The numerical analysis of eigenvalue problem solutions in the multigroup neutron diffusion theory
Energy Technology Data Exchange (ETDEWEB)
Woznicki, Z.I. [Institute of Atomic Energy, Otwock-Swierk (Poland)
1994-12-31
The main goal of this paper is to present a general iteration strategy for solving the discrete form of multidimensional neutron diffusion equations equivalent mathematically to an eigenvalue problem. Usually a solution method is based on different levels of iterations. The presented matrix formalism allows us to visualize explicitly how the used matrix splitting influences the matrix structure in an eigenvalue problem to be solved as well as the interdependence between inner and outer iteration within global iterations. Particular interactive strategies are illustrated by numerical results obtained for several reactor problems. (author). 21 refs, 32 figs, 15 tabs.
The numerical analysis of eigenvalue problem solutions in multigroup neutron diffusion theory
Energy Technology Data Exchange (ETDEWEB)
Woznicki, Z.I. [Institute of Atomic Energy, Otwock-Swierk (Poland)
1995-12-31
The main goal of this paper is to present a general iteration strategy for solving the discrete form of multidimensional neutron diffusion equations equivalent mathematically to an eigenvalue problem. Usually a solution method is based on different levels of iterations. The presented matrix formalism allows us to visualize explicitly how the used matrix splitting influences the matrix structure in an eigenvalue problem to be solved as well as the interdependence between inner and outer iterations within global iterations. Particular iterative strategies are illustrated by numerical results obtained for several reactor problems. (author). 21 refs, 35 figs, 16 tabs.
Quadratic eigenvalue problems.
Energy Technology Data Exchange (ETDEWEB)
Walsh, Timothy Francis; Day, David Minot
2007-04-01
In this report we will describe some nonlinear eigenvalue problems that arise in the areas of solid mechanics, acoustics, and coupled structural acoustics. We will focus mostly on quadratic eigenvalue problems, which are a special case of nonlinear eigenvalue problems. Algorithms for solving the quadratic eigenvalue problem will be presented, along with some example calculations.
1987-06-01
and f. Let us consider the problem of finding the minimal constant C. We are thus interested in 2~ IVA u dx (1.24) C = sup . u2u (0 (F =0 (u dx"<u(O...fournir des bornes superieures ou inferieures, C.R. Acad. Sci., Paris 235, 995-997. .V Prodi, G. (1962]: Theoremi di tipo locale per il sistema de Navier
Convergence analysis of two-node CMFD method for two-group neutron diffusion eigenvalue problem
Jeong, Yongjin; Park, Jinsu; Lee, Hyun Chul; Lee, Deokjung
2015-12-01
In this paper, the nonlinear coarse-mesh finite difference method with two-node local problem (CMFD2N) is proven to be unconditionally stable for neutron diffusion eigenvalue problems. The explicit current correction factor (CCF) is derived based on the two-node analytic nodal method (ANM2N), and a Fourier stability analysis is applied to the linearized algorithm. It is shown that the analytic convergence rate obtained by the Fourier analysis compares very well with the numerically measured convergence rate. It is also shown that the theoretical convergence rate is only governed by the converged second harmonic buckling and the mesh size. It is also noted that the convergence rate of the CCF of the CMFD2N algorithm is dependent on the mesh size, but not on the total problem size. This is contrary to expectation for eigenvalue problem. The novel points of this paper are the analytical derivation of the convergence rate of the CMFD2N algorithm for eigenvalue problem, and the convergence analysis based on the analytic derivations.
Convergence analysis of two-node CMFD method for two-group neutron diffusion eigenvalue problem
Energy Technology Data Exchange (ETDEWEB)
Jeong, Yongjin; Park, Jinsu [Ulsan National Institute of Science and Technology, UNIST-gil 50, Eonyang-eup, Ulju-gun, Ulsan, 689-798 (Korea, Republic of); Lee, Hyun Chul [Korea Atomic Energy Research Institute, 111 Daedeok-daero 989 beon-gil, Yuseong-gu, Daejeon 305-353 (Korea, Republic of); Lee, Deokjung, E-mail: deokjung@unist.ac.kr [Ulsan National Institute of Science and Technology, UNIST-gil 50, Eonyang-eup, Ulju-gun, Ulsan, 689-798 (Korea, Republic of)
2015-12-01
In this paper, the nonlinear coarse-mesh finite difference method with two-node local problem (CMFD2N) is proven to be unconditionally stable for neutron diffusion eigenvalue problems. The explicit current correction factor (CCF) is derived based on the two-node analytic nodal method (ANM2N), and a Fourier stability analysis is applied to the linearized algorithm. It is shown that the analytic convergence rate obtained by the Fourier analysis compares very well with the numerically measured convergence rate. It is also shown that the theoretical convergence rate is only governed by the converged second harmonic buckling and the mesh size. It is also noted that the convergence rate of the CCF of the CMFD2N algorithm is dependent on the mesh size, but not on the total problem size. This is contrary to expectation for eigenvalue problem. The novel points of this paper are the analytical derivation of the convergence rate of the CMFD2N algorithm for eigenvalue problem, and the convergence analysis based on the analytic derivations.
Parallel Symmetric Eigenvalue Problem Solvers
2015-05-01
Plemmons G. Golub and A. Sameh. High-speed computing : scientific appli- cations and algorithm design. University of Illinois Press, Champaign, Illinois , 1988...16. SECURITY CLASSIFICATION OF: Sparse symmetric eigenvalue problems arise in many computational science and engineering applications such as...Eigenvalue Problem Solvers Report Title Sparse symmetric eigenvalue problems arise in many computational science and engineering applications such as
Subspace Methods for Eigenvalue Problems
Hochstenbach, Michiel Erik
2003-01-01
This thesis treats a number of aspects of subspace methods for various eigenvalue problems. Vibrations and their corresponding eigenvalues (or frequencies) arise in science, engineering, and daily life. Matrix eigenvalue problems come from a large number of areas, such as chemistry, mechanics, dyn
Random eigenvalue problems revisited
Indian Academy of Sciences (India)
S Adhikari
2006-08-01
The description of real-life engineering structural systems is associated with some amount of uncertainty in specifying material properties, geometric parameters, boundary conditions and applied loads. In the context of structural dynamics it is necessary to consider random eigenvalue problems in order to account for these uncertainties. Within the engineering literature, current methods to deal with such problems are dominated by approximate perturbation methods. Some exact methods to obtain joint distribution of the natural frequencies are reviewed and their applicability in the context of real-life engineering problems is discussed. A new approach based on an asymptotic approximation of multi-dimensional integrals is proposed. A closed-form expression for general order joint moments of arbitrary numbers of natural frequencies of linear stochastic systems is derived. The proposed method does not employ the ‘small randomness’ assumption usually used in perturbation based methods. Joint distributions of the natural frequencies are investigated using numerical examples and the results are compared with Monte Carlo simulation.
On the buckling eigenvalue problem
Energy Technology Data Exchange (ETDEWEB)
Antunes, Pedro R S, E-mail: pant@cii.fc.ul.pt [Departamento de Matematica, Universidade Lusofona de Humanidades e Tecnologias, Av. do Campo Grande, 376, 1749-024 Lisboa (Portugal); Group of Mathematical Physics of the University of Lisbon, Complexo Interdisciplinar, Av. Professor Gama Pinto 2, P-1649-003 Lisboa (Portugal)
2011-05-27
We prove a density result which allows us to justify the application of the method of fundamental solutions to solve the buckling eigenvalue problem of a plate. We address an example of an analytic convex domain for which the first eigenfunction does change the sign and present a large-scale numerical study with polygons providing numerical evidence to some new conjectures.
Highly indefinite multigrid for eigenvalue problems
Energy Technology Data Exchange (ETDEWEB)
Borges, L.; Oliveira, S.
1996-12-31
Eigenvalue problems are extremely important in understanding dynamic processes such as vibrations and control systems. Large scale eigenvalue problems can be very difficult to solve, especially if a large number of eigenvalues and the corresponding eigenvectors need to be computed. For solving this problem a multigrid preconditioned algorithm is presented in {open_quotes}The Davidson Algorithm, preconditioning and misconvergence{close_quotes}. Another approach for solving eigenvalue problems is by developing efficient solutions for highly indefinite problems. In this paper we concentrate on the use of new highly indefinite multigrid algorithms for the eigenvalue problem.
THE EIGENVALUE PROBLEM FOR THE LAPLACIAN EQUATIONS
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
This article studies the Dirichlet eigenvalue problem for the Laplacian equations △u = -λu, x ∈Ω, u = 0, x ∈ (δ)Ω, where Ω (∩) Rn is a smooth bounded convex domain. By using the method of appropriate barrier function combined with the maximum principle, authors obtain a sharp lower bound of the difference of the first two eigenvalues for the Dirichlet eigenvalue problem. This study improves the result of S.T.Yau et al.
Iterative approach for the eigenvalue problems
Indian Academy of Sciences (India)
J Datta; P K Bera
2011-01-01
An approximation method based on the iterative technique is developed within the framework of linear delta expansion (LDE) technique for the eigenvalues and eigenfunctions of the one-dimensional and three-dimensional realistic physical problems. This technique allows us to obtain the coefficient in the perturbation series for the eigenfunctions and the eigenvalues directly by knowing the eigenfunctions and the eigenvalues of the unperturbed problems in quantum mechanics. Examples are presented to support this. Hence, the LDE technique can be used for non-perturbative as well as perturbative systems to find approximate solutions of eigenvalue problems.
Multiparameter eigenvalue problems Sturm-Liouville theory
Atkinson, FV
2010-01-01
One of the masters in the differential equations community, the late F.V. Atkinson contributed seminal research to multiparameter spectral theory and Sturm-Liouville theory. His ideas and techniques have long inspired researchers and continue to stimulate discussion. With the help of co-author Angelo B. Mingarelli, Multiparameter Eigenvalue Problems: Sturm-Liouville Theory reflects much of Dr. Atkinson's final work.After covering standard multiparameter problems, the book investigates the conditions for eigenvalues to be real and form a discrete set. It gives results on the determinants of fun
Multiparameter eigenvalue problems and expansion theorems
Volkmer, Hans
1988-01-01
This book provides a self-contained treatment of two of the main problems of multiparameter spectral theory: the existence of eigenvalues and the expansion in series of eigenfunctions. The results are first obtained in abstract Hilbert spaces and then applied to integral operators and differential operators. Special attention is paid to various definiteness conditions which can be imposed on multiparameter eigenvalue problems. The reader is not assumed to be familiar with multiparameter spectral theory but should have some knowledge of functional analysis, in particular of Brower's degree of maps.
Pattern selection as a nonlinear eigenvalue problem
Büchel, P
1996-01-01
A unique pattern selection in the absolutely unstable regime of driven, nonlinear, open-flow systems is reviewed. It has recently been found in numerical simulations of propagating vortex structures occuring in Taylor-Couette and Rayleigh-Benard systems subject to an externally imposed through-flow. Unlike the stationary patterns in systems without through-flow the spatiotemporal structures of propagating vortices are independent of parameter history, initial conditions, and system length. They do, however, depend on the boundary conditions in addition to the driving rate and the through-flow rate. Our analysis of the Ginzburg-Landau amplitude equation elucidates how the pattern selection can be described by a nonlinear eigenvalue problem with the frequency being the eigenvalue. Approaching the border between absolute and convective instability the eigenvalue problem becomes effectively linear and the selection mechanism approaches that of linear front propagation. PACS: 47.54.+r,47.20.Ky,47.32.-y,47.20.Ft
A note on quasilinear elliptic eigenvalue problems
Directory of Open Access Journals (Sweden)
Gianni Arioli
1999-11-01
Full Text Available We study an eigenvalue problem by a non-smooth critical point theory. Under general assumptions, we prove the existence of at least one solution as a minimum of a constrained energy functional. We apply some results on critical point theory with symmetry to provide a multiplicity result.
The eigenvalue problem in phase space.
Cohen, Leon
2017-07-27
We formulate the standard quantum mechanical eigenvalue problem in quantum phase space. The equation obtained involves the c-function that corresponds to the quantum operator. We use the Wigner distribution for the phase space function. We argue that the phase space eigenvalue equation obtained has, in addition to the proper solutions, improper solutions. That is, solutions for which no wave function exists which could generate the distribution. We discuss the conditions for ascertaining whether a position momentum function is a proper phase space distribution. We call these conditions psi-representability conditions, and show that if these conditions are imposed, one extracts the correct phase space eigenfunctions. We also derive the phase space eigenvalue equation for arbitrary phase space distributions functions. © 2017 Wiley Periodicals, Inc. © 2017 Wiley Periodicals, Inc.
Inverse Eigenvalue Problem in Structural Dynamics Design
Institute of Scientific and Technical Information of China (English)
Huiqing Xie; Hua Dai
2006-01-01
A kind of inverse eigenvalue problem in structural dynamics design is considered. The problem is formulated as an optimization problem. The properties of this problem are analyzed, and the existence of the optimum solution is proved. The directional derivative of the objective function is obtained and a necessary condition for a point to be a local minimum point is given. Then a numerical algorithm for solving the problem is presented and a plane-truss problem is discussed to show the applications of the theories and the algorithm.
Generalized Inverse Eigenvalue Problem for Centrohermitian Matrices
Institute of Scientific and Technical Information of China (English)
刘仲云; 谭艳祥; 田兆录
2004-01-01
In this paper we first consider the existence and the general form of solution to the following generalized inverse eigenvalue problem(GIEP) : given a set of n-dimension complex vectors { xj }jm = 1 and a set of complex numbers { λj} jm = 1, find two n × n centrohermitian matrices A, B such that { xj }jm = 1 and { λj }jm= 1 are the generalized eigenvectors and generalized eigenvalues of Ax = λBx, respectively. We then discuss the optimal approximation problem for the GIEP. More concretely, given two arbitrary matrices, A-, B- ∈Cn×n , we find two matrices A* and B* such that the matrix (A* ,B* ) is closest to (A- ,B-) in the Frobenius norm, where the matrix (A*, B* ) is the solution to the GIEP. We show that the expression of the solution of the optimal approximation is unique and derive the expression for it.
Sparse Regression as a Sparse Eigenvalue Problem
Moghaddam, Baback; Gruber, Amit; Weiss, Yair; Avidan, Shai
2008-01-01
We extend the l0-norm "subspectral" algorithms for sparse-LDA [5] and sparse-PCA [6] to general quadratic costs such as MSE in linear (kernel) regression. The resulting "Sparse Least Squares" (SLS) problem is also NP-hard, by way of its equivalence to a rank-1 sparse eigenvalue problem (e.g., binary sparse-LDA [7]). Specifically, for a general quadratic cost we use a highly-efficient technique for direct eigenvalue computation using partitioned matrix inverses which leads to dramatic x103 speed-ups over standard eigenvalue decomposition. This increased efficiency mitigates the O(n4) scaling behaviour that up to now has limited the previous algorithms' utility for high-dimensional learning problems. Moreover, the new computation prioritizes the role of the less-myopic backward elimination stage which becomes more efficient than forward selection. Similarly, branch-and-bound search for Exact Sparse Least Squares (ESLS) also benefits from partitioned matrix inverse techniques. Our Greedy Sparse Least Squares (GSLS) generalizes Natarajan's algorithm [9] also known as Order-Recursive Matching Pursuit (ORMP). Specifically, the forward half of GSLS is exactly equivalent to ORMP but more efficient. By including the backward pass, which only doubles the computation, we can achieve lower MSE than ORMP. Experimental comparisons to the state-of-the-art LARS algorithm [3] show forward-GSLS is faster, more accurate and more flexible in terms of choice of regularization
Iterative solution of the reduced eigenvalue problem
Energy Technology Data Exchange (ETDEWEB)
Sauer, G. (Technischer Ueberwachungs-Verein Bayern e.V., Muenchen (Germany, F.R.))
1991-04-01
The Guyan method of reducing the stiffness and mass matrices of large linear structures introduces errors in the reduced mass matrix. These errors cannot be completely avoided even if the analysis coordinates are chosen optimally. However, they can be elimiated by iterating on the eigenvectors found from the Guyan reduced matrices. The necessary iteration steps follow directly from the eigenvalue problem. The resulting iteration procedures are presented and applied to two test problems showing that the iterations enable the exact eigensolutions to be extracted. All errors from the Guyan reduced matrices are removed or substantially decreased. (orig.).
Frequency response as a surrogate eigenvalue problem in topology optimization
DEFF Research Database (Denmark)
Andreassen, Erik; Ferrari, Federico; Sigmund, Ole
2017-01-01
This article discusses the use of frequency response surrogates for eigenvalue optimization problems in topology optimization that may be used to avoid solving the eigenvalue problem. The motivation is to avoid complications that arise from multiple eigenvalues and the computational complexity as...
Ternary diffusion path in terms of eigenvalues and eigenvectors
Ram-Mohan, L. R.; Dayananda, Mysore A.
2016-04-01
Based on the transfer matrix methodology, a new analysis is presented for the description of slopes of the ternary diffusion path for a solid-solid diffusion couple. Concentration profiles and diffusion paths for isothermal, ternary diffusion couples are examined in the context of eigenvalues and eigenvectors obtained from the diagonalisation of the ? ternary interdiffusion coefficients employed for their representation. New relations are derived relating the decoupled interdiffusion fluxes to combinations of concentration gradients through the major and minor eigenvalues, and the diffusion path becomes parallel to the major eigenvector at each path end. General expressions for the slope of the ternary diffusion path at any section of the couple are also derived in terms of eigenvalue and eigenvector parameters. Expressions for the path slope at the Matano plane involve only concentrations, major and minor eigenvalues and eigenvector parameters. New constraints relating the eigenvalues and the concentration gradients of the individual components are also presented at selected sections, where the diffusion path is parallel to the straight line joining the terminal composition points on an isotherm. Applications of the various relations are illustrated with the aid of a hypothetical couple and an experimental Cu-Ni-Zn diffusion couple.
Polynomial Eigenvalue Solutions to Minimal Problems in Computer Vision.
