presented. It designed for the current drive problem in tokamak with an arbitrary magnetic equilibrium. This tool allows self-consistent calculations of the bootstrap current in presence of other external current sources. RF current drive for arbitrary type of waves may be used. Several moments of the electron distribution function are determined, like the exact and effective fractions of trapped electrons, the plasma current, absorbed RF power, runaway and magnetic ripple loss rates and non-thermal Bremsstrahlung. Advanced numerical techniques have been used to make it the first fully implicit (reverse time) 3-D solver, particularly well designed for implementation in a chain of code for realistic current drive calculations in high {beta}{sub p} plasmas. All the details of the physics background and the numerical scheme are presented, as well a some examples to illustrate main code capabilities. Several important numerical points are addressed concerning code stability and potential numerical and physical limitations. (authors)
International Nuclear Information System (INIS)
Decker, J.; Peysson, Y.
2004-12-01
A new original code for solving the 3-D relativistic and bounce-averaged electron drift kinetic equation is presented. It designed for the current drive problem in tokamak with an arbitrary magnetic equilibrium. This tool allows self-consistent calculations of the bootstrap current in presence of other external current sources. RF current drive for arbitrary type of waves may be used. Several moments of the electron distribution function are determined, like the exact and effective fractions of trapped electrons, the plasma current, absorbed RF power, runaway and magnetic ripple loss rates and non-thermal Bremsstrahlung. Advanced numerical techniques have been used to make it the first fully implicit (reverse time) 3-D solver, particularly well designed for implementation in a chain of code for realistic current drive calculations in high β p plasmas. All the details of the physics background and the numerical scheme are presented, as well a some examples to illustrate main code capabilities. Several important numerical points are addressed concerning code stability and potential numerical and physical limitations. (authors)
Cuenod, Charles-André; Favetto, Benjamin; Genon-Catalot, Valentine; Rozenholc, Yves; Samson, Adeline
2011-09-01
Dynamic Contrast Enhanced imaging (DCE-imaging) following a contrast agent bolus allows the extraction of information on tissue micro-vascularization. The dynamic signals obtained from DCE-imaging are modeled by pharmacokinetic compartmental models which integrate the Arterial Input Function. These models use ordinary differential equations (ODEs) to describe the exchanges between the arterial and capillary plasma and the extravascular-extracellular space. Their least squares fitting takes into account measurement noises but fails to deal with unpredictable fluctuations due to external/internal sources of variations (patients' anxiety, time-varying parameters, measurement errors in the input function, etc.). Adding Brownian components to the ODEs leads to stochastic differential equations (SDEs). In DCE-imaging, SDEs are discretely observed with an additional measurement noise. We propose to estimate the parameters of these noisy SDEs by maximum likelihood, using the Kalman filter. In DCE-imaging, the contrast agent injected in vein arrives in plasma with an unknown time delay. The delay parameter induces a change-point in the drift of the SDE and ODE models, which is estimated also. Estimations based on the SDE and ODE pharmacokinetic models are compared to real DCE-MRI data. They show that the use of SDE provides robustness in the estimation results. A simulation study confirms these results. Copyright © 2011 Elsevier Inc. All rights reserved.
Overgaard, Rune Viig; Holford, Nick; Rytved, Klaus A; Madsen, Henrik
2007-02-01
To describe the pharmacodynamic effects of recombinant human interleukin-21 (IL-21) on core body temperature in cynomolgus monkeys using basic mechanisms of heat regulation. A major effort was devoted to compare the use of ordinary differential equations (ODEs) with stochastic differential equations (SDEs) in pharmacokinetic pharmacodynamic (PKPD) modelling. A temperature model was formulated including circadian rhythm, metabolism, heat loss, and a thermoregulatory set-point. This model was formulated as a mixed-effects model based on SDEs using NONMEM. The effects of IL-21 were on the set-point and the circadian rhythm of metabolism. The model was able to describe a complex set of IL-21 induced phenomena, including 1) disappearance of the circadian rhythm, 2) no effect after first dose, and 3) high variability after second dose. SDEs provided a more realistic description with improved simulation properties, and further changed the model into one that could not be falsified by the autocorrelation function. The IL-21 induced effects on thermoregulation in cynomolgus monkeys are explained by a biologically plausible model. The quality of the model was improved by the use of SDEs.
Flux-probability distributions from the master equation for radiation transport in stochastic media
International Nuclear Information System (INIS)
Franke, Brian C.; Prinja, Anil K.
2011-01-01
We present numerical investigations into the accuracy of approximations in the master equation for radiation transport in discrete binary random media. Our solutions of the master equation yield probability distributions of particle flux at each element of phase space. We employ the Levermore-Pomraning interface closure and evaluate the effectiveness of closures for the joint conditional flux distribution for estimating scattering integrals. We propose a parameterized model for this joint-pdf closure, varying between correlation neglect and a full-correlation model. The closure is evaluated for a variety of parameter settings. Comparisons are made with benchmark results obtained through suites of fixed-geometry realizations of random media in rod problems. All calculations are performed using Monte Carlo techniques. Accuracy of the approximations in the master equation is assessed by examining the probability distributions for reflection and transmission and by evaluating the moments of the pdfs. The results suggest the correlation-neglect setting in our model performs best and shows improved agreement in the atomic-mix limit. (author)
Dolgov, Sergey; Khoromskij, Boris N.; Litvinenko, Alexander; Matthies, Hermann G.