Kukelova, Zuzana; Bujnak, Martin; Pajdla, Tomas
2012-07-01
We present a method for solving systems of polynomial equations appearing in computer vision. This method is based on polynomial eigenvalue solvers and is more straightforward and easier to implement than the state-of-the-art Gröbner basis method since eigenvalue problems are well studied, easy to understand, and efficient and robust algorithms for solving these problems are available. We provide a characterization of problems that can be efficiently solved as polynomial eigenvalue problems (PEPs) and present a resultant-based method for transforming a system of polynomial equations to a polynomial eigenvalue problem. We propose techniques that can be used to reduce the size of the computed polynomial eigenvalue problems. To show the applicability of the proposed polynomial eigenvalue method, we present the polynomial eigenvalue solutions to several important minimal relative pose problems.
DERIVATIVES OF EIGENPAIRS OF SYMMETRIC QUADRATIC EIGENVALUE PROBLEM
Institute of Scientific and Technical Information of China (English)
无
2005-01-01
Derivatives of eigenvalues and eigenvectors with respect to parameters in symmetric quadratic eigenvalue problem are studied. The first and second order derivatives of eigenpairs are given. The derivatives are calculated in terms of the eigenvalues and eigenvectors of the quadratic eigenvalue problem, and the use of state space representation is avoided, hence the cost of computation is greatly reduced. The efficiency of the presented method is demonstrated by considering a spring-mass-damper system.
Noncommutative Algebraic Equations and Noncommutative Eigenvalue Problem
Schwarz, A
2000-01-01
We analyze the perturbation series for noncommutative eigenvalue problem $AX=X\\lambda$ where $\\lambda$ is an element of a noncommutative ring, $ A$ is a matrix and $X$ is a column vector with entries from this ring. As a corollary we obtain a theorem about the structure of perturbation series for Tr $x^r$ where $x$ is a solution of noncommutative algebraic equation (for $r=1$ this theorem was proved by Aschieri, Brace, Morariu, and Zumino, hep-th/0003228, and used to study Born-Infeld lagrangian for the gauge group $U(1)^k$).
Preconditioned Krylov subspace methods for eigenvalue problems
Energy Technology Data Exchange (ETDEWEB)
Wu, Kesheng; Saad, Y.; Stathopoulos, A. [Univ. of Minnesota, Minneapolis, MN (United States)
1996-12-31
Lanczos algorithm is a commonly used method for finding a few extreme eigenvalues of symmetric matrices. It is effective if the wanted eigenvalues have large relative separations. If separations are small, several alternatives are often used, including the shift-invert Lanczos method, the preconditioned Lanczos method, and Davidson method. The shift-invert Lanczos method requires direct factorization of the matrix, which is often impractical if the matrix is large. In these cases preconditioned schemes are preferred. Many applications require solution of hundreds or thousands of eigenvalues of large sparse matrices, which pose serious challenges for both iterative eigenvalue solver and preconditioner. In this paper we will explore several preconditioned eigenvalue solvers and identify the ones suited for finding large number of eigenvalues. Methods discussed in this paper make up the core of a preconditioned eigenvalue toolkit under construction.
A Two-Level Method for Nonsymmetric Eigenvalue Problems
Institute of Scientific and Technical Information of China (English)
Karel Kolman
2005-01-01
A two-level discretization method for eigenvalue problems is studied. Compared to the standard Galerkin finite element discretization technique performed on a fine grid this method discretizes the eigenvalue problem on a coarse grid and obtains an improved eigenvector (eigenvalue) approximation by solving only a linear problem on the fine grid (or two linear problems for the case of eigenvalue approximation of nonsymmetric problems). The improved solution has the asymptotic accuracy of the Galerkin discretization solution. The link between the method and the iterated Galerkin method is established. Error estimates for the general nonsymmetric case are derived.
Inverse Eigenvalue Problems for Two Special Acyclic Matrices
Directory of Open Access Journals (Sweden)
Debashish Sharma
2016-03-01
Full Text Available In this paper, we study two inverse eigenvalue problems (IEPs of constructing two special acyclic matrices. The first problem involves the reconstruction of matrices whose graph is a path, from given information on one eigenvector of the required matrix and one eigenvalue of each of its leading principal submatrices. The second problem involves reconstruction of matrices whose graph is a broom, the eigen data being the maximum and minimum eigenvalues of each of the leading principal submatrices of the required matrix. In order to solve the problems, we use the recurrence relations among leading principal minors and the property of simplicity of the extremal eigenvalues of acyclic matrices.
EIGENVALUE PROBLEM OF A LARGE SCALE INDEFINITE GYROSCOPIC DYNAMIC SYSTEM
Institute of Scientific and Technical Information of China (English)
SUI Yong-feng; ZHONG Wan-xie
2006-01-01
Gyroscopic dynamic system can be introduced to Hamiltonian system. Based on an adjoint symplectic subspace iteration method of Hamiltonian gyroscopic system,an adjoint symplectic subspace iteration method of indefinite Hamiltonian function gyroscopic system was proposed to solve the eigenvalue problem of indefinite Hamiltonian function gyroscopic system. The character that the eigenvalues of Hamiltonian gyroscopic system are only pure imaginary or zero was used. The eigenvalues that Hamiltonian function is negative can be separated so that the eigenvalue problem of positive definite Hamiltonian function system was presented, and an adjoint symplectic subspace iteration method of positive definite Hamiltonian function system was used to solve the separated eigenvalue problem. Therefore, the eigenvalue problem of indefinite Hamiltonian function gyroscopic system was solved, and two numerical examples were given to demonstrate that the eigensolutions converge exactly.
Wu, Sheng-Jhih; Chu, Moody T.
2017-08-01
An inverse eigenvalue problem usually entails two constraints, one conditioned upon the spectrum and the other on the structure. This paper investigates the problem where triple constraints of eigenvalues, singular values, and diagonal entries are imposed simultaneously. An approach combining an eclectic mix of skills from differential geometry, optimization theory, and analytic gradient flow is employed to prove the solvability of such a problem. The result generalizes the classical Mirsky, Sing-Thompson, and Weyl-Horn theorems concerning the respective majorization relationships between any two of the arrays of main diagonal entries, eigenvalues, and singular values. The existence theory fills a gap in the classical matrix theory. The problem might find applications in wireless communication and quantum information science. The technique employed can be implemented as a first-step numerical method for constructing the matrix. With slight modification, the approach might be used to explore similar types of inverse problems where the prescribed entries are at general locations.
High-precision methods in eigenvalue problems and their applications
Akulenko, Leonid D
2004-01-01
This book presents a survey of analytical, asymptotic, numerical, and combined methods of solving eigenvalue problems. It considers the new method of accelerated convergence for solving problems of the Sturm-Liouville type as well as boundary-value problems with boundary conditions of the first, second, and third kind. The authors also present high-precision asymptotic methods for determining eigenvalues and eigenfunctions of higher oscillation modes and consider numerous eigenvalue problems that appear in oscillation theory, acoustics, elasticity, hydrodynamics, geophysics, quantum mechanics, structural mechanics, electrodynamics, and microelectronics.
Convergence of adaptive finite element methods for eigenvalue problems
Garau, Eduardo M.; Morin, Pedro; Zuppa, Carlos
2008-01-01
In this article we prove convergence of adaptive finite element methods for second order elliptic eigenvalue problems. We consider Lagrange finite elements of any degree and prove convergence for simple as well as multiple eigenvalues under a minimal refinement of marked elements, for all reasonable marking strategies, and starting from any initial triangulation.
The interior transmission problem and bounds on transmission eigenvalues
Hitrik, Michael; Ola, Petri; Päivärinta, Lassi
2010-01-01
We study the interior transmission eigenvalue problem for sign-definite multiplicative perturbations of the Laplacian in a bounded domain. We show that all but finitely many complex transmission eigenvalues are confined to a parabolic neighborhood of the positive real axis.
Local and Parallel Finite Element Algorithms for Eigenvalue Problems
Institute of Scientific and Technical Information of China (English)
Jinchao Xu; Aihui Zhou
2002-01-01
Some new local and parallel finite element algorithms are proposed and analyzed in this paper for eigenvalue problems. With these algorithms, the solution of an eigenvalue problem on a fine grid is reduced to the solution of an eigenvalue problem on a relatively coarse grid together with solutions of some linear algebraic systems on fine grid by using some local and parallel procedure. A theoretical tool for analyzing these algorithms is some local error estimate that is also obtained in this paper for finite element approximations of eigenvectors on general shape-regular grids.
Symmetric finite volume schemes for eigenvalue problems in arbitrary dimensions
Institute of Scientific and Technical Information of China (English)
2008-01-01
Based on a linear finite element space,two symmetric finite volume schemes for eigenvalue problems in arbitrary dimensions are constructed and analyzed.Some relationships between the finite element method and the finite difference method are addressed,too.
Symmetric finite volume schemes for eigenvalue problems in arbitrary dimensions
Institute of Scientific and Technical Information of China (English)
DAI Xiaoying; YANG Zhang; ZHOU Aihui
2008-01-01
Based on a linear finite element space, two symmetric finite volume schemes for eigenvalue problems in arbitrary dimensions are constructed and analyzed. Some relationships between the finite element method and the finite difference method are addressed, too.
On a quasilinear elliptic eigenvalue problem with constraint
Institute of Scientific and Technical Information of China (English)
无
2004-01-01
Via construction of pseudo gradient vector field and descending flow argument, we prove the existence of one positive, one negative and one sign-changing solutions for a quasilinear elliptic eigenvalue problem with constraint.
A generalized eigenvalue problem solution for an uncoupled multicomponent system
Energy Technology Data Exchange (ETDEWEB)
Diago-Cisneros, L; Fernandez-Anaya, G; Bonfanti-Escalera, G [Departamento de Fisica y Matematicas, Universidad Iberoamericana, CP 01219, DF Mexico (Mexico)], E-mail: ldiago@fisica.uh.cu
2008-09-15
Meaningful and well-founded physical quantities are convincingly determined by eigenvalue problem solutions emerging from a second-order N-coupled system of differential equations, known as the Sturm-Liouville matrix boundary problem. Via the generalized Schur decomposition procedure and imposing to the multicomponent system to be decoupled, which is a widely accepted remarkable physical situation, we have unambiguously demonstrated a simultaneously triangularizable scenario for (2Nx2N) matrices content in a generalized eigenvalue equation.
Energy Technology Data Exchange (ETDEWEB)
Ceolin, C.; Schramm, M.; Vilhena, M.T.; Bodmann, B.E.J., E-mail: celina.ceolin@gmail.com, E-mail: marceloschramm@hotmail.com, E-mail: vilhena@pq.cnpq.br, E-mail: bardo.bodmann@ufrgs.br [Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, RS (Brazil). Programa de Pos-Graduacao em Engenharia Mecanica
2013-07-01
In this work the authors solved the steady state neutron diffusion equation for a multi-layer slab assuming the multi-group energy model. The method to solve the equation system is based on a expansion in Taylor Series, which was proven to be useful in [1] [2] [3]. The results obtained can be used as initial condition for neutron space kinetics problems. The neutron scalar flux was expanded in a power series, and the coefficients were found by using the ordinary differential equation and the boundary and interface conditions. The effective multiplication factor k was evaluated using the power method [4]. We divided the domain into several slabs to guarantee the convergence with a low truncation order. We present the formalism together with some numerical simulations. (author)
Energy Technology Data Exchange (ETDEWEB)
Ceolin, Celina; Schramm, Marcelo; Bodmann, Bardo Ernst Josef; Vilhena, Marco Tullio Mena Barreto de [Universidade Federal do Rio Grande do Sul, Porto Alegre (Brazil). Programa de Pos-Graduacao em Engenharia Mecanica; Bogado Leite, Sergio de Queiroz [Comissao Nacional de Energia Nuclear, Rio de Janeiro (Brazil)
2014-11-15
In this work the authors solved the steady state neutron diffusion equation for a multi-layer slab assuming the multi-group energy model. The method to solve the equation system is based on an expansion in Taylor Series resulting in an analytical expression. The results obtained can be used as initial condition for neutron space kinetics problems. The neutron scalar flux was expanded in a power series, and the coefficients were found by using the ordinary differential equation and the boundary and interface conditions. The effective multiplication factor k was evaluated using the power method. We divided the domain into several slabs to guarantee the convergence with a low truncation order. We present the formalism together with some numerical simulations.
Existence of a principal eigenvalue for the Tricomi problem
Directory of Open Access Journals (Sweden)
Daniela Lupo
2000-10-01
Full Text Available The existence of a principal eigenvalue is established for the Tricomi problem in normal domains; that is, the existence of a positive eigenvalue of minimum modulus with an associated positive eigenfunction. The argument here uses prior results of the authors on the generalized solvability in weighted Sobolev spaces and associated maximum/minimum principles cite{[LP2]} coupled with known results of Krein-Rutman type.
NUMERICAL SOLUTIONS OF AN EIGENVALUE PROBLEM IN UNBOUNDED DOMAINS
Institute of Scientific and Technical Information of China (English)
Han Houde; Zhou Zhenya; Zheng Chunxiong
2005-01-01
A coupling method of finite element and infinite large element is proposed for the numerical solution of an eigenvalue problem in unbounded domains in this paper. With some conditions satisfied, the considered problem is proved to have discrete spectra. Several numerical experiments are presented. The results demonstrate the feasibility of the proposed method.
An Inverse Eigenvalue Problem for Jacobi Matrices
Directory of Open Access Journals (Sweden)
Zhengsheng Wang
2011-01-01
eigenvectors. The solvability of the problem is discussed, and some sufficient conditions for existence of the solution of this problem are proposed. Furthermore, a numerical algorithm and two examples are presented.
Nonlinear eigenvalue problems with semipositone structure
Directory of Open Access Journals (Sweden)
Alfonso Castro
2000-10-01
Full Text Available In this paper we summarize the developments of semipositone problems to date, including very recent results on semipositone systems. We also discuss applications and open problems.
An eigenvalue problem for the associated Askey-Wilson polynomials
Bruder, Andrea; Suslov, Sergei K
2012-01-01
To derive an eigenvalue problem for the associated Askey-Wilson polynomials, we consider an auxiliary function in two variables which is related to the associated Askey-Wilson polynomials introduced by Ismail and Rahman. The Askey-Wilson operator, applied in each variable separately, maps this function to the ordinary Askey-Wilson polynomials with different sets of parameters. A third Askey-Wilson operator is found with the help of a computer algebra program which links the two, and an eigenvalue problem is stated.
A multilevel finite element method for Fredholm integral eigenvalue problems
Xie, Hehu; Zhou, Tao
2015-12-01
In this work, we proposed a multigrid finite element (MFE) method for solving the Fredholm integral eigenvalue problems. The main motivation for such studies is to compute the Karhunen-Loève expansions of random fields, which play an important role in the applications of uncertainty quantification. In our MFE framework, solving the eigenvalue problem is converted to doing a series of integral iterations and eigenvalue solving in the coarsest mesh. Then, any existing efficient integration scheme can be used for the associated integration process. The error estimates are provided, and the computational complexity is analyzed. It is noticed that the total computational work of our method is comparable with a single integration step in the finest mesh. Several numerical experiments are presented to validate the efficiency of the proposed numerical method.
On the eigenvalue spectrum for time-delayed Floquet problems
Just, Wolfram
2000-08-01
A linear homogeneous scalar differential-difference equation with harmonic time dependence is investigated. The associated eigenvalue problem is solved in terms of a continued fraction expansion for the characteristic equation. The dependence of the largest eigenvalue on the system parameters, being relevant for stability of periodic states in delay systems, is discussed in detail. The competition between the two timescales, the delay and the external period cause intricate structures. The result suggests features to improve control of chaos by time-delayed feedback schemes with time-dependent control amplitudes.
A filtering method for the interval eigenvalue problem
DEFF Research Database (Denmark)
Hladik, Milan; Daney, David; Tsigaridas, Elias
2011-01-01
We consider the general problem of computing intervals that contain the real eigenvalues of interval matrices. Given an outer approximation (superset) of the real eigenvalue set of an interval matrix, we propose a filtering method that iteratively improves the approximation. Even though our method...... is based on a sufficient regularity condition, it is very efficient in practice and our experimental results suggest that it improves, in general, significantly the initial outer approximation. The proposed method works for general, as well as for symmetric interval matrices....
Biharmonic eigen-value problems and Lp estimates
Directory of Open Access Journals (Sweden)
Chaitan P. Gupta
1990-01-01
Full Text Available Biharmonic eigen-values arise in the study of static equilibrium of an elastic body which has been suitably secured at the boundary. This paper is concerned mainly with the existence of and Lp-estimates for the solutions of certain biharmonic boundary value problems which are related to the first eigen-values of the associated biharmonic operators. The methods used in this paper consist of the a-priori estimates due to Agmon-Douglas-Nirenberg and P. L. Lions along with the Fredholm theory for compact operators.
Boundary and eigenvalue problems in mathematical physics
Sagan, Hans
1989-01-01
This well-known text uses a limited number of basic concepts and techniques - Hamilton's principle, the theory of the first variation and Bernoulli's separation method - to develop complete solutions to linear boundary value problems associated with second order partial differential equations such as the problems of the vibrating string, the vibrating membrane, and heat conduction. It is directed to advanced undergraduate and beginning graduate students in mathematics, applied mathematics, physics, and engineering who have completed a course in advanced calculus. In the first three chapters,
Explicit solution for an infinite dimensional generalized inverse eigenvalue problem
Directory of Open Access Journals (Sweden)
Kazem Ghanbari
2001-01-01
Full Text Available We study a generalized inverse eigenvalue problem (GIEP, Ax=λBx, in which A is a semi-infinite Jacobi matrix with positive off-diagonal entries ci>0, and B= diag (b0,b1,…, where bi≠0 for i=0,1,…. We give an explicit solution by establishing an appropriate spectral function with respect to a given set of spectral data.
Kulshreshtha, Kshitij; Nataraj, Neela
2005-08-01
The paper deals with a parallel implementation of a mixed finite element method of approximation of eigenvalues and eigenvectors of fourth order eigenvalue problems with variable/constant coefficients. The implementation has been done in Silicon Graphics Origin 3800, a four processor Intel Xeon Symmetric Multiprocessor and a beowulf cluster of four Intel Pentium III PCs. The generalised eigenvalue problem obtained after discretization using the mixed finite element method is solved using the package LANSO. The numerical results obtained are compared with existing results (if available). The time, speedup comparisons in different environments for some examples of practical and research interest and importance are also given.
Plain strain problem of poroelasticity using eigenvalue approach
Indian Academy of Sciences (India)
Rajneesh Kumar; Aseem Miglani; N R Garg
2000-09-01
A plain strain problem of an isotropic elastic liquid-saturated porous medium in poroelasticity has been studied. The eigenvalue approach using the Laplace and Fourier transforms has been employed and these transforms have been inverted by using a numerical technique. An application of infinite space with concentrated force at the origin has been presented to illustrate the utility of the approach. The displacement and stress components in the physical domain are obtained numerically. The results are shown graphically and can be used for a broad class of problems related to liquid-saturated porous media.
Eigenvalues of a diffusion process with a critical point
Dekker, H.; Kampen, N.G. van
1979-01-01
The eigenvalues of a Fokker-Planck equation involving a critical point have been computed by means of a simple discretization technique. The results smoothly connect the monostable case above the critical point with the bistable case below it.
An Inverse Eigenvalue Problem for Damped Gyroscopic Second-Order Systems
Directory of Open Access Journals (Sweden)
Yongxin Yuan
2009-01-01
analytical mass and stiffness matrices, so that ( has a prescribed subset of eigenvalues and eigenvectors, is considered. Necessary and sufficient conditions under which this quadratic inverse eigenvalue problem is solvable are specified.
HIGH ACCURACY ANALYSIS OF ELLIPTIC EIGENVALUE PROBLEM FOR THE WILSON NONCONFORMING FINITE ELEMENT
Institute of Scientific and Technical Information of China (English)
吴冬生
2001-01-01
In this paper, the Wilson nonconforming finite element is considered for solving elliptic eigen-value problems. Based on an interpolation postprocessing, superconvergence estimates of both eigenfunction and eigenvalue are obtained.
A New Inexact Inverse Subspace Iteration for Generalized Eigenvalue Problems
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Fatemeh Mohammad
2014-05-01
Full Text Available In this paper, we represent an inexact inverse subspace iteration method for computing a few eigenpairs of the generalized eigenvalue problem $Ax = \\lambda Bx$[Q.~Ye and P.~Zhang, Inexact inverse subspace iteration for generalized eigenvalue problems, Linear Algebra and its Application, 434 (2011 1697-1715]. In particular, the linear convergence property of the inverse subspace iteration is preserved.
Levenberg-Marquardt method for the eigenvalue complementarity problem.
Chen, Yuan-yuan; Gao, Yan
2014-01-01
The eigenvalue complementarity problem (EiCP) is a kind of very useful model, which is widely used in the study of many problems in mechanics, engineering, and economics. The EiCP was shown to be equivalent to a special nonlinear complementarity problem or a mathematical programming problem with complementarity constraints. The existing methods for solving the EiCP are all nonsmooth methods, including nonsmooth or semismooth Newton type methods. In this paper, we reformulate the EiCP as a system of continuously differentiable equations and give the Levenberg-Marquardt method to solve them. Under mild assumptions, the method is proved globally convergent. Finally, some numerical results and the extensions of the method are also given. The numerical experiments highlight the efficiency of the method.
Approximation on computing partial sum of nonlinear differential eigenvalue problems
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
In computing the electronic structure and energy band in a system of multi-particles, quite a large number of problems are referred to the acquisition of obtaining the partial sum of densities and energies using the “first principle”. In the conventional method, the so-called self-consistency approach is limited to a small scale because of high computing complexity. In this paper, the problem of computing the partial sum for a class of nonlinear differential eigenvalue equations is changed into the constrained functional minimization. By space decomposition and perturbation method, a secondary approximating formula for the minimal is provided. It is shown that this formula is more precise and its quantity of computation can be reduced significantly
A multi-step method for partial eigenvalue assignment problem of high order control systems
Liu, Hao; Xu, Jiajia
2017-09-01
In this paper, we consider the partial eigenvalue assignment problem of high order control systems. Based on the orthogonality relations, we propose a new method for solving this problem by which the undesired eigenvalues are moved to desired values and keep the remaining eigenvalues unchanged. Using the inverse of Cauchy matrix, we give the solvable condition and the explicit solutions to this problem. Numerical examples show that our method is effective.
Indefinite Eigenvalue Problems for p-Laplacian Operators with Potential Terms on Networks
Directory of Open Access Journals (Sweden)
Jea-Hyun Park
2014-01-01
Full Text Available We address some forward and inverse problems involving indefinite eigenvalues for discrete p-Laplacian operators with potential terms. These indefinite eigenvalues are the discrete analogues of p-Laplacians on Riemannian manifolds with potential terms. We first define and discuss some fundamental properties of the indefinite eigenvalue problems for discrete p-Laplacian operators with potential terms with respect to some given weight functions. We then discuss resonance problems, anti-minimum principles, and inverse conductivity problems for the discrete p-Laplacian operators with potential terms involving the smallest indefinite eigenvalues.
Several concepts to investigate strongly nonnormal eigenvalue problems
Dorsselaer, J.L.M. van
2001-01-01
Eigenvalue analysis plays an important role in understanding physical phenomena. However, if one deals with strongly nonnormal matrices or operators, the eigenvalues alone may not tell the full story. A popular tool which can be useful to get more insight in the reliability or sensitivity of eigenva
Numerical study of three-parameter matrix eigenvalue problem by Rayleigh quotient method
Bora, Niranjan; Baruah, Arun Kumar
2016-06-01
In this paper, an attempt is done to find approximate eigenvalues and the corresponding eigenvectors of three-parameter matrix eigenvalue problem by extending Rayleigh Quotient Iteration Method (RQIM), which is generally used to solve generalized eigenvalue problems of the form Ax = λBx. Convergence criteria of RQIM will be derived in terms of matrix 2-norm. We will test the computational efficiency of the Method analytically with the help of numerical examples. All calculations are done in MATLAB software.
Quantum inequalities and "quantum interest" as eigenvalue problems
Fewster, C J; Fewster, Christopher J.; Teo, Edward
2000-01-01
Quantum inequalities (QI's) provide lower bounds on the averaged energy density of a quantum field. We show how the QI's for massless scalar fields in even dimensional Minkowski space may be reformulated in terms of the positivity of a certain self-adjoint operator - a generalised Schroedinger operator with the energy density as the potential - and hence as an eigenvalue problem. We use this idea to verify that the energy density produced by a moving mirror in two dimensions is compatible with the QI's for a large class of mirror trajectories. In addition, we apply this viewpoint to the `quantum interest conjecture' of Ford and Roman, which asserts that the positive part of an energy density always overcompensates for any negative components. For various simple models in two and four dimensions we obtain the best possible bounds on the `quantum interest rate' and on the maximum delay between a negative pulse and a compensating positive pulse. Perhaps surprisingly, we find that - in four dimensions - it is imp...
Costiner, Sorin; Taasan, Shlomo
1994-01-01
This paper presents multigrid (MG) techniques for nonlinear eigenvalue problems (EP) and emphasizes an MG algorithm for a nonlinear Schrodinger EP. The algorithm overcomes the mentioned difficulties combining the following techniques: an MG projection coupled with backrotations for separation of solutions and treatment of difficulties related to clusters of close and equal eigenvalues; MG subspace continuation techniques for treatment of the nonlinearity; an MG simultaneous treatment of the eigenvectors at the same time with the nonlinearity and with the global constraints. The simultaneous MG techniques reduce the large number of self consistent iterations to only a few or one MG simultaneous iteration and keep the solutions in a right neighborhood where the algorithm converges fast.
Multigrid method applied to the solution of an elliptic, generalized eigenvalue problem
Energy Technology Data Exchange (ETDEWEB)
Alchalabi, R.M. [BOC Group, Murray Hill, NJ (United States); Turinsky, P.J. [North Carolina State Univ., Raleigh, NC (United States)
1996-12-31
The work presented in this paper is concerned with the development of an efficient MG algorithm for the solution of an elliptic, generalized eigenvalue problem. The application is specifically applied to the multigroup neutron diffusion equation which is discretized by utilizing the Nodal Expansion Method (NEM). The underlying relaxation method is the Power Method, also known as the (Outer-Inner Method). The inner iterations are completed using Multi-color Line SOR, and the outer iterations are accelerated using Chebyshev Semi-iterative Method. Furthermore, the MG algorithm utilizes the consistent homogenization concept to construct the restriction operator, and a form function as a prolongation operator. The MG algorithm was integrated into the reactor neutronic analysis code NESTLE, and numerical results were obtained from solving production type benchmark problems.
PRECONDITIONING BLOCK LANCZOS ALGORITHM FOR SOLVING SYMMETRIC EIGENVALUE PROBLEMS
Institute of Scientific and Technical Information of China (English)
Hua Dai; Peter Lancaster
2000-01-01
A preconditioned iterative method for computing a few eigenpairs of large sparse symmetric matrices is presented in this paper. The proposed method which combines the preconditioning techniques with the efficiency of block Lanczos algorithm is suitable for determination of the extreme eigenvalues as well as their multiplicities. The global convergence and the asymptotically quadratic convergence of the new method are also demonstrated.
Antar, B. N.
1976-01-01
A numerical technique is presented for locating the eigenvalues of two point linear differential eigenvalue problems. The technique is designed to search for complex eigenvalues belonging to complex operators. With this method, any domain of the complex eigenvalue plane could be scanned and the eigenvalues within it, if any, located. For an application of the method, the eigenvalues of the Orr-Sommerfeld equation of the plane Poiseuille flow are determined within a specified portion of the c-plane. The eigenvalues for alpha = 1 and R = 10,000 are tabulated and compared for accuracy with existing solutions.
Menon, Ravishankar; Gerstoft, Peter; Hodgkiss, William S
2012-11-01
Cross-correlations of diffuse noise fields can be used to extract environmental information. The influence of directional sources (usually ships) often results in a bias of the travel time estimates obtained from the cross-correlations. Using an array of sensors, insights from random matrix theory on the behavior of the eigenvalues of the sample covariance matrix (SCM) in an isotropic noise field are used to isolate the diffuse noise component from the directional sources. A sequential hypothesis testing of the eigenvalues of the SCM reveals eigenvalues dominated by loud sources that are statistical outliers for the assumed diffuse noise model. Travel times obtained from cross-correlations using only the diffuse noise component (i.e., by discarding or attenuating the outliers) converge to the predicted travel times based on the known array sensor spacing and measured sound speed at the site and are stable temporally (i.e., unbiased estimates). Data from the Shallow Water 2006 experiment demonstrates the effectiveness of this approach and that the signal-to-noise ratio builds up as the square root of time, as predicted by theory.
On a Non-Symmetric Eigenvalue Problem Governing Interior Structural–Acoustic Vibrations
Directory of Open Access Journals (Sweden)
Heinrich Voss
2016-06-01
Full Text Available Small amplitude vibrations of a structure completely filled with a fluid are considered. Describing the structure by displacements and the fluid by its pressure field, the free vibrations are governed by a non-self-adjoint eigenvalue problem. This survey reports on a framework for taking advantage of the structure of the non-symmetric eigenvalue problem allowing for a variational characterization of its eigenvalues. Structure-preserving iterative projection methods of the the Arnoldi and of the Jacobi–Davidson type and an automated multi-level sub-structuring method are reviewed. The reliability and efficiency of the methods are demonstrated by a numerical example.
Optical reflection from planetary surfaces as an operator-eigenvalue problem
Wildey, R.L.
1986-01-01
The understanding of quantum mechanical phenomena has come to rely heavily on theory framed in terms of operators and their eigenvalue equations. This paper investigates the utility of that technique as related to the reciprocity principle in diffuse reflection. The reciprocity operator is shown to be unitary and Hermitian; hence, its eigenvectors form a complete orthonormal basis. The relevant eigenvalue is found to be infinitely degenerate. A superposition of the eigenfunctions found from solution by separation of variables is inadequate to form a general solution that can be fitted to a one-dimensional boundary condition, because the difficulty of resolving the reciprocity operator into a superposition of independent one-dimensional operators has yet to be overcome. A particular lunar application in the form of a failed prediction of limb-darkening of the full Moon from brightness versus phase illustrates this problem. A general solution is derived which fully exploits the determinative powers of the reciprocity operator as an unresolved two-dimensional operator. However, a solution based on a sum of one-dimensional operators, if possible, would be much more powerful. A close association is found between the reciprocity operator and the particle-exchange operator of quantum mechanics, which may indicate the direction for further successful exploitation of the approach based on the operational calculus. ?? 1986 D. Reidel Publishing Company.
Integral Transforms and a Class of Singular S-Hermitian Eigenvalue Problems
Dijksma, A.; Snoo, H.S.V. de
1973-01-01
For a class of singular S-hermitian eigenvalue problems we show that the corresponding integral transforms are surjective. This class was discussed by us earlier and is more restricted than the one, which has been considered by others.
Directory of Open Access Journals (Sweden)
Pengzhan Huang
2011-01-01
Full Text Available Several stabilized finite element methods for the Stokes eigenvalue problem based on the lowest equal-order finite element pair are numerically investigated. They are penalty, regular, multiscale enrichment, and local Gauss integration method. Comparisons between them are carried out, which show that the local Gauss integration method has good stability, efficiency, and accuracy properties, and it is a favorite method among these methods for the Stokes eigenvalue problem.
Directory of Open Access Journals (Sweden)
Yidu Yang
2012-01-01
Full Text Available This paper discusses highly finite element algorithms for the eigenvalue problem of electric field. Combining the mixed finite element method with the Rayleigh quotient iteration method, a new multi-grid discretization scheme and an adaptive algorithm are proposed and applied to the eigenvalue problem of electric field. Theoretical analysis and numerical results show that the computational schemes established in the paper have high efficiency.
An integrable Poisson map generated from the eigenvalue problem of the Lotka-Volterra hierarchy
Energy Technology Data Exchange (ETDEWEB)
Wu Yongtang [Department of Computer Science, Hong Kong Baptist University, Kowloon Tong, Hong Kong (China); Wang Hongye [Department of Mathematics, Zhengzhou University, Henan (China); Du Dianlou [Department of Mathematics, Zhengzhou University, Henan (China)]. E-mail: ddl@zzu.edu.cn
2002-05-03
A 3x3 discrete eigenvalue problem associated with the Lotka-Volterra hierarchy is studied and the corresponding nonlinearized one, an integrable Poisson map with a Lie-Poisson structure, is also presented. Moreover, a 2x2 nonlinearized eigenvalue problem, which also begets the Lotka-Volterra hierarchy, is proved to be a reduction of the Poisson map on the leaves of the symplectic foliation. (author)
A Two-Scale Discretization Scheme for Mixed Variational Formulation of Eigenvalue Problems
Directory of Open Access Journals (Sweden)
Yidu Yang
2012-01-01
Full Text Available This paper discusses highly efficient discretization schemes for mixed variational formulation of eigenvalue problems. A new finite element two-scale discretization scheme is proposed by combining the mixed finite element method with the shifted-inverse power method for solving matrix eigenvalue problems. With this scheme, the solution of an eigenvalue problem on a fine grid Kh is reduced to the solution of an eigenvalue problem on a much coarser grid KH and the solution of a linear algebraic system on the fine grid Kh. Theoretical analysis shows that the scheme has high efficiency. For instance, when using the Mini element to solve Stokes eigenvalue problem, the resulting solution can maintain an asymptotically optimal accuracy by taking H=O(h4, and when using the Pk+1-Pk element to solve eigenvalue problems of electric field, the calculation results can maintain an asymptotically optimal accuracy by taking H=O(h3. Finally, numerical experiments are presented to support the theoretical analysis.
TWO-DIMENSIONAL APPROXIMATION OF EIGENVALUE PROBLEMS IN SHELL THEORY: FLEXURAL SHELLS
Institute of Scientific and Technical Information of China (English)
无
2000-01-01
The eigenvalue problem for a thin linearly elastic shell, of thickness 2e, clamped along its lateral surface is considered. Under the geometric assumption on the middle surface of the shell that the space of inextensional displacements is non-trivial, the authors obtain, as ε→0,the eigenvalue problem for the two-dimensional"flexural shell"model if the dimension of the space is infinite. If the space is finite dimensional, the limits of the eigenvalues could belong to the spectra of both flexural and membrane shells. The method consists of rescaling the variables and studying the problem over a fixed domain. The principal difficulty lies in obtaining suitable a priori estimates for the scaled eigenvalues.
EvArnoldi: A New Algorithm for Large-Scale Eigenvalue Problems.
Tal-Ezer, Hillel
2016-05-19
Eigenvalues and eigenvectors are an essential theme in numerical linear algebra. Their study is mainly motivated by their high importance in a wide range of applications. Knowledge of eigenvalues is essential in quantum molecular science. Solutions of the Schrödinger equation for the electrons composing the molecule are the basis of electronic structure theory. Electronic eigenvalues compose the potential energy surfaces for nuclear motion. The eigenvectors allow calculation of diople transition matrix elements, the core of spectroscopy. The vibrational dynamics molecule also requires knowledge of the eigenvalues of the vibrational Hamiltonian. Typically in these problems, the dimension of Hilbert space is huge. Practically, only a small subset of eigenvalues is required. In this paper, we present a highly efficient algorithm, named EvArnoldi, for solving the large-scale eigenvalues problem. The algorithm, in its basic formulation, is mathematically equivalent to ARPACK ( Sorensen , D. C. Implicitly Restarted Arnoldi/Lanczos Methods for Large Scale Eigenvalue Calculations ; Springer , 1997 ; Lehoucq , R. B. ; Sorensen , D. C. SIAM Journal on Matrix Analysis and Applications 1996 , 17 , 789 ; Calvetti , D. ; Reichel , L. ; Sorensen , D. C. Electronic Transactions on Numerical Analysis 1994 , 2 , 21 ) (or Eigs of Matlab) but significantly simpler.
Quadratic partial eigenvalue assignment problem with time delay for active vibration control
Pratt, J. M.; Singh, K. V.; Datta, B. N.