2015-01-01
We apply the tensor train (TT) decomposition to construct the tensor product polynomial chaos expansion (PCE) of a random field, to solve the stochastic elliptic diffusion PDE with the stochastic Galerkin discretization, and to compute some
SITU, Rong
2005-01-01
Derivation of Ito's formulas, Girsanov's theorems and martingale representation theorem for stochastic DEs with jumpsApplications to population controlReflecting stochastic DE techniqueApplications to the stock market. (Backward stochastic DE approach)Derivation of Black-Scholes formula for market with and without jumpsNon-linear filtering problems with jumps.
International Nuclear Information System (INIS)
Chen, Feng; Han, Yuecai
2013-01-01
The existence of time-periodic stochastic motions of an incompressible fluid is obtained. Here the fluid is subject to a time-periodic body force and an additional time-periodic stochastic force that is produced by a rigid body moves periodically stochastically with the same period in the fluid
Energy Technology Data Exchange (ETDEWEB)
Chen, Feng, E-mail: chenfengmath@163.com, E-mail: hanyc@jlu.edu.cn; Han, Yuecai, E-mail: chenfengmath@163.com, E-mail: hanyc@jlu.edu.cn [School of Mathematics, Jilin University, Changchun 130012 (China)
2013-12-15
The existence of time-periodic stochastic motions of an incompressible fluid is obtained. Here the fluid is subject to a time-periodic body force and an additional time-periodic stochastic force that is produced by a rigid body moves periodically stochastically with the same period in the fluid.
Kittell, D. E.; Yarrington, C. D.; Lechman, J. B.; Baer, M. R.
2018-05-01
A new paradigm is introduced for modeling reactive shock waves in heterogeneous solids at the continuum level. Inspired by the probability density function methods from turbulent reactive flows, it is hypothesized that the unreacted material microstructures lead to a distribution of heat release rates from chemical reaction. Fluctuations in heat release, rather than velocity, are coupled to the reactive Euler equations which are then solved via the Riemann problem. A numerically efficient, one-dimensional hydrocode is used to demonstrate this new approach, and simulation results of a representative impact calculation (inert flyer into explosive target) are discussed.
International Nuclear Information System (INIS)
Wehner, M.F.
1983-01-01
A path-integral solution is derived for processes described by nonlinear Fokker-Plank equations together with externally imposed boundary conditions. This path-integral solution is written in the form of a path sum for small time steps and contains, in addition to the conventional volume integral, a surface integral which incorporates the boundary conditions. A previously developed numerical method, based on a histogram representation of the probability distribution, is extended to a trapezoidal representation. This improved numerical approach is combined with the present path-integral formalism for restricted processes and is show t give accurate results. 35 refs., 5 figs
Chakrabarti, Anindya S.
2016-01-01
We present a model of technological evolution due to interaction between multiple countries and the resultant effects on the corresponding macro variables. The world consists of a set of economies where some countries are leaders and some are followers in the technology ladder. All of them potentially gain from technological breakthroughs. Applying Lotka-Volterra (LV) equations to model evolution of the technology frontier, we show that the way technology diffuses creates repercussions in the partner economies. This process captures the spill-over effects on major macro variables seen in the current highly globalized world due to trickle-down effects of technology.
International Nuclear Information System (INIS)
Appleby, J A D; Wu, H
2008-01-01
In this paper we consider functional differential equations subjected to either instantaneous state-dependent noise, or to a white noise perturbation. The drift of the equations depend linearly on the current value and on the maximum of the solution. The functional term always provides positive feedback, while the instantaneous term can be mean-reverting or can exhibit positive feedback. We show in the white noise case that if the instantaneous term is mean reverting and dominates the history term, then solutions are recurrent, and upper bounds on the a.s. growth rate of the partial maxima of the solution can be found. When the instantaneous term is weaker, or is of positive feedback type, we determine necessary and sufficient conditions on the diffusion coefficient which ensure the exact exponential growth of solutions. An application of these results to an inefficient financial market populated by reference traders and speculators is given, in which the difference between the current instantaneous returns and maximum of the returns over the last few time units is used to determine trading strategies.
Probability and Cumulative Density Function Methods for the Stochastic Advection-Reaction Equation
Energy Technology Data Exchange (ETDEWEB)
Barajas-Solano, David A.; Tartakovsky, Alexandre M.