2009-08-01
Partial pole assignment in active vibration control refers to reassigning a small set of unwanted eigenvalues of the quadratic eigenvalue problem (QEP) associated with the second order system of a vibrating structure, by using feedback control force, to suitably chosen location without altering the remaining large number of eigenvalues and eigenvectors. There are several challenges of solving this quadratic partial eigenvalue assignment problem (QPEVAP) in a computational setting which the traditional pole-placement problems for first-order control systems do not have to deal with. In order to these challenges, there has been some work in recent years to solve QPEVAP in a computationally viable way. However, these works do not take into account of the practical phenomenon of the time-delay effect in the system. In this paper, a new "direct and partial modal" approach of the quadratic partial eigenvalue assignment problem with time-delay is proposed. The approach works directly in the quadratic system without requiring transformation to a standard state-space system and requires the knowledge of only a small number of eigenvalues and eigenvectors that can be computed or measured in practice. Two illustrative examples are presented in the context of active vibration control with constant time-delay to illustrate the success of our proposed approach. Future work includes generalization of this approach to a more practical complex time-delay system and extension of this work to the multi-input problem.
Inverse Eigenvalue Problems for a Structure with Linear Parameters
Institute of Scientific and Technical Information of China (English)
WU Liang-sheng; YANG Jia-hua; WEI Yuan-qian; MEN Hao; YANG Qing-kun; LIU Zhen-yu
2005-01-01
The inverse design method of a dynamic system with linear parameters has been studied. For some specified eigenvalues and eigenvectors, the design parameter vector which is often composed of whole or part of coefficients of spring and mass of the system can be obtained and the rigidity and mass matrices of an initially designed structure can be reconstructed through solving linear algebra equations. By using implicit function theorem, the conditions of existence and uniqueness of the solution are also deduced. The theory and method can be used for inverse vibration design of complex structure system.
HOMOTOPY SOLUTION OF THE INVERSE GENERALIZED EIGENVALUE PROBLEMS IN STRUCTURAL DYNAMICS
Institute of Scientific and Technical Information of China (English)
李书; 王波; 胡继忠
2004-01-01
The structural dynamics problems, such as structural design, parameter identification and model correction, are considered as a kind of the inverse generalized eigenvalue problems mathematically. The inverse eigenvalue problems are nonlinear. In general, they could be transformed into nonlinear equations to solve. The structural dynamics inverse problems were treated as quasi multiplicative inverse eigenalue problems which were solved by homotopy method for nonlinear equations. This method had no requirements for initial value essentially because of the homotopy path to solution. Numerical examples were presented to illustrate the homotopy method.
Murphy, W D; Bernabe, M L
1978-08-01
The Prony method is extended to handle the nonsymmetric algebraic eigenvalue problem and improved to search automatically for the number of dominant eigenvalues. A simple iterative algorithm is given to compute the associated eigenvectors. Resolution studies using the QR method are made in order to determine the accuracy of the matrix approximation. Numerical results are given for both simple well defined resonators and more complex advanced designs containing multiple propagation geometries and misaligned mirrors.
A Hardy Inequality with Remainder Terms in the Heisenberg Group and the Weighted Eigenvalue Problem
Directory of Open Access Journals (Sweden)
Dou Jingbo
2007-01-01
Full Text Available Based on properties of vector fields, we prove Hardy inequalities with remainder terms in the Heisenberg group and a compact embedding in weighted Sobolev spaces. The best constants in Hardy inequalities are determined. Then we discuss the existence of solutions for the nonlinear eigenvalue problems in the Heisenberg group with weights for the -sub-Laplacian. The asymptotic behaviour, simplicity, and isolation of the first eigenvalue are also considered.
A Hardy Inequality with Remainder Terms in the Heisenberg Group and the Weighted Eigenvalue Problem
Directory of Open Access Journals (Sweden)
Zixia Yuan
2007-12-01
Full Text Available Based on properties of vector fields, we prove Hardy inequalities with remainder terms in the Heisenberg group and a compact embedding in weighted Sobolev spaces. The best constants in Hardy inequalities are determined. Then we discuss the existence of solutions for the nonlinear eigenvalue problems in the Heisenberg group with weights for the p-sub-Laplacian. The asymptotic behaviour, simplicity, and isolation of the first eigenvalue are also considered.
Institute of Scientific and Technical Information of China (English)
Qiumei Huang; Yidu Yang
2008-01-01
In this paper,we introduce a new extrapolation formula by combining Richardson extrapolation and Sloan iteration algorithms.Using this extrapolation formula,we obtain some asymptotic expansions of the Galerkin finite element method for semi-simple eigenvalue problems of Fredholm integral equations of the second kind and improve the accuracy of the numerical approximations of the corresponding eigenvalues.Some numerical experiments are carried out to demonstrate the effectiveness of OUr new method and to confirm our theoretical results.
Real dqds for the nonsymmetric tridiagonal eigenvalue problem
Ferreira, Carla
2012-01-01
We present a new transform, triple dqds, to help to compute the eigenvalues of a real tridiagonal matrix C using real arithmetic. The algorithm uses the real dqds transform to shift by a real number and triple dqds to shift by a complex conjugate pair. We present what seems to be a new criteria for splitting the current pair L,U. The algorithm rejects any transform which suffers from excessive element growth and then tries a new transform. Our numerical tests show that the algorithm is about 100 times faster than the Ehrlich-Aberth method of D. A. Bini, L. Gemignani and F. Tisseur. Our code is comparable in performance to a complex dqds code and is sometimes 3 times faster.
A numerical method for eigenvalue problems in modeling liquid crystals
Energy Technology Data Exchange (ETDEWEB)
Baglama, J.; Farrell, P.A.; Reichel, L.; Ruttan, A. [Kent State Univ., OH (United States); Calvetti, D. [Stevens Inst. of Technology, Hoboken, NJ (United States)
1996-12-31
Equilibrium configurations of liquid crystals in finite containments are minimizers of the thermodynamic free energy of the system. It is important to be able to track the equilibrium configurations as the temperature of the liquid crystals decreases. The path of the minimal energy configuration at bifurcation points can be computed from the null space of a large sparse symmetric matrix. We describe a new variant of the implicitly restarted Lanczos method that is well suited for the computation of extreme eigenvalues of a large sparse symmetric matrix, and we use this method to determine the desired null space. Our implicitly restarted Lanczos method determines adoptively a polynomial filter by using Leja shifts, and does not require factorization of the matrix. The storage requirement of the method is small, and this makes it attractive to use for the present application.
Eigenvalue problems of Atkinson, Feller and Krein, and their mutual relationship
Directory of Open Access Journals (Sweden)
Hans Volkmer
2005-04-01
Full Text Available It is shown that every regular Krein-Feller eigenvalue problem can be transformed to a semidefinite Sturm-Liouville problem introduced by Atkinson. This makes it possible to transfer results between the corresponding theories. In particular, Prufer angle methods become available for Krein-Feller problems.
INVERSE EIGENVALUE PROBLEM OF HERMITIAN GENERALIZED ANTI-HAMILTONIAN MATRICES%HGAH矩阵的逆特征值问题
Institute of Scientific and Technical Information of China (English)
张忠志; Liu Changrong
2004-01-01
In this paper, the inverse eigenvalue problem of Hermitian generalized anti-Hamiltonian matrices and relevant optimal approximate problem are considered. The necessary and sufficient conditions of the solvability for inverse eigenvalue problem and an expression of the general solution of the problem are derived. The solution of the relevant optimal approximate problem is given.
Brahma, Sanjoy; Datta, Biswa
2009-07-01
The partial quadratic eigenvalue assignment problem (PQEVAP) concerns the reassignment of a small number of undesirable eigenvalues of a quadratic matrix pencil, while leaving the remaining large number of eigenvalues and the corresponding eigenvectors unchanged. The problem arises in controlling undesirable resonance in vibrating structures and in stabilizing control systems. The solution of this problem requires computations of a pair of feedback matrices. For practical effectiveness, these feedback matrices must be computed in such a way that their norms and the condition number of the closed-loop eigenvector matrix are as small as possible. These considerations give rise to the minimum norm partial quadratic eigenvalue assignment problem (MNPQEVAP) and the robust partial quadratic eigenvalue assignment problem (RPQEVAP), respectively. In this paper we propose new optimization based algorithms for solving these problems. The problems are solved directly in a second-order setting without resorting to a standard first-order formulation so as to avoid the inversion of a possibly ill-conditioned matrix and the loss of exploitable structures of the original model. The algorithms require the knowledge of only the open-loop eigenvalues to be replaced and their corresponding eigenvectors. The remaining open-loop eigenvalues and their corresponding eigenvectors are kept unchanged. The invariance of the large number of eigenvalues and eigenvectors under feedback is guaranteed by a proven mathematical result. Furthermore, the gradient formulas needed to solve the problems by using the quasi-Newton optimization technique employed are computed in terms of the known quantities only. Above all, the proposed methods do not require the reduction of the model order or the order of the controller, even when the underlying finite element model has a very large degree of freedom. These attractive features, coupled with minimal computational requirements, such as solutions of small
A POSTERIORI ERROR ANALYSIS OF NONCONFORMING METHODS FOR THE EIGENVALUE PROBLEM
Institute of Scientific and Technical Information of China (English)
Youai LI
2009-01-01
This paper extends the unifying theory for a posteriori error analysis of the nonconforming finite element methods to the second order elliptic eigenvalue problem. In particular, the author proposes the a posteriori error estimator for nonconforming methods of the eigenvalue problems and prove its reliability and efficiency based on two assumptions concerning both the weak continuity and the weak orthogonality of the nonconforming finite element spaces, respectively. In addition, the author examines these two assumptions for those nonconforming methods checked in literature for the Laplace, Stokes, and the linear elasticity problems.
Institute of Scientific and Technical Information of China (English)
吴颖; 罗亚军; 杨晓雪
2003-01-01
We present a novel formalism for energy eigenvalue problems when the corresponding Hamiltonians can be expressed as a function of an angular momentum. The problems are turned into finding operator polynomials by solving a c-number differential equation. Simple and efficient computer-aided analytical and numerical methods may be developed based on the formalism.
A Jacobi-Davidson type method for a right definite two-parameter eigenvalue problem
Hochstenbach, M.; Plestenjak, B.
2001-01-01
We present a new numerical iterative method for computing selected eigenpairs of a right definite two-parameter eigenvalue problem. The method works even without good initial approximations and is able to tackle large problems that are too expensive for existing methods. The new method is similar
An Approach to Some Non-Classical Eigenvalue Problems of Structural Dynamics
Directory of Open Access Journals (Sweden)
Sandi Horea
2015-12-01
Full Text Available Two main shortcomings of common formulations, encountered in the literature concerning the linear problems of structural dynamics are revealed: the implicit, not discussed, postulation, of the use of Kelvin – Voigt constitutive laws (which is often infirmed by experience and the calculation difficulties involved by the attempts to use other constitutive laws. In order to overcome these two categories of shortcomings, the use of the bilateral Laplace – Carson transformation is adopted. Instead of the dependence on time, t, of a certain function f (t, the dependence of its image f# (p on the complex parameter p = χ + iω (ω: circular frequency will occur. This leads to the formulation of associated non-classical eigenvalue problems. The basic relations satisfied by the eigenvalues λr#(p and the eigenvectors vr#(p of dynamic systems are examined (among other, the property of orthogonality of eigenvectors is replaced by the property of pseudo-orthogonality. The case of points p = p’, where multiple eigenvalues occur and where, as a rule, chains of principal vectors are to be considered, is discussed. An illustrative case, concerning a non-classical eigenvalue problem, is presented. Plots of variation along the ω axis, for the real and imaginary components of eigenvalues and eigenvectors, are presented. A brief final discussion closes the paper.
TWO-GRID DISCRETIZATION SCHEMES OF THE NONCONFORMING FEM FOR EIGENVALUE PROBLEMS
Institute of Scientific and Technical Information of China (English)
Yidu Yang
2009-01-01
This paper extends the two-grid discretization scheme of the conforming finite elements proposed by Xu and Zhou (Math. Comput., 70 (2001), pp.17-25) to the nonconforming finite elements for eigenvalue problems. In particular, two two-grid discretization schemes based on Rayleigh quotient technique are proposed. By using these new schemes, the solution of an eigenvalue problem on a fine mesh is reduced to that on a much coarser mesh together with the solution of a linear algebraic system on the fine mesh. The resulting solution still maintains an asymptotically optimal accuracy. Comparing with the two-grid discretization scheme of the conforming finite elements, the main advantages of our new schemes are twofold when the mesh size is small enough. First, the lower bounds of the exact eigenvalues in our two-grid discretization schemes can be obtained. Second, the first eigenvalue given by the new schemes has much better accuracy than that obtained by solving the eigenvalue problems on the fine mesh directly.
Kochunas, Brendan; Fitzgerald, Andrew; Larsen, Edward
2017-09-01
A central problem in nuclear reactor analysis is calculating solutions of steady-state k-eigenvalue problems with thermal hydraulic feedback. In this paper we propose and utilize a model problem that permits the theoretical analysis of iterative schemes for solving such problems. To begin, we discuss a model problem (with nonlinear cross section feedback) and its justification. We proceed with a Fourier analysis for source iteration schemes applied to the model problem. Then we analyze commonly-used iteration schemes involving non-linear diffusion acceleration and feedback. For each scheme we show (1) that they are conditionally stable, (2) the conditions that lead to instability, and (3) that traditional relaxation approaches can improve stability. Lastly, we propose a new iteration scheme that theory predicts is an improvement upon the existing methods.
An Implementation and Evaluation of the AMLS Method for SparseEigenvalue Problems
Energy Technology Data Exchange (ETDEWEB)
Gao, Weiguo; Li, Xiaoye S.; Yang, Chao; Bai, Zhaojun
2006-02-14
We describe an efficient implementation and present aperformance study of an algebraic multilevel sub-structuring (AMLS)method for sparse eigenvalue problems. We assess the time and memoryrequirements associated with the key steps of the algorithm, and compareitwith the shift-and-invert Lanczos algorithm in computational cost. Oureigenvalue problems come from two very different application areas: theaccelerator cavity design and the normal mode vibrational analysis of thepolyethylene particles. We show that the AMLS method, when implementedcarefully, is very competitive with the traditional method in broadapplication areas, especially when large numbers of eigenvalues aresought.
A NUMERICAL CALCULATION METHOD FOR EIGENVALUE PROBLEMS OF NONLINEAR INTERNAL WAVES
Institute of Scientific and Technical Information of China (English)
SHI Xin-gang; FAN Zhi-song; LIU Hai-long
2009-01-01
Generally speaking, the background shear current U(z)must be taken into account in eigenvalue problems of nonlinear internal waves in ocean, as is different from those of linear internal waves. A numerical calculation method for eigenvalue problems of nonlinear internal waves is presented in this paper on the basis of the Thompson-Haskell's calculation method. As an application of this method, at a station (21°N, 117°15′E) in the South China Sea, a modal structure and parameters of nonlinear internal waves are calculated, and the results closely agree with the calculated results based on observation by Yang et al..
Perturbation of a Multiple Eigenvalue in the Benard Problem for Two Fluid Layers.
1984-12-01
EIGENVAWUE IN THlE BENARtD PROBLEM FOR TWO FLUID LAYERS Ca O~ Yuriko Renardy and Michael Renardy MUathematics Research Center University of Wisconsin...OF WISCONSIN - MADISON MATHEMATICS RESEARCH CENTER PERTUBBATION OF A MULTIPLE EIGENVALUE IN THE BENARD PROBLEM FOR TWO FLUID LAYERS Yuriko Renardy and...PROBLEM FOR TWO FLUID LAYERS Yuriko Renardy and Michael Renardy 1. INTRODUCTION In the B6nard problem for one fluid, "exchange of stabilities" holds
Minimization and error estimates for a class of the nonlinear Schrodinger eigenvalue problems
Institute of Scientific and Technical Information of China (English)
MurongJIANG; JiachangSUN
2000-01-01
It is shown that the nonlinear eigenvaiue problem can be transformed into a constrained functional problem. The corresponding minimal function is a weak solution of this nonlinear problem. In this paper, one type of the energy functional for a class of the nonlinear SchrSdinger eigenvalue problems is proposed, the existence of the minimizing solution is proved and the error estimate is given out.
Cakoni, Fioralba; Haddar, Houssem
2013-10-01
In inverse scattering theory, transmission eigenvalues can be seen as the extension of the notion of resonant frequencies for impenetrable objects to the case of penetrable dielectrics. The transmission eigenvalue problem is a relatively late arrival to the spectral theory of partial differential equations. Its first appearance was in 1986 in a paper by Kirsch who was investigating the denseness of far-field patterns for scattering solutions of the Helmholtz equation or, in more modern terminology, the injectivity of the far-field operator [1]. The paper of Kirsch was soon followed by a more systematic study by Colton and Monk in the context of developing the dual space method for solving the inverse scattering problem for acoustic waves in an inhomogeneous medium [2]. In this paper they showed that for a spherically stratified media transmission eigenvalues existed and formed a discrete set. Numerical examples were also given showing that in principle transmission eigenvalues could be determined from the far-field data. This first period of interest in transmission eigenvalues was concluded with papers by Colton et al in 1989 [3] and Rynne and Sleeman in 1991 [4] showing that for an inhomogeneous medium (not necessarily spherically stratified) transmission eigenvalues, if they existed, formed a discrete set. For the next seventeen years transmission eigenvalues were ignored. This was mainly due to the fact that, with the introduction of various sampling methods to determine the shape of an inhomogeneous medium from far-field data, transmission eigenvalues were something to be avoided and hence the fact that transmission eigenvalues formed at most a discrete set was deemed to be sufficient. In addition, questions related to the existence of transmission eigenvalues or the structure of associated eigenvectors were recognized as being particularly difficult due to the nonlinearity of the eigenvalue problem and the special structure of the associated transmission
A Nonlinera Krylov Accelerator for the Boltzmann k-Eigenvalue Problem
Calef, Matthew T; Warsa, James S; Berndt, Markus; Carlson, Neil N
2011-01-01
We compare variants of Anderson Mixing with the Jacobian-Free Newton-Krylov and Broyden methods applied to the k-eigenvalue formulation of the linear Boltzmann transport equation. We present evidence that one variant of Anderson Mixing finds solutions in the fewest number of iterations. We examine and strengthen theoretical results of Anderson Mixing applied to linear problems.
Photonic Band Structure of Dispersive Metamaterials Formulated as a Hermitian Eigenvalue Problem
Raman, Aaswath
2010-02-26
We formulate the photonic band structure calculation of any lossless dispersive photonic crystal and optical metamaterial as a Hermitian eigenvalue problem. We further show that the eigenmodes of such lossless systems provide an orthonormal basis, which can be used to rigorously describe the behavior of lossy dispersive systems in general. © 2010 The American Physical Society.
Bonito, Andrea
2011-01-01
We propose and analyze an approximation technique for the Maxwell eigenvalue problem using H1-conforming finite elements. The key idea consists of considering a mixed method controlling the divergence of the electric field in a fractional Sobolev space H-α with α ∈ (1/2, 1). The method is shown to be convergent and spectrally correct. © 2011 American Mathematical Society.
The linearization of boundary eigenvalue problems and reproducing kernel Hilbert spaces
Ćurgus, Branko; Dijksma, Aad; Read, Tom
2001-01-01
The boundary eigenvalue problems for the adjoint of a symmetric relation S in a Hilbert space with finite, not necessarily equal, defect numbers, which are related to the selfadjoint Hilbert space extensions of S are characterized in terms of boundary coefficients and the reproducing kernel Hilbert
Some remarks on the optimization of eigenvalue problems involving the p-Laplacian
Directory of Open Access Journals (Sweden)
Wacław Pielichowski
2008-01-01
Full Text Available Given a bounded domain \\(\\Omega \\subset \\mathbb{R}^n\\, numbers \\(p \\gt 1\\, \\(\\alpha \\geq 0\\ and \\(A \\in [0,|\\Omega |]\\, consider the optimization problem: find a subset \\(D \\subset \\Omega \\, of measure \\(A\\, for which the first eigenvalue of the operator \\(u\\mapsto -\\text{div} (|\
Eigenvalues of a baroclinic stability problem with Ekman damping
Antar, B. N.; Fowlis, W. W.
1980-01-01
An analytical solution is presented for the baroclinic stability problem of a Boussinesq fluid in a beta-plane channel with Ekman suction boundary conditions. All of the modes, stable and unstable, belonging to this problem are identified. It is found that an unstable mode exists for only a certain range of values of the Burger number. The value of the Burger number at the upper limit of this range increases as the Ekman number decreases. Beyond this upper limit only a damped mode exists. It is also found that this transition in parameter space from the unstable to the stable mode occurs in a discontinuous manner.
Eigenvalues of the time—dependent fluid flow problem I
Directory of Open Access Journals (Sweden)
El-Sayed M. Zayed
1990-01-01
Full Text Available The direct and inverse boundary value problems for the linear unsteady viscous fluid flow through a closed conduit of a circular annular cross-section Ω with arbitrary time-dependent pressure gradient under the third boundary conditions have been investigated.
Determination of Electromagnetic Source Direction as an Eigenvalue Problem
Martínez-Oliveros, Juan C; Bale, Stuart D; Krucker, Säm
2012-01-01
Low-frequency solar and interplanetary radio bursts are generated at frequencies below the ionospheric plasma cutoff and must therefore be measured in space, with deployable antenna systems. The problem of measuring both the general direction and polarization of an electromagnetic source is commonly solved by iterative fitting methods such as linear regression that deal simultaneously with both directional and polarization parameters. We have developed a scheme that separates the problem of deriving the source direction from that of determining the polarization, avoiding iteration in a multi-dimensional manifold. The crux of the method is to first determine the source direction independently of concerns as to its polarization. Once the source direction is known, its direct characterization in terms of Stokes vectors in a single iteration if desired, is relatively simple. This study applies the source-direction determination to radio signatures of flares received by STEREO. We studied two previously analyzed r...
Positive solutions and eigenvalues of nonlocal boundary-value problems
Directory of Open Access Journals (Sweden)
Jifeng Chu
2005-07-01
Full Text Available We study the ordinary differential equation $x''+lambda a(tf(x=0$ with the boundary conditions $x(0=0$ and $x'(1=int_{eta}^{1}x'(sdg(s$. We characterize values of $lambda$ for which boundary-value problem has a positive solution. Also we find appropriate intervals for $lambda$ so that there are two positive solutions.
Eigenvalues of boundary value problems for higher order differential equations
Wong, Patricia J. Y.; Agarwal, Ravi P.
1996-01-01
We shall consider the boundary value problem y ( n ) + λ Q ( t , y , y 1 , ⋅ ⋅ ⋅ , y ( n − 2 ) ) = λ P ( t , y , y 1 , ⋅ ⋅ ⋅ , y ( n − 1 ) ) , n ≥ 2 , t ∈ ( 0 , 1 ) , y ( i ) ( 0 ) = 0 , 0 ≤ i ≤ n − 3 , α y ( n − 2 ) ( 0 ) − β y ( n − 1 ) ( 0 ) = 0 , γ y ( n − 2 ) ( 1 ) + δ y ( n...
Eigenvalues of boundary value problems for higher order differential equations
Patricia J. Y. Wong; Agarwal, Ravi P.
1996-01-01
We shall consider the boundary value problem y ( n ) + λ Q ( t , y , y 1 , ⋅ ⋅ ⋅ , y ( n − 2 ) ) = λ P ( t , y , y 1 , ⋅ ⋅ ⋅ , y ( n − 1 ) ) , n ≥ 2 , t ∈ ( 0 , 1 ) , y ( i ) ( 0 ) = 0 , 0 ≤ i ≤ n − 3 , α y ( n − 2 ) ( 0 ) − β y ( n − 1 ) ( 0 ) = 0 , γ y ( n − 2 ) ( 1 ) + δ y ( n...
Nodal Solutions for a Nonlinear Fourth-Order Eigenvalue Problem
Institute of Scientific and Technical Information of China (English)
Ru Yun MA; Bevan THOMPSON
2008-01-01
We are concerned with determining the values of λ, for which there exist nodal solutions of the fourth-order boundary value problem y =λa(x)f(y),00 for all u ≠0. We give conditions on the ratio f (s)/s,at infinity and zero, that guarantee the existence of nodal solutions.The proof of our main results is based upon bifurcation techniques.
Periodic-parabolic eigenvalue problems with a large parameter and degeneration
Daners, Daniel; Thornett, Christopher
2016-07-01
We consider a periodic-parabolic eigenvalue problem with a non-negative potential λm vanishing on a non-cylindrical domain Dm satisfying conditions similar to those for the parabolic maximum principle. We show that the limit as λ → ∞ leads to a periodic-parabolic problem on Dm having a periodic-parabolic principal eigenvalue and eigenfunction which are unique in some sense. We substantially improve a result from [Du and Peng, Trans. Amer. Math. Soc. 364 (2012), p. 6039-6070]. At the same time we offer a different approach based on a periodic-parabolic initial boundary value problem. The results are motivated by an analysis of the asymptotic behaviour of positive solutions to semilinear logistic periodic-parabolic problems with temporal and spacial degeneracies.
A Projection free method for Generalized Eigenvalue Problem with a nonsmooth Regularizer.
Hwang, Seong Jae; Collins, Maxwell D; Ravi, Sathya N; Ithapu, Vamsi K; Adluru, Nagesh; Johnson, Sterling C; Singh, Vikas
2015-12-01
Eigenvalue problems are ubiquitous in computer vision, covering a very broad spectrum of applications ranging from estimation problems in multi-view geometry to image segmentation. Few other linear algebra problems have a more mature set of numerical routines available and many computer vision libraries leverage such tools extensively. However, the ability to call the underlying solver only as a "black box" can often become restrictive. Many 'human in the loop' settings in vision frequently exploit supervision from an expert, to the extent that the user can be considered a subroutine in the overall system. In other cases, there is additional domain knowledge, side or even partial information that one may want to incorporate within the formulation. In general, regularizing a (generalized) eigenvalue problem with such side information remains difficult. Motivated by these needs, this paper presents an optimization scheme to solve generalized eigenvalue problems (GEP) involving a (nonsmooth) regularizer. We start from an alternative formulation of GEP where the feasibility set of the model involves the Stiefel manifold. The core of this paper presents an end to end stochastic optimization scheme for the resultant problem. We show how this general algorithm enables improved statistical analysis of brain imaging data where the regularizer is derived from other 'views' of the disease pathology, involving clinical measurements and other image-derived representations.
{\\it Ab initio} nuclear structure - the large sparse matrix eigenvalue problem
Vary, James P; Ng, Esmond; Yang, Chao; Sosonkina, Masha
2009-01-01
The structure and reactions of light nuclei represent fundamental and formidable challenges for microscopic theory based on realistic strong interaction potentials. Several {\\it ab initio} methods have now emerged that provide nearly exact solutions for some nuclear properties. The {\\it ab initio} no core shell model (NCSM) and the no core full configuration (NCFC) method, frame this quantum many-particle problem as a large sparse matrix eigenvalue problem where one evaluates the Hamiltonian matrix in a basis space consisting of many-fermion Slater determinants and then solves for a set of the lowest eigenvalues and their associated eigenvectors. The resulting eigenvectors are employed to evaluate a set of experimental quantities to test the underlying potential. For fundamental problems of interest, the matrix dimension often exceeds $10^{10}$ and the number of nonzero matrix elements may saturate available storage on present-day leadership class facilities. We survey recent results and advances in solving t...
Solving eigenvalue problems on curved surfaces using the Closest Point Method
Macdonald, Colin B.; Brandman, Jeremy; Ruuth, Steven J.
2011-09-01
Eigenvalue problems are fundamental to mathematics and science. We present a simple algorithm for determining eigenvalues and eigenfunctions of the Laplace-Beltrami operator on rather general curved surfaces. Our algorithm, which is based on the Closest Point Method, relies on an embedding of the surface in a higher-dimensional space, where standard Cartesian finite difference and interpolation schemes can be easily applied. We show that there is a one-to-one correspondence between a problem defined in the embedding space and the original surface problem. For open surfaces, we present a simple way to impose Dirichlet and Neumann boundary conditions while maintaining second-order accuracy. Convergence studies and a series of examples demonstrate the effectiveness and generality of our approach.
Solving eigenvalue problems on curved surfaces using the Closest Point Method
Macdonald, Colin B; Ruuth, Steven J
2011-01-01
Eigenvalue problems are fundamental to mathematics and science. We present a simple algorithm for determining eigenvalues and eigenfunctions of the Laplace--Beltrami operator on rather general curved surfaces. Our algorithm, which is based on the Closest Point Method, relies on an embedding of the surface in a higher-dimensional space, where standard Cartesian finite difference and interpolation schemes can be easily applied. We show that there is a one-to-one correspondence between a problem defined in the embedding space and the original surface problem. For open surfaces, we present a simple way to impose Dirichlet and Neumann boundary conditions while maintaining second-order accuracy. Convergence studies and a series of examples demonstrate the effectiveness and generality of our approach.
Solving eigenvalue problems on curved surfaces using the Closest Point Method
Macdonald, Colin B.
2011-06-01
Eigenvalue problems are fundamental to mathematics and science. We present a simple algorithm for determining eigenvalues and eigenfunctions of the Laplace-Beltrami operator on rather general curved surfaces. Our algorithm, which is based on the Closest Point Method, relies on an embedding of the surface in a higher-dimensional space, where standard Cartesian finite difference and interpolation schemes can be easily applied. We show that there is a one-to-one correspondence between a problem defined in the embedding space and the original surface problem. For open surfaces, we present a simple way to impose Dirichlet and Neumann boundary conditions while maintaining second-order accuracy. Convergence studies and a series of examples demonstrate the effectiveness and generality of our approach. © 2011 Elsevier Inc.
Solving Large Scale Nonlinear Eigenvalue Problem in Next-Generation Accelerator Design
Energy Technology Data Exchange (ETDEWEB)
Liao, Ben-Shan; Bai, Zhaojun; /UC, Davis; Lee, Lie-Quan; Ko, Kwok; /SLAC
2006-09-28
A number of numerical methods, including inverse iteration, method of successive linear problem and nonlinear Arnoldi algorithm, are studied in this paper to solve a large scale nonlinear eigenvalue problem arising from finite element analysis of resonant frequencies and external Q{sub e} values of a waveguide loaded cavity in the next-generation accelerator design. They present a nonlinear Rayleigh-Ritz iterative projection algorithm, NRRIT in short and demonstrate that it is the most promising approach for a model scale cavity design. The NRRIT algorithm is an extension of the nonlinear Arnoldi algorithm due to Voss. Computational challenges of solving such a nonlinear eigenvalue problem for a full scale cavity design are outlined.
Ltaief, Hatem
2012-01-01
This paper proposes an efficient implementation of the generalized symmetric eigenvalue problem on multicore architecture. Based on a four-stage approach and tile algorithms, the original problem is first transformed into a standard symmetric eigenvalue problem by computing the Cholesky factorization of the right hand side symmetric definite positive matrix (first stage), and applying the inverse of the freshly computed triangular Cholesky factors to the original dense symmetric matrix of the problem (second stage). Calculating the eigenpairs of the resulting problem is then equivalent to the eigenpairs of the original problem. The computation proceeds by reducing the updated dense symmetric matrix to symmetric band form (third stage). The band structure is further reduced by applying a bulge chasing procedure, which annihilates the extra off-diagonal entries using orthogonal transformations (fourth stage). More details on the third and fourth stage can be found in Haidar et al. [Accepted at SC\\'11, November 2011]. The eigenvalues are then calculated from the tridiagonal form using the standard LAPACK QR algorithm (i.e., DTSEQR routine), while the complex and challenging eigenvector computations will be addressed in a companion paper. The tasks from the various stages can concurrently run in an out-of-order fashion. The data dependencies are cautiously tracked by the dynamic runtime system environment QUARK, which ensures the dependencies are not violated for numerical correctness purposes. The obtained tile four-stage generalized symmetric eigenvalue solver significantly outperforms the state-of-the-art numerical libraries (up to 21-fold speed up against multithreaded LAPACK with optimized multithreaded MKL BLAS and up to 4-fold speed up against the corresponding routine from the commercial numerical software Intel MKL) on four sockets twelve cores AMD system with a 24000×24000 matrix size. © 2012 The authors and IOS Press. All rights reserved.
Mode decomposition of nonlinear eigenvalue problems and application in flow stability
Institute of Scientific and Technical Information of China (English)
高军; 罗纪生
2014-01-01
Direct numerical simulations are carried out with different disturbance forms introduced into the inlet of a flat plate boundary layer with the Mach number 4.5. According to the biorthogonal eigenfunction system of the linearized Navier-Stokes equations and the adjoint equations, the decomposition of the direct numerical simulation results into the discrete normal mode is easily realized. The decomposition coefficients can be solved by doing the inner product between the numerical results and the eigenfunctions of the adjoint equations. For the quadratic polynomial eigenvalue problem, the inner product operator is given in a simple form, and it is extended to an N th-degree polynomial eigenvalue problem. The examples illustrate that the simplified mode decomposition is available to analyze direct numerical simulation results.
Júdice, Joaquim; Raydan, Marcos; Rosa, Silvério; Santos, Sandra
2008-04-01
This paper is devoted to the eigenvalue complementarity problem (EiCP) with symmetric real matrices. This problem is equivalent to finding a stationary point of a differentiable optimization program involving the Rayleigh quotient on a simplex (Queiroz et al., Math. Comput. 73, 1849-1863, 2004). We discuss a logarithmic function and a quadratic programming formulation to find a complementarity eigenvalue by computing a stationary point of an appropriate merit function on a special convex set. A variant of the spectral projected gradient algorithm with a specially designed line search is introduced to solve the EiCP. Computational experience shows that the application of this algorithm to the logarithmic function formulation is a quite efficient way to find a solution to the symmetric EiCP.
Diffeomorphism invariant eigenvalue problem for metric perturbations in a bounded region
Marachevsky, V N; Marachevsky, Valeri; Vassilevich, Dmitri
1995-01-01
We suggest a method of construction of general diffeomorphism invariant boundary conditions for metric fluctuations. The case of d+1 dimensional Euclidean disk is studied in detail. The eigenvalue problem for the Laplace operator on metric perturbations is reduced to that on d-dimensional vector, tensor and scalar fields. Explicit form of the eigenfunctions of the Laplace operator is derived. We also study restrictions on boundary conditions which are imposed by hermiticity of the Laplace operator.
Lyapunov inequalities for the periodic boundary value problem at higher eigenvalues
Canada, Antonio
2009-01-01
This paper is devoted to provide some new results on Lyapunov type inequalities for the periodic boundary value problem at higher eigenvalues. Our main result is derived from a detailed analysis on the number and distribution of zeros of nontrivial solutions and their first derivatives, together with the study of some special minimization problems, where the Lagrange multiplier Theorem plays a fundamental role. This allows to obtain the optimal constants. Our applications include the Hill's equation where we give some new conditions on its stability properties and also the study of periodic and nonlinear problems at resonance where we show some new conditions which allow to prove the existence and uniqueness of solutions.
A case against a divide and conquer approach to the nonsymmetric eigenvalue problem
Energy Technology Data Exchange (ETDEWEB)
Jessup, E.R.
1991-12-01
Divide and conquer techniques based on rank-one updating have proven fast, accurate, and efficient in parallel for the real symmetric tridiagonal and unitary eigenvalue problems and for the bidiagonal singular value problem. Although the divide and conquer mechanism can also be adapted to the real nonsymmetric eigenproblem in a straightforward way, most of the desirable characteristics of the other algorithms are lost. In this paper, we examine the problems of accuracy and efficiency that can stand in the way of a nonsymmetric divide and conquer eigensolver based on low-rank updating. 31 refs., 2 figs.
Alzahrani, Faris S.; Abbas, Ibrahim A.
2016-08-01
The present paper is devoted to the study of a two-dimensional thermal shock problem with weak, normal and strong conductivity using the eigenvalue approach. The governing equations are taken in the context of the new consideration of heat conduction with fractional order generalized thermoelasticity with the Lord-Shulman model (LS model). The bounding surface of the half-space is taken to be traction free and subjected to a time-dependent thermal shock. The Laplace and the exponential Fourier transform techniques are used to obtain the analytical solutions in the transformed domain by the eigenvalue approach. Numerical computations have been done for copper-like material for weak, normal and strong conductivity and the results are presented graphically to estimate the effects of the fractional order parameter.