2018-01-01
We present a cumulative density function (CDF) method for the probabilistic analysis of $d$-dimensional advection-dominated reactive transport in heterogeneous media. We employ a probabilistic approach in which epistemic uncertainty on the spatial heterogeneity of Darcy-scale transport coefficients is modeled in terms of random fields with given correlation structures. Our proposed CDF method employs a modified Large-Eddy-Diffusivity (LED) approach to close and localize the nonlocal equations governing the one-point PDF and CDF of the concentration field, resulting in a $(d + 1)$ dimensional PDE. Compared to the classsical LED localization, the proposed modified LED localization explicitly accounts for the mean-field advective dynamics over the phase space of the PDF and CDF. To illustrate the accuracy of the proposed closure, we apply our CDF method to one-dimensional single-species reactive transport with uncertain, heterogeneous advection velocities and reaction rates modeled as random fields.
Directory of Open Access Journals (Sweden)
Lin Hu
2011-01-01
Full Text Available A class of drift-implicit one-step schemes are proposed for the neutral stochastic delay differential equations (NSDDEs driven by Poisson processes. A general framework for mean-square convergence of the methods is provided. It is shown that under certain conditions global error estimates for a method can be inferred from estimates on its local error. The applicability of the mean-square convergence theory is illustrated by the stochastic θ-methods and the balanced implicit methods. It is derived from Theorem 3.1 that the order of the mean-square convergence of both of them for NSDDEs with jumps is 1/2. Numerical experiments illustrate the theoretical results. It is worth noting that the results of mean-square convergence of the stochastic θ-methods and the balanced implicit methods are also new.
International Nuclear Information System (INIS)
Neate, A D; Truman, A
2005-01-01
The inviscid limit of the Burgers equation, with body forces white noise in time, is discussed in terms of the level surfaces of the minimizing Hamilton-Jacobi function and the classical mechanical caustic and their algebraic pre-images under the classical mechanical flow map. The problem is analysed in terms of a reduced (one-dimensional) action function using a circle of ideas due to Arnol'd, Cayley and Klein. We characterize those parts of the caustic which are singular, and give an explicit expression for the cusp density on caustics and level surfaces. By considering the double points of level surfaces we find an explicit formula for the Maxwell set in the two-dimensional polynomial case, and we extend this to higher dimensions using a double discriminant of the reduced action, solving a long-standing problem for Hamiltonian dynamical systems. When the pre-level surface touches the pre-caustic, the geometry (number of cusps) on the level surface changes infinitely rapidly causing 'real turbulence'. Using an idea of Klein, it is shown that the geometry (number of swallowtails) on the caustic also changes infinitely rapidly when the real part of the pre-caustic touches its complex counterpart, causing 'complex turbulence'. These are both inherently stochastic in nature, and we determine their intermittence in terms of the recurrent behaviour of two processes
Mortensen, Stig B; Klim, Søren; Dammann, Bernd; Kristensen, Niels R; Madsen, Henrik; Overgaard, Rune V
2007-10-01
The non-linear mixed-effects model based on stochastic differential equations (SDEs) provides an attractive residual error model, that is able to handle serially correlated residuals typically arising from structural mis-specification of the true underlying model. The use of SDEs also opens up for new tools for model development and easily allows for tracking of unknown inputs and parameters over time. An algorithm for maximum likelihood estimation of the model has earlier been proposed, and the present paper presents the first general implementation of this algorithm. The implementation is done in Matlab and also demonstrates the use of parallel computing for improved estimation times. The use of the implementation is illustrated by two examples of application which focus on the ability of the model to estimate unknown inputs facilitated by the extension to SDEs. The first application is a deconvolution-type estimation of the insulin secretion rate based on a linear two-compartment model for C-peptide measurements. In the second application the model is extended to also give an estimate of the time varying liver extraction based on both C-peptide and insulin measurements.
Syahidatul Ayuni Mazlan, Mazma; Rosli, Norhayati; Jauhari Arief Ichwan, Solachuddin; Suhaity Azmi, Nina
2017-09-01
A stochastic model is introduced to describe the growth of cancer affected by anti-cancer therapeutics of Chondroitin Sulfate (CS). The parameters values of the stochastic model are estimated via maximum likelihood function. The numerical method of Euler-Maruyama will be employed to solve the model numerically. The efficiency of the stochastic model is measured by comparing the simulated result with the experimental data.
International Nuclear Information System (INIS)
Montes T, J.L.; Cortes C, C.C.
1992-08-01
The objective of the report is to determine manually or by means of a calculation sheet, the coefficients α and β of the average enrichment equation as function of the fuel burnt (B) using the Lineal Reactivity Pattern, with information generated by the RECORD code of the FMS package. (Author)
Stochastic stability of four-wheel-steering system
International Nuclear Information System (INIS)
Huang Dongwei; Wang Hongli; Zhu Zhiwen; Feng Zhang
2007-01-01
A four-wheel-steering system subjected to white noise excitations was reduced to a two-degree-of-freedom quasi-non-integrable-Hamiltonian system. Subsequently we obtained an one-dimensional Ito stochastic differential equation for the averaged Hamiltonian of the system by using the stochastic averaging method for quasi-non-integrable-Hamiltonian systems. Thus, the stochastic stability of four-wheel-steering system was analyzed by analyzing the sample behaviors of the averaged Hamiltonian at the boundary H = 0 and calculating its Lyapunov exponent. An example given at the end demonstrated that the conclusion obtained is of considerable significance