Haidar, Azzam
2012-01-01
Classical solvers for the dense symmetric eigenvalue problem suffer from the first step, which involves a reduction to tridiagonal form that is dominated by the cost of accessing memory during the panel factorization. The solution is to reduce the matrix to a banded form, which then requires the eigenvalues of the banded matrix to be computed. The standard divide and conquer algorithm can be modified for this purpose. The paper combines this insight with tile algorithms that can be scheduled via a dynamic runtime system to multicore architectures. A detailed analysis of performance and accuracy is included. Performance improvements of 14-fold and 4-fold speedups are reported relative to LAPACK and Intel\\'s Math Kernel Library.
POSITIVE SOLUTIONS OF A NONLINEAR THREE-POINT EIGENVALUE PROBLEM WITH INTEGRAL BOUNDARY CONDITIONS
Directory of Open Access Journals (Sweden)
FAOUZI HADDOUCHI
2015-11-01
Full Text Available In this paper, we study the existence of positive solutions of a three-point integral boundary value problem (BVP for the following second-order differential equation u''(t + \\lambda a(tf(u(t = 0; 0 0 is a parameter, 0 <\\eta < 1, 0 <\\alpha < 1/{\\eta}. . By using the properties of the Green's function and Krasnoselskii's fixed point theorem on cones, the eigenvalue intervals of the nonlinear boundary value problem are considered, some sufficient conditions for the existence of at least one positive solutions are established.
Chen, Yong
2010-01-01
For an indecomposable $3\\times 3$ stochastic matrix (i.e., 1-step transition probability matrix) with coinciding negative eigenvalues, a new necessary and sufficient condition of the imbedding problem for time homogeneous Markov chains is shown by means of an alternate parameterization of the transition rate matrix (i.e., intensity matrix, infinitesimal generator), which avoids calculating matrix logarithm or matrix square root. In addition, an implicit description of the imbedding problem for the $3\\times 3$ stochastic matrix in Johansen [J. Lond. Math. Soc., 8, 345-351. (1974)] is pointed out.
Lin, Lin
2016-01-01
We present the first systematic work for deriving a posteriori error estimates for general non-polynomial basis functions in an interior penalty discontinuous Galerkin (DG) formulation for solving eigenvalue problems associated with second order linear operators. Eigenvalue problems of such types play important roles in scientific and engineering applications, particularly in theoretical chemistry, solid state physics and material science. Based on the framework developed in [{\\it L. Lin, B. Stamm, http://dx.doi.org/10.1051/m2an/2015069}] for second order PDEs, we develop residual type upper and lower bound error estimates for measuring the a posteriori error for eigenvalue problems. The main merit of our method is that the method is parameter-free, in the sense that all but one solution-dependent constants appearing in the upper and lower bound estimates are explicitly computable by solving local and independent eigenvalue problems, and the only non-computable constant can be reasonably approximated by a com...
Zhidkov, E P; Solovieva, T M
2001-01-01
The spectral problems with the eigenvalue-depending operator usually appear when the relative variants of the Schroedinger equation are considered in the impulse space. The eigenvalues and eigenfunctions calculation error caused by the numerical solving of such equations is the sum of the error entering the approximation of a continuous equation by the discrete equations systems with the help of the Bubnov-Galerkine method and the iterative method one. It is shown that the iterative method error is one-two order smaller than the problem of the discretisation one. Hence, the eigenvalues and eigenfunctions calculation accuracy of the spectral problem with the eigenvalue-depending operator is not worse than the linear spectral problem solution accuracy.
Solutions of fractional diffusion problems
Directory of Open Access Journals (Sweden)
Rabha W. Ibrahim
2010-10-01
Full Text Available Using the concept of majorant functions, we prove the existence and uniqueness of holomorphic solutions to nonlinear fractional diffusion problems. The analytic continuation of these solutions is studied and the singularity for two cases are posed.
A Posterior Error Analysis for the Nonconforming Discretization of Stokes Eigenvalue Problem
Institute of Scientific and Technical Information of China (English)
Shang Hui JIA; Fu Sheng LUO; He Hu XIE
2014-01-01
In this paper, we present a posteriori error estimator for the nonconforming finite element approximation, including using Crouzeix-Raviart element and extended Crouzeix-Raviart element, of the Stokes eigenvalue problem. With the technique of Helmholtz decomposition, we first give out a posteriori error estimator and prove it as the global upper bound and local lower bound of the approximation error. Then, by deleting a jump term in the indicator, another simpler but equivalent indicator is obtained. Some numerical experiments are provided to verify our analysis.
Li, Huai-Fan
2013-01-01
We take advantage of the Sturm-Liouville eigenvalue problem to analytically study the holographic insulator/superconductor phase transition in the probe limit. The interesting point is that this analytical method can not only estimate the most stable mode of the phase transition, but also the second stable mode. We find that this analytical method perfectly matches with other numerical methods, such as the shooting method. Besides, we argue that only Dirichlet boundary condition of the trial function is enough under certain circumstances, which will lead to a more precise estimation. This relaxation for the boundary condition of the trial function is first observed in this paper as far as know.
Spectral Method with the Tensor-Product Nodal Basis for the Steklov Eigenvalue Problem
Directory of Open Access Journals (Sweden)
Xuqing Zhang
2013-01-01
Full Text Available This paper discusses spectral method with the tensor-product nodal basis at the Legendre-Gauss-Lobatto points for solving the Steklov eigenvalue problem. A priori error estimates of spectral method are discussed, and based on the work of Melenk and Wohlmuth (2001, a posterior error estimator of the residual type is given and analyzed. In addition, this paper combines the shifted-inverse iterative method and spectral method to establish an efficient scheme. Finally, numerical experiments with MATLAB program are reported.
Recurrence relation for the 6j-symbol of suq(2) as a symmetric eigenvalue problem
Khavkine, Igor
2015-08-01
A well-known recurrence relation for the 6j-symbol of the quantum group suq(2) is realized as a tridiagonal, symmetric eigenvalue problem. This formulation can be used to implement an efficient numerical evaluation algorithm, taking advantage of existing specialized numerical packages. For convenience, all formulas relevant for such an implementation are collected in Appendix A. This realization is a byproduct of an alternative proof of the recurrence relation, which generalizes a classical (q = 1) result of Schulten and Gordon and uses the diagrammatic spin network formalism of Temperley-Lieb recoupling theory to simplify intermediate calculations.
A Structured Approach to Solve the Inverse Eigenvalue Problem for a Beam with Added Mass
Directory of Open Access Journals (Sweden)
Farhad Mir Hosseini
2014-01-01
Full Text Available The problem of determining the eigenvalues of a vibrational system having multiple lumped attachments has been investigated extensively. However, most of the research conducted in this field focuses on determining the natural frequencies of the combined system assuming that the characteristics of the combined vibrational system are known (forward problem. A problem of great interest from the point of view of engineering design is the ability to impose certain frequencies on the vibrational system or to avoid certain frequencies by modifying the characteristics of the vibrational system (inverse problem. In this paper, a method to impose two natural frequencies on a dynamical system consisting of an Euler-Bernoulli beam and carrying a single mass attachment is evaluated.
Energy Technology Data Exchange (ETDEWEB)
Stathopoulos, A.; Fischer, C.F. [Vanderbilt Univ., Nashville, TN (United States); Saad, Y.
1994-12-31
The solution of the large, sparse, symmetric eigenvalue problem, Ax = {lambda}x, is central to many scientific applications. Among many iterative methods that attempt to solve this problem, the Lanczos and the Generalized Davidson (GD) are the most widely used methods. The Lanczos method builds an orthogonal basis for the Krylov subspace, from which the required eigenvectors are approximated through a Rayleigh-Ritz procedure. Each Lanczos iteration is economical to compute but the number of iterations may grow significantly for difficult problems. The GD method can be considered a preconditioned version of Lanczos. In each step the Rayleigh-Ritz procedure is solved and explicit orthogonalization of the preconditioned residual ((M {minus} {lambda}I){sup {minus}1}(A {minus} {lambda}I)x) is performed. Therefore, the GD method attempts to improve convergence and robustness at the expense of a more complicated step.
Pak, Chan-gi; Lung, Shu
2009-01-01
Modern airplane design is a multidisciplinary task which combines several disciplines such as structures, aerodynamics, flight controls, and sometimes heat transfer. Historically, analytical and experimental investigations concerning the interaction of the elastic airframe with aerodynamic and in retia loads have been conducted during the design phase to determine the existence of aeroelastic instabilities, so called flutter .With the advent and increased usage of flight control systems, there is also a likelihood of instabilities caused by the interaction of the flight control system and the aeroelastic response of the airplane, known as aeroservoelastic instabilities. An in -house code MPASES (Ref. 1), modified from PASES (Ref. 2), is a general purpose digital computer program for the analysis of the closed-loop stability problem. This program used subroutines given in the International Mathematical and Statistical Library (IMSL) (Ref. 3) to compute all of the real and/or complex conjugate pairs of eigenvalues of the Hessenberg matrix. For high fidelity configuration, these aeroelastic system matrices are large and compute all eigenvalues will be time consuming. A subspace iteration method (Ref. 4) for complex eigenvalues problems with nonsymmetric matrices has been formulated and incorporated into the modified program for aeroservoelastic stability (MPASES code). Subspace iteration method only solve for the lowest p eigenvalues and corresponding eigenvectors for aeroelastic and aeroservoelastic analysis. In general, the selection of p is ranging from 10 for wing flutter analysis to 50 for an entire aircraft flutter analysis. The application of this newly incorporated code is an experiment known as the Aerostructures Test Wing (ATW) which was designed by the National Aeronautic and Space Administration (NASA) Dryden Flight Research Center, Edwards, California to research aeroelastic instabilities. Specifically, this experiment was used to study an instability
Directory of Open Access Journals (Sweden)
Jie Liu
2014-01-01
discusses the nonconforming rotated Q1 finite element computable upper bound a posteriori error estimate of the boundary value problem established by M. Ainsworth and obtains efficient computable upper bound a posteriori error indicators for the eigenvalue problem associated with the boundary value problem. We extend the a posteriori error estimate to the Steklov eigenvalue problem and also derive efficient computable upper bound a posteriori error indicators. Finally, through numerical experiments, we verify the validity of the a posteriori error estimate of the boundary value problem; meanwhile, the numerical results show that the a posteriori error indicators of the eigenvalue problem and the Steklov eigenvalue problem are effective.
Numerical approximation on computing partial sum of nonlinear Schroedinger eigenvalue problems
Institute of Scientific and Technical Information of China (English)
JiachangSUN; DingshengWANG; 等
2001-01-01
In computing electronic structure and energy band in the system of multiparticles,quite a large number of problems are to obtain the partial sum of the densities and energies by using “First principle”。In the ordinary method,the so-called self-consistency approach,the procedure is limited to a small scale because of its high computing complexity.In this paper,the problem of computing the partial sum for a class of nonlinear Schroedinger eigenvalue equations is changed into the constrained functional minimization.By space decompostion and Rayleigh-Schroedinger method,one approximating formula for the minimal is provided.The numerical experiments show that this formula is more precise and its quantity of computation is smaller.
Jarlebring, Elias; Michiels, Wim
2012-01-01
The partial Schur factorization can be used to represent several eigenpairs of a matrix in a numerically robust way. Different adaptions of the Arnoldi method are often used to compute partial Schur factorizations. We propose here a technique to compute a partial Schur factorization of a nonlinear eigenvalue problem (NEP). The technique is inspired by the algorithm in [8], now called the infinite Arnoldi method. The infinite Arnoldi method is a method designed for NEPs, and can be interpreted as Arnoldi's method applied to a linear infinite-dimensional operator, whose reciprocal eigenvalues are the solutions to the NEP. As a first result we show that the invariant pairs of the operator are equivalent to invariant pairs of the NEP. We characterize the structure of the invariant pairs of the operator and show how one can carry out a modification of the infinite Arnoldi method by respecting the structure. This also allows us to naturally add the feature known as locking. We nest this algorithm with an outer iter...
Amirkhanov, I V; Zhidkova, I E; Vasilev, S A
2000-01-01
Asymptotics of eigenfunctions and eigenvalues has been obtained for a singular perturbated relativistic analog of Schr`dinger equation. A singular convergence of asymptotic expansions of the boundary problems to degenerated problems is shown for a nonrelativistic Schr`dinger equation. The expansions obtained are in a good agreement with a numeric experiment.
Graph theory approach to the eigenvalue problem of large space structures
Reddy, A. S. S. R.; Bainum, P. M.
1981-01-01
Graph theory is used to obtain numerical solutions to eigenvalue problems of large space structures (LSS) characterized by a state vector of large dimensions. The LSS are considered as large, flexible systems requiring both orientation and surface shape control. Graphic interpretation of the determinant of a matrix is employed to reduce a higher dimensional matrix into combinations of smaller dimensional sub-matrices. The reduction is implemented by means of a Boolean equivalent of the original matrices formulated to obtain smaller dimensional equivalents of the original numerical matrix. Computation time becomes less and more accurate solutions are possible. An example is provided in the form of a free-free square plate. Linearized system equations and numerical values of a stiffness matrix are presented, featuring a state vector with 16 components.
A High-Performance Numerical Library for Solving Eigenvalue Problems: FEAST Solver v2.0 User's Guide
Polizzi, Eric
2012-01-01
The FEAST solver package is a free high-performance numerical library for solving the standard or generalized eigenvalue problem, and obtaining all the eigenvalues and eigenvectors within a given search interval. It is based on an innovative fast and stable numerical algorithm presented in Phys. Rev B Vol.79, p115112 (2009) - named the FEAST algorithm - which deviates fundamentally from the traditional Krylov subspace iteration based techniques (Arnoldi and Lanczos algorithms) or other Davidson-Jacobi techniques. The FEAST algorithm takes its inspiration from the density-matrix representation and contour integration technique in quantum mechanics. It is free from orthogonalization procedures, and its main computational tasks consist of solving very few inner independent linear systems with multiple right-hand sides and one reduced eigenvalue problem orders of magnitude smaller than the original one. The FEAST algorithm combines simplicity and efficiency and offers many important capabilities for achieving hig...
Transmission eigenvalues for elliptic operators
Hitrik, Michael; Ola, Petri; Päivärinta, Lassi
2010-01-01
A reduction of the transmission eigenvalue problem for multiplicative sign-definite perturbations of elliptic operators with constant coefficients to an eigenvalue problem for a non-selfadjoint compact operator is given. Sufficient conditions for the existence of transmission eigenvalues and completeness of generalized eigenstates for the transmission eigenvalue problem are derived. In the trace class case, the generic existence of transmission eigenvalues is established.
Institute of Scientific and Technical Information of China (English)
ZHANG Zhong-zhi; HAN Xu-li
2005-01-01
By using the characteristic properties of the anti-Hermitian generalized anti-Hamiltonian matrices, we prove some necessary and sufficient conditions of the solvability for algebra inverse eigenvalue problem of anti-Hermitian generalized anti-Hamiltonian matrices, and obtain a general expression of the solution to this problem. By using the properties of the orthogonal projection matrix, we also obtain the expression of the solution to optimal approximate problem of an n× n complex matrix under spectral restriction.
实对称五对角矩阵逆特征值问题%INVERSE EIGENVALUE PROBLEM FOR REAL SYMMETRIC FIVE-DIAGONAL MATRIX
Institute of Scientific and Technical Information of China (English)
王正盛
2002-01-01
In this paper, a kind of inverse eigenvalue problem which is the recon-struction of real symmetric five-diagonal matrix by three eigenvalues and corre-sponding eigenvectors is proposed. The solvability of the problem is disucssedand some sufficient and necessary conditions for existence of solution of thisproblem are given. Furthermore numerical algorithm and some numerical experi-ments are given.
Directory of Open Access Journals (Sweden)
Serguei I. Iakovlev
2013-01-01
Full Text Available It is shown that any \\(\\mu \\in \\mathbb{C}\\ is an infinite multiplicity eigenvalue of the Steklov smoothing operator \\(S_h\\ acting on the space \\(L^1_{loc}(\\mathbb{R}\\. For \\(\\mu \
Eigenvalue Problem for Nonlinear Fractional Differential Equations with Integral Boundary Conditions
Directory of Open Access Journals (Sweden)
Guotao Wang
2014-01-01
Full Text Available By employing known Guo-Krasnoselskii fixed point theorem, we investigate the eigenvalue interval for the existence and nonexistence of at least one positive solution of nonlinear fractional differential equation with integral boundary conditions.
MARG2D code. 1. Eigenvalue problem for two dimensional Newcomb equation
Energy Technology Data Exchange (ETDEWEB)
Tokuda, Shinji [Japan Atomic Energy Research Inst., Naka, Ibaraki (Japan). Naka Fusion Research Establishment; Watanabe, Tomoko
1997-10-01
A new method and a code MARG2D have been developed to solve the 2-dimensional Newcomb equation which plays an important role in the magnetohydrodynamic (MHD) stability analysis in an axisymmetric toroidal plasma such as a tokamak. In the present formulation, an eigenvalue problem is posed for the 2-D Newcomb equation, where the weight function (the kinetic energy integral) and the boundary conditions at rational surfaces are chosen so that an eigenfunction correctly behaves as the linear combination of the small solution and the analytical solutions around each of the rational surfaces. Thus, the difficulty on solving the 2-D Newcomb equation has been resolved. By using the MARG2D code, the ideal MHD marginally stable state can be identified for a 2-D toroidal plasma. The code is indispensable on computing the outer-region matching data necessary for the resistive MHD stability analysis. Benchmark with ERATOJ, an ideal MHD stability code, has been carried out and the MARG2D code demonstrates that it indeed identifies both stable and marginally stable states against ideal MHD motion. (author)
Positive Solutions for Two-Point Semipositone Right Focal Eigenvalue Problem
Directory of Open Access Journals (Sweden)
Yuguo Lin
2007-10-01
Full Text Available Krasnoselskii's fixed-point theorem in a cone is used to discuss the existence of positive solutions to semipositone right focal eigenvalue problems (Ã¢ÂˆÂ’1nÃ¢ÂˆÂ’pu(n(t=ÃŽÂ»f(t,u(t,u'(t,Ã¢Â€Â¦,u(pÃ¢ÂˆÂ’1(t, u(i(0=0, 0Ã¢Â‰Â¤iÃ¢Â‰Â¤pÃ¢ÂˆÂ’1, u(i(1=0, pÃ¢Â‰Â¤iÃ¢Â‰Â¤nÃ¢ÂˆÂ’1, where nÃ¢Â‰Â¥2, 1Ã¢Â‰Â¤pÃ¢Â‰Â¤nÃ¢ÂˆÂ’1 is fixed, f:[0,1]ÃƒÂ—[0,Ã¢ÂˆÂžpÃ¢Â†Â’(Ã¢ÂˆÂ’Ã¢ÂˆÂž,Ã¢ÂˆÂž is continuous with f(t,u1,u2,Ã¢Â€Â¦,upÃ¢Â‰Â¥Ã¢ÂˆÂ’M for some positive constant M.
1980-09-29
FOUNDATIONS OF EIGENVALUE DISTRIBUTION THEORY FOR GENERAL A NON--ETC(U) SEP 80 M MARCUS, M GOLDBERG, M NEWMAN AFOSR-79-0127 UNCLASSIFIED AFOSR-TR-80...September 1980 Title of Research: Foundations of Eigenvalue Distribution Theory for General & Nonnegative Matrices, Stability Criteria for Hyperbolic
Cotta, R. M.; Naveira-Cotta, C. P.; Knupp, D. C.; Zotin, J. L. Z.; Pontes, P. C.
2016-09-01
This lecture offers an updated review on the Generalized Integral Transform Technique (GITT), with focus on handling complex geometries, coupled problems, and nonlinear convection-diffusion, so as to illustrate some new application paradigms. Special emphasis is given to demonstrating novel developments, such as a single domain reformulation strategy that simplifies the treatment of complex geometries, an integral balance scheme in handling multiscale problems, the adoption of convective eigenvalue problems in dealing with strongly convective formulations, and the direct integral transformation of nonlinear convection-diffusion problems based on nonlinear eigenvalue problems. Representative application examples are then provided that employ recent extensions on the Generalized Integral Transform Technique (GITT), and a few numerical results are reported to illustrate the convergence characteristics of the proposed eigenfunction expansions.
A Variational Approach to the Isoperimetric Inequality for the Robin Eigenvalue Problem
Bucur, Dorin; Giacomini, Alessandro
2010-12-01
The isoperimetric inequality for the first eigenvalue of the Laplace operator with Robin boundary conditions was recently proved by Daners in the context of Lipschitz sets. This paper introduces a new approach to the isoperimetric inequality, based on the theory of special functions of bounded variation (SBV). We extend the notion of the first eigenvalue λ1 for general domains with finite volume (possibly unbounded and with irregular boundary), and we prove that the balls are the unique minimizers of λ1 among domains with prescribed volume.
Existence and comparison of smallest eigenvalues for a fractional boundary-value problem
Directory of Open Access Journals (Sweden)
Paul W. Eloe
2014-02-01
Full Text Available The theory of $u_0$-positive operators with respect to a cone in a Banach space is applied to the fractional linear differential equations $$ D_{0+}^{\\alpha} u+\\lambda_1p(tu=0\\quad\\text{and}\\quad D_{0+}^{\\alpha} u+\\lambda_2q(tu=0, $$ $0< t< 1$, with each satisfying the boundary conditions $u(0=u(1=0$. The existence of smallest positive eigenvalues is established, and a comparison theorem for smallest positive eigenvalues is obtained.
Indian Academy of Sciences (India)
Rahul Sharma; Subbajit Nandy; S P Bhattacharyya
2006-06-01
An energy-dependent partitioning scheme is explored for extracting a small number of eigenvalues of a real symmetric matrix with the help of genetic algorithm. The proposed method is tested with matrices of different sizes (30 × 30 to 1000 × 1000). Comparison is made with Löwdin's strategy for solving the problem. The relative advantages and disadvantages of the GA-based method are analyzed.
Photonic crystal fibres: mapping Maxwell's equations onto a Schrödinger equation eigenvalue problem
DEFF Research Database (Denmark)
Mortensen, Niels Asger
2006-01-01
We consider photonic crystal fibres (PCFs) made from arbitrary base materials and introduce a short-wavelength approximation which allows for a mapping of the Maxwell's equations onto a dimensionless eigenvalue equations which has the form of the Schröding equation in quantum mechanics. The mappi...
Sleijpen, G.L.G.; Vorst, H.A. van der
1995-01-01
We discuss a new method for the iterative computation of a portion of the spectrum of a large sparse matrix.The matrix may be complex and non-normal.The method also delivers the Schur vectors associated with the computed eigenvalues. The eigenvectors can easily be computed from the Schur vectors,
The Jacobi-Davidson method for eigenvalue problems as an accelerated inexact Newton scheme
Sleijpen, G.L.G.; Vorst, H.A. van der
1995-01-01
We discuss a new method for the iterative computation of a portion of the spectrum of a large sparse matrix. The matrix may be complex and non-normal. The method also delivers the Schur vectors associated with the computed eigenvalues. The eigenvectors can easily be computed from the Schur vectors,
A POSTERIORI ERROR ESTIMATES IN ADINI FINITE ELEMENT FOR EIGENVALUE PROBLEMS
Institute of Scientific and Technical Information of China (English)
Yi-du Yang
2000-01-01
In this paper, we discuss a posteriori error estimates of the eigenvalue λh given by Adini nonconforming finite element. We give an assymptotically exact error estimator of the λh. We prove that the order of convergence of the λh is just 2and the λh converge from below for sufficiently small h.
Singular Sturm-Liouville problems whose coefficients depend rationally on the eigenvalue parameter
Hassi, Seppo; Moller, M; de Snoo, H
2004-01-01
Let -Domega((.), z)D + q be a differential operator in L-2(0, infinity) whose leading coefficient contains the eigenvalue parameter z. For the case that omega((.), z) has the particular form omega(t, z) = p(t) + c(t)(2)/(z - r (t)), z is an element of C \\ R, and the coefficient functions satisfy cer
The Jacobi-Davidson method for eigenvalue problems as an accelerated inexact Newton scheme
Sleijpen, G.L.G.; Vorst, H.A. van der
2001-01-01
We discuss a new method for the iterative computation of a portion of the spectrum of a large sparse matrix. The matrix may be complex and non-normal. The method also delivers the Schur vectors associated with the computed eigenvalues. The eigenvectors can easily be computed from the Schur vectors,
Sleijpen, G.L.G.; Vorst, H.A. van der
2006-01-01
We discuss a new method for the iterative computation of a portion of the spectrum of a large sparse matrix.The matrix may be complex and non-normal.The method also delivers the Schur vectors associated with the computed eigenvalues. The eigenvectors can easily be computed from the Schur vectors, an
A generalized Jacobi-Davidson iteration method for linear eigenvalue problems
Sleijpen, G.L.G.; Vorst, H.A. van der
1998-01-01
In this paper we propose a new method for the iterative computation of a few of the extremal eigenvalues of a symmetric matrix and their associated eigenvectors. The method is based on an old and almost unknown method of Jacobi. Jacobi's approach, combined with Davidson's method, leads to a new meth
Eigenvalue problems for a quasilinear elliptic equation on ℝN
Directory of Open Access Journals (Sweden)
Marilena N. Poulou
2005-01-01
Full Text Available We prove the existence of a simple, isolated, positive principal eigenvalue for the quasilinear elliptic equation −Δpu=λg(x|u|p−2u, x∈ℝN, lim|x|→+∞u(x=0, where Δpu=div(|∇u|p−2∇u is the p-Laplacian operator and the weight function g(x, being bounded, changes sign and is negative and away from zero at infinity.
On the Asymptotic of an Eigenvalue Problem with 2 Interior Singularities
Indian Academy of Sciences (India)
A Neamaty; S Haghaieghy
2009-11-01
In this paper we consider the linear differential equation of the form $$-y''(x)+q(x)y(x)= y(x),\\quad -∞ < a < x < b < ∞$$ where satisfies Dirichlet boundary conditions and is a real-valued function which has even number of singularities at $c_1,\\ldots,c_{2n}\\in(a, b)$. We will study the asymptotic eigenvalue near the singularity points.
Energy Technology Data Exchange (ETDEWEB)
Bottcher, C.; Strayer, M.R. [Oak Ridge National Lab., TN (United States); Werby, M.F. [Naval Research Lab. Detachment, Stennis Space Center, MS (United States)
1993-10-01
The Helmholtz-Poincare Wave Equation (H-PWE) arises in many areas of classical wave scattering theory. In particular it can be found for the cases of acoustical scattering from submerged bounded objects and electromagnetic scattering from objects. The extended boundary integral equations (EBIE) method is derived from considering both the exterior and interior solutions of the H-PWE`s. This coupled set of expressions has the advantage of not only offering a prescription for obtaining a solution for the exterior scattering problem, but it also obviates the problem of irregular values corresponding to fictitious interior eigenvalues. Once the coupled equations are derived, they can by obtained in matrix form be expanding all relevant terms in partial wave expansions, including a biorthogonal expansion of the Green function. However some freedom of choice in the choice of the surface expansion is available since the unknown surface quantities may be expanded in a variety of ways to long as closure is obtained. Out of many possible choices, we develop an optimal method to obtain such expansions which is based on the optimum eigenfunctions related to the surface of the object. In effect, we convert part of the problem (that associated with the Fredholms integral equation of the first kind) an eigenvalue problem of a related Hermition operator. The methodology will be explained in detail and examples will be presented.
Institute of Scientific and Technical Information of China (English)
Chong-hua Yu; O. Axelsson
2000-01-01
In this paper, an algorithm for computing some of the largest (smallest) generalized eigenvalues with corresponding eigenvectors of a sparse symmetric positive definite matrix pencil is presented. The algorithm uses an iteration function and inverse power iteration process to get the largest one first, then executes m-1Lanczos-like steps to get initial approximations of the next m - 1 ones, without computing any Ritz pair, for which a procedure combining Rayleigh quotient iteration with shifted inverse power iteration is used to obtain more accurate eigenvalues and eigenvectors. This algorithm keeps the advantages of preserving sparsity of the original matrices as in Lanczos method and RQI and converges with a higher rate than the method described in[12] and provides a simple technique to compute initial approximate pairs which are guaranteed to converge to the wanted m largest eigenpairs using RQI. In addition, it avoids some of the disadvantages of Lanczos and RQI, for solving extreme eigenproblems. When symmetric positive definfite linear systems must be solved in the process, an algebraic multilevel iteration method (AMLI) is applied. The algorithm is fully parallelizable.
BOUNDARY VALUE PROBLEMS, PARTIAL DIFFERENTIAL EQUATIONS ), (* PARTIAL DIFFERENTIAL EQUATIONS , BOUNDARY VALUE PROBLEMS), (*NUMERICAL ANALYSIS, BOUNDARY VALUE PROBLEMS), FUNCTIONS(MATHEMATICS), DIFFERENCE EQUATIONS
Olivier, C. P.; Herbst, B. M.; Molchan, M. A.
2012-06-01
Deconinck and Kutz (2006 J. Comput. Phys. 219 296-321) developed an efficient algorithm for solving the Zakharov-Shabat eigenvalue problem with periodic potentials numerically. It is natural to use the same algorithm for solving the problem for non-periodic potential (decaying potentials defined over the whole real line) using large periods. In this paper, we propose the use of a specific value of the Floquet exponent. Our numerical results indicate that it can produce accurate results long before the period becomes large enough for the analytical convergence results of Gardner (1997 J. Reine Angew. Math. 491 149-81) to be valid. We also illustrate the rather complicated path to convergence of some nonlinear Schrödinger potentials.
The general solution of the eigenvalue problem for a high-gain FEL
Saldin, E L; Yurkov, M V
2001-01-01
The exact solution of the eigenvalue equation for a high-gain FEL derived in Xie (Nucl. Instr. and Meth. A 445 (2000) 59) is generalized in order to include the space charge effects. This solution is valid not only for natural undulator focusing, but also for alternating-gradient focusing under some condition that is presented. At such, the obtained solution includes all the important effects in the system of axially homogeneous electron beam and undulator: diffraction, betatron motion, energy spread, space charge and frequency detuning. It is valid for ground TEM sub 0 sub 0 mode as well as for high-order modes and can be used for calculation of high-gain FEL amplifiers operating in the wavelength regions from far infrared down to X-ray. In addition, a computationally efficient approximate solution for TEM sub 0 sub 0 mode is derived providing high accuracy (better than 1% in the whole range of parameters). It can be used for quick optimization of FEL amplifiers.
An efficient solver for large structured eigenvalue problems in relativistic quantum chemistry
Shiozaki, Toru
2015-01-01
We report an efficient program for computing the eigenvalues and symmetry-adapted eigenvectors of very large quaternionic (or Hermitian skew-Hamiltonian) matrices, using which structure-preserving diagonalization of matrices of dimension N > 10000 is now routine on a single computer node. Such matrices appear frequently in relativistic quantum chemistry owing to the time-reversal symmetry. The implementation is based on a blocked version of the Paige-Van Loan algorithm [D. Kressner, BIT 43, 775 (2003)], which allows us to use the Level 3 BLAS subroutines for most of the computations. Taking advantage of the symmetry, the program is faster by up to a factor of two than state-of-the-art implementations of complex Hermitian diagonalization; diagonalizing a 12800 x 12800 matrix took 42.8 (9.5) and 85.6 (12.6) minutes with 1 CPU core (16 CPU cores) using our symmetry-adapted solver and Intel MKL's ZHEEV that is not structure-preserving, respectively. The source code is publicly available under the FreeBSD license.
Luukko, P. J. J.; Räsänen, E.
2013-03-01
We present a code for solving the single-particle, time-independent Schrödinger equation in two dimensions. Our program utilizes the imaginary time propagation (ITP) algorithm, and it includes the most recent developments in the ITP method: the arbitrary order operator factorization and the exact inclusion of a (possibly very strong) magnetic field. Our program is able to solve thousands of eigenstates of a two-dimensional quantum system in reasonable time with commonly available hardware. The main motivation behind our work is to allow the study of highly excited states and energy spectra of two-dimensional quantum dots and billiard systems with a single versatile code, e.g., in quantum chaos research. In our implementation we emphasize a modern and easily extensible design, simple and user-friendly interfaces, and an open-source development philosophy. Catalogue identifier: AENR_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AENR_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: GNU General Public License version 3 No. of lines in distributed program, including test data, etc.: 11310 No. of bytes in distributed program, including test data, etc.: 97720 Distribution format: tar.gz Programming language: C++ and Python. Computer: Tested on x86 and x86-64 architectures. Operating system: Tested under Linux with the g++ compiler. Any POSIX-compliant OS with a C++ compiler and the required external routines should suffice. Has the code been vectorised or parallelized?: Yes, with OpenMP. RAM: 1 MB or more, depending on system size. Classification: 7.3. External routines: FFTW3 (http://www.fftw.org), CBLAS (http://netlib.org/blas), LAPACK (http://www.netlib.org/lapack), HDF5 (http://www.hdfgroup.org/HDF5), OpenMP (http://openmp.org), TCLAP (http://tclap.sourceforge.net), Python (http://python.org), Google Test (http://code.google.com/p/googletest/) Nature of problem: Numerical calculation
Boundary integral equation methods in eigenvalue problems of elastodynamics and thin plates
Kitahara, M
1985-01-01
The boundary integral equation (BIE) method has been used more and more in the last 20 years for solving various engineering problems. It has important advantages over other techniques for numerical treatment of a wide class of boundary value problems and is now regarded as an indispensable tool for potential problems, electromagnetism problems, heat transfer, fluid flow, elastostatics, stress concentration and fracture problems, geomechanical problems, and steady-state and transient electrodynamics.In this book, the author gives a complete, thorough and detailed survey of the method. It pro
Energy Technology Data Exchange (ETDEWEB)
Guerin, P
2007-12-15
The neutronic simulation of a nuclear reactor core is performed using the neutron transport equation, and leads to an eigenvalue problem in the steady-state case. Among the deterministic resolution methods, diffusion approximation is often used. For this problem, the MINOS solver based on a mixed dual finite element method has shown his efficiency. In order to take advantage of parallel computers, and to reduce the computing time and the local memory requirement, we propose in this dissertation two domain decomposition methods for the resolution of the mixed dual form of the eigenvalue neutron diffusion problem. The first approach is a component mode synthesis method on overlapping sub-domains. Several Eigenmodes solutions of a local problem solved by MINOS on each sub-domain are taken as basis functions used for the resolution of the global problem on the whole domain. The second approach is a modified iterative Schwarz algorithm based on non-overlapping domain decomposition with Robin interface conditions. At each iteration, the problem is solved on each sub domain by MINOS with the interface conditions deduced from the solutions on the adjacent sub-domains at the previous iteration. The iterations allow the simultaneous convergence of the domain decomposition and the eigenvalue problem. We demonstrate the accuracy and the efficiency in parallel of these two methods with numerical results for the diffusion model on realistic 2- and 3-dimensional cores. (author)
The Eigenvalue Method for Extremal Problems on Infinite Vertex-Transitive Graphs
DeCorte, P.E.B.
2015-01-01
This thesis is about maximum independent set and chromatic number problems on certain kinds of infinite graphs. A typical example comes from the Witsenhausen problem: For $n \\geq 2$, let $S^{n-1} := \\{ x \\in \\R^n : \\|x\\|_2 =1 \\}$ be the unit sphere in $\\R^n$, and let $G=(V,E)$ be the graph with $V =
Some properties of eigenvalues and generalized eigenvectors of one boundary-value problem
Olgar, Hayati; Mukhtarov, Oktay; Aydemir, Kadriye
2016-08-01
We investigate a discontinuous boundary value problem which consists of a Sturm-Liouville equation with piece-wise continuous potential together with eigenparameter-dependent boundary conditions and supplementary transmission conditions. We establish some spectral properties of the considered problem. In particular it is shown that the generalized eigen-functions form a Riesz basis of the adequate Hilbert space.
Benner, Peter; Dolgov, Sergey; Khoromskaia, Venera; Khoromskij, Boris N.
2017-04-01
In this paper, we propose and study two approaches to approximate the solution of the Bethe-Salpeter equation (BSE) by using structured iterative eigenvalue solvers. Both approaches are based on the reduced basis method and low-rank factorizations of the generating matrices. We also propose to represent the static screen interaction part in the BSE matrix by a small active sub-block, with a size balancing the storage for rank-structured representations of other matrix blocks. We demonstrate by various numerical tests that the combination of the diagonal plus low-rank plus reduced-block approximation exhibits higher precision with low numerical cost, providing as well a distinct two-sided error estimate for the smallest eigenvalues of the Bethe-Salpeter operator. The complexity is reduced to O (Nb2) in the size of the atomic orbitals basis set, Nb, instead of the practically intractable O (Nb6) scaling for the direct diagonalization. In the second approach, we apply the quantized-TT (QTT) tensor representation to both, the long eigenvectors and the column vectors in the rank-structured BSE matrix blocks, and combine this with the ALS-type iteration in block QTT format. The QTT-rank of the matrix entities possesses almost the same magnitude as the number of occupied orbitals in the molecular systems, No
On an Inverse Eigenvalue Problem for a Semilinear Sturm-Liouville Operator
Zhidkov, P E
2005-01-01
The following problem is considered: $-u''+f(u)=\\lambda u, x\\in (0,1), u=u(x), u(0)=1, u'(0)=u(1)=0,$ where $\\lambda $ is a spectral parameter. We study the inverse problem: for a given part of the spectrum $\\lambda _n\\to +\\infty $ we seek odd $f$. We obtain a description of the whole class of solutions of this problem. In addition, we show that there exists at most one function $f$ such that an auxiliary function is nondecreasing.
Problems with Discontinuous Diffusion/Dispersion Coefficients
Directory of Open Access Journals (Sweden)
Stefano Ferraris
2012-01-01
accurate on smooth solutions and based on a special numerical treatment of the diffusion/dispersion coefficients that makes its application possible also when such coefficients are discontinuous. Numerical experiments confirm the convergence of the numerical approximation and show a good behavior on a set of benchmark problems in two space dimensions.
Wenner, Michael T.
Obtaining the solution to the linear Boltzmann equation is often is often a daunting task. The time-independent form is an equation of six independent variables which cannot be solved analytically in all but some special problems. Instead, numerical approaches have been devised. This work focuses on improving Monte Carlo methods for its solution in eigenvalue form. First, a statistical method of stationarity detection called the KPSS test adapted as a Monte Carlo eigenvalue source convergence test. The KPSS test analyzes the source center of mass series which was chosen since it should be indicative of overall source behavior, and is physically easy to understand. A source center of mass plot alone serves as a good visual source convergence diagnostic. The KPSS test and three different information theoretic diagnostics were implemented into the well known KENOV.a code inside of the SCALE (version 5) code package from Oak Ridge National Laboratory and compared through analysis of a simple problem and several difficult source convergence benchmarks. Results showed that the KPSS test can add to the overall confidence by identifying more problematic simulations than without its usage. Not only this, the source center of mass information on hand visually aids in the understanding of the problem physics. The second major focus of this dissertation concerned variance reduction methodologies for Monte Carlo eigenvalue problems. The CADIS methodology, based on importance sampling, was adapted to the eigenvalue problems. It was shown that the straight adaption of importance sampling can provide a significant variance reduction in determination of keff (in cases studied up to 30%?). A modified version of this methodology was developed which utilizes independent deterministic importance simulations. In this new methodology, each particle is simulated multiple times, once to every other discretized source region utilizing the importance for that region only. Since each particle
Hamiltonian矩阵特征值问题的Lanczos-型算法%Lanczos Algorithms-Type for Hamiltonian Eigenvalue Problems
Institute of Scientific and Technical Information of China (English)
郭蔚
2001-01-01
In the paper,we applicate Lanczos algorithms-type to Hamiltonian eigenvalue problems and give an error analysis in iterative procedure.%应用Lanczos-型算法求Hamiltonian矩阵的特征根问题，并且给出了在迭代过程中的误差估计．
Institute of Scientific and Technical Information of China (English)
张俊
2011-01-01
This paper discusses Wilson nonconforming element of eigenvalue problem of SchrSdinger equation and its error estimates.%讨论三维Schrǒdinger方程的特征值问题的Wilson元离散，并给出相应的误差估计．
M. Genseberger (Menno)
2008-01-01
htmlabstractMost computational work in Jacobi-Davidson [9], an iterative method for large scale eigenvalue problems, is due to a so-called correction equation. In [5] a strategy for the approximate solution of the correction equation was proposed. This strategy is based on a domain decomposition
Khapaev, M. M.; Khapaeva, T. M.
2016-10-01
A functional-based variational method is proposed for finding the eigenfunctions and eigenvalues in the Sturm-Liouville problem with Dirichlet boundary conditions at the left endpoint and Neumann conditions at the right endpoint. Computations are performed for three potentials: sin(( x-π)2/π), cos(4 x), and a high nonisosceles triangle.
Parallel Sparse Linear System and Eigenvalue Problem Solvers: From Multicore to Petascale Computing
2015-06-01
problems that achieve high performance on a single multicore node and clusters of many multicore nodes. Further, we demonstrate both the superior ...the superior robustness and parallel scalability of our solvers compared to other publicly available parallel solvers for these two fundamental...LU‐ and algebraic multigrid‐preconditioned Krylov subspace methods. This has been demonstrated in previous annual reports of this
Hamed, Haikel Ben; Bennacer, Rachid
2008-08-01
This work consists in evaluating algebraically and numerically the influence of a disturbance on the spectral values of a diagonalizable matrix. Thus, two approaches will be possible; to use the theorem of disturbances of a matrix depending on a parameter, due to Lidskii and primarily based on the structure of Jordan of the no disturbed matrix. The second approach consists in factorizing the matrix system, and then carrying out a numerical calculation of the roots of the disturbances matrix characteristic polynomial. This problem can be a standard model in the equations of the continuous media mechanics. During this work, we chose to use the second approach and in order to illustrate the application, we choose the Rayleigh-Bénard problem in Darcy media, disturbed by a filtering through flow. The matrix form of the problem is calculated starting from a linear stability analysis by a finite elements method. We show that it is possible to break up the general phenomenon into other elementary ones described respectively by a disturbed matrix and a disturbance. A good agreement between the two methods was seen. To cite this article: H.B. Hamed, R. Bennacer, C. R. Mecanique 336 (2008).
Inverse eigenvalue problems in vibration absorption: Passive modification and active control
Mottershead, John E.; Ram, Yitshak M.
2006-01-01
The abiding problem of vibration absorption has occupied engineering scientists for over a century and there remain abundant examples of the need for vibration suppression in many industries. For example, in the automotive industry the resolution of noise, vibration and harshness (NVH) problems is of extreme importance to customer satisfaction. In rotorcraft it is vital to avoid resonance close to the blade passing speed and its harmonics. An objective of the greatest importance, and extremely difficult to achieve, is the isolation of the pilot's seat in a helicopter. It is presently impossible to achieve the objectives of vibration absorption in these industries at the design stage because of limitations inherent in finite element models. Therefore, it is necessary to develop techniques whereby the dynamic of the system (possibly a car or a helicopter) can be adjusted after it has been built. There are two main approaches: structural modification by passive elements and active control. The state of art of the mathematical theory of vibration absorption is presented and illustrated for the benefit of the reader with numerous simple examples.
An Inverse Eigenvalue Problem for a Vibrating String with Two Dirichlet Spectra
Rundell, William
2013-04-23
A classical inverse problem is "can you hear the density of a string clamped at both ends?" The mathematical model gives rise to an inverse Sturm-Liouville problem for the unknown density ñ, and it is well known that the answer is negative: the Dirichlet spectrum from the clamped end-point conditions is insufficient. There are many known ways to add additional information to gain a positive answer, and these include changing one of the boundary conditions and recomputing the spectrum or giving the energy in each eigenmode-the so-called norming constants. We make the assumption that neither of these changes are possible. Instead we will add known mass-densities to the string in a way we can prescribe and remeasure the Dirichlet spectrum. We will not be able to answer the uniqueness question in its most general form, but will give some insight to what "added masses" should be chosen and how this can lead to a reconstruction of the original string density. © 2013 Society for Industrial and Applied Mathematics.
Energy Technology Data Exchange (ETDEWEB)
Yamazaki, Ichitaro; Wu, Kesheng; Simon, Horst
2008-10-27
The original software package TRLan, [TRLan User Guide], page 24, implements the thick restart Lanczos method, [Wu and Simon 2001], page 24, for computing eigenvalues {lambda} and their corresponding eigenvectors v of a symmetric matrix A: Av = {lambda}v. Its effectiveness in computing the exterior eigenvalues of a large matrix has been demonstrated, [LBNL-42982], page 24. However, its performance strongly depends on the user-specified dimension of a projection subspace. If the dimension is too small, TRLan suffers from slow convergence. If it is too large, the computational and memory costs become expensive. Therefore, to balance the solution convergence and costs, users must select an appropriate subspace dimension for each eigenvalue problem at hand. To free users from this difficult task, nu-TRLan, [LNBL-1059E], page 23, adjusts the subspace dimension at every restart such that optimal performance in solving the eigenvalue problem is automatically obtained. This document provides a user guide to the nu-TRLan software package. The original TRLan software package was implemented in Fortran 90 to solve symmetric eigenvalue problems using static projection subspace dimensions. nu-TRLan was developed in C and extended to solve Hermitian eigenvalue problems. It can be invoked using either a static or an adaptive subspace dimension. In order to simplify its use for TRLan users, nu-TRLan has interfaces and features similar to those of TRLan: (1) Solver parameters are stored in a single data structure called trl-info, Chapter 4 [trl-info structure], page 7. (2) Most of the numerical computations are performed by BLAS, [BLAS], page 23, and LAPACK, [LAPACK], page 23, subroutines, which allow nu-TRLan to achieve optimized performance across a wide range of platforms. (3) To solve eigenvalue problems on distributed memory systems, the message passing interface (MPI), [MPI forum], page 23, is used. The rest of this document is organized as follows. In Chapter 2 [Installation
Adaptive computation for convection dominated diffusion problems
Institute of Scientific and Technical Information of China (English)
CHEN Zhiming; JI Guanghua
2004-01-01
We derive sharp L∞(L1) a posteriori error estimate for the convection dominated diffusion equations of the form αu/αt+div(vu)-εΔu=g. The derived estimate is insensitive to the diffusionparameter ε→0. The problem is discretized implicitly in time via the method of characteristics and in space via continuous piecewise linear finite elements. Numerical experiments are reported to show the competitive behavior of the proposed adaptive method.
Aktosun, Tuncay; Gintides, Drossos; Papanicolaou, Vassilis G.
2011-11-01
The recovery of a spherically symmetric wave speed v is considered in a bounded spherical region of radius b from the set of the corresponding transmission eigenvalues for which the corresponding eigenfunctions are also spherically symmetric. If the integral of 1/v on the interval [0, b] is less than b, assuming that there exists at least one v corresponding to the data, it is shown that v is uniquely determined by the data consisting of such transmission eigenvalues and their ‘multiplicities’, where the ‘multiplicity’ is defined as the multiplicity of the transmission eigenvalue as a zero of a key quantity. When that integral is equal to b, the unique recovery is obtained when the data contain one additional piece of information. Some similar results are presented for the unique determination of the potential from the transmission eigenvalues with ‘multiplicities’ for a related Schrödinger equation.
Haidar, Azzam
2011-01-01
This paper introduces a novel implementation in reducing a symmetric dense matrix to tridiagonal form, which is the preprocessing step toward solving symmetric eigenvalue problems. Based on tile algorithms, the reduction follows a two-stage approach, where the tile matrix is first reduced to symmetric band form prior to the final condensed structure. The challenging trade-off between algorithmic performance and task granularity has been tackled through a grouping technique, which consists of aggregating fine-grained and memory-aware computational tasks during both stages, while sustaining the application\\'s overall high performance. A dynamic runtime environment system then schedules the different tasks in an out-of-order fashion. The performance for the tridiagonal reduction reported in this paper is unprecedented. Our implementation results in up to 50-fold and 12-fold improvement (130 Gflop/s) compared to the equivalent routines from LAPACK V3.2 and Intel MKL V10.3, respectively, on an eight socket hexa-core AMD Opteron multicore shared-memory system with a matrix size of 24000×24000. Copyright 2011 ACM.
Energy Technology Data Exchange (ETDEWEB)
Geist, G.A. [Oak Ridge National Lab., TN (United States). Computer Science and Mathematics Div.; Howell, G.W. [Florida Inst. of Tech., Melbourne, FL (United States). Dept. of Applied Mathematics; Watkins, D.S. [Washington State Univ., Pullman, WA (United States). Dept. of Pure and Applied Mathematics
1997-11-01
The BR algorithm, a new method for calculating the eigenvalues of an upper Hessenberg matrix, is introduced. It is a bulge-chasing algorithm like the QR algorithm, but, unlike the QR algorithm, it is well adapted to computing the eigenvalues of the narrowband, nearly tridiagonal matrices generated by the look-ahead Lanczos process. This paper describes the BR algorithm and gives numerical evidence that it works well in conjunction with the Lanczos process. On the biggest problems run so far, the BR algorithm beats the QR algorithm by a factor of 30--60 in computing time and a factor of over 100 in matrix storage space.
The Schrodinger Eigenvalue March
Tannous, C.; Langlois, J.
2011-01-01
A simple numerical method for the determination of Schrodinger equation eigenvalues is introduced. It is based on a marching process that starts from an arbitrary point, proceeds in two opposite directions simultaneously and stops after a tolerance criterion is met. The method is applied to solving several 1D potential problems including symmetric…
Diagonalizable quadratic eigenvalue problems
Lancaster, Peter; Zaballa, Ion
2009-05-01
A system is defined to be an n×n matrix function L(λ)=λ2M+λD+K where M,D,K∈C and M is nonsingular. First, a careful review is made of the possibility of direct decoupling to a diagonal (real or complex) system by applying congruence or strict equivalence transformations to L(λ). However, the main contribution is a complete description of the much wider class of systems which can be decoupled by applying congruence or strict equivalence transformations to a linearization of a system while preserving the structure of L(λ). The theory is liberally illustrated with examples.
Generalized Inverse Eigenvalue Problem for Centrohermitian Matrices%关于中心厄米特矩阵的广义反特征值问题
Institute of Scientific and Technical Information of China (English)
刘仲云; 谭艳祥; 田兆录
2004-01-01
In this paper we first consider the existence and the general form of solution to the following generalized inverse eigenvalue problem(GIEP): given a set of n-dimension complex vectors {xj}mj= 1 and a set of complex numbers {λj}jm=1, find two n × n centrohermitian matrices A, B such that {xj}jm = 1 and {λj}jm= 1 are the generalized eigenvectors and generalized eigenvalues of Ax = λBx, respectively. We then discuss the optimal approximation problem for the GIEP. More concretely, given two arbitrary matrices, A, E ∈Cn×n, we find two matrices A* and B* such that the matrix (A* ,B* ) is closest to (A ,B) in the Frobenius norm, where the matrix (A *,B*) is the solution to the GIEP. We show that the expression of the solution of the optimal approximation is unique and derive the expression for it.
球面区域上buckling特征值的万有估计%UNIVERSAL BOUNDS ON EIGENVALUES OF THE BUCKLING PROBLEM ON SPHERICAL DOMAINS
Institute of Scientific and Technical Information of China (English)
黄广月; 李兴校; 曹林芬
2011-01-01
We study the eigenvalues of buckling problem on domains in the unit sphere.By introducing a new parameter and using Cauchy inequality,we optimize the inequality obtained by Wang and Xia in[12].%本文研究了球面域上的buckling特征值问题.通过引入新的参数和使用Cauchy不等式,优化了Wang-Xia在文献[1 2]中的不等式.
Energy Technology Data Exchange (ETDEWEB)
CREUTZ, M.
2006-01-26
It is popular to discuss low energy physics in lattice gauge theory ill terms of the small eigenvalues of the lattice Dirac operator. I play with some ensuing pitfalls in the interpretation of these eigenvalue spectra. In short, thinking about the eigenvalues of the Dirac operator in the presence of gauge fields can give some insight, for example the elegant Banks-Casher picture for chiral symmetry breaking. Nevertheless, care is necessary because the problem is highly non-linear. This manifests itself in the non-intuitive example of how adding flavors enhances rather than suppresses low eigenvalues. Issues involving zero mode suppression represent one facet of a set of connected unresolved issues. Are there non-perturbative ambiguities in quantities such as the topological susceptibility? How essential are rough gauge fields, i.e. gauge fields on which the winding number is ambiguous? How do these issues interplay with the quark masses? I hope the puzzles presented here will stimulate more thought along these lines.
The origin and nature of spurious eigenvalues in the spectral tau method
Energy Technology Data Exchange (ETDEWEB)
Dawkins, P.T. [Lamar Univ., Beaumont, TX (United States). Dept. of Mathematics; Dunbar, S.R. [Univ. of Nebraska, Lincoln, NE (United States). Dept. of Mathematics and Statistics; Douglass, R.W. [Idaho National Engineering Lab., Idaho Falls, ID (United States)
1998-12-10
The Chebyshev-tau spectral method for approximating eigenvalues of boundary value problems may produce spurious eigenvalues with large positive real parts, even when all true eigenvalues of the problem are known to have negative real parts. The authors explain the origin and nature of the spurious eigenvalues in an example problem. The explanation will demonstrate that the large positive eigenvalues are an approximation of infinite eigenvalues in a nearby generalized eigenvalue problem.
Perturbation Theory of Embedded Eigenvalues
DEFF Research Database (Denmark)
Engelmann, Matthias
We study problems connected to perturbation theory of embedded eigenvalues in two different setups. The first part deals with second order perturbation theory of mass shells in massive translation invariant Nelson type models. To this end an expansion of the eigenvalues w.r.t. fiber parameter up...... project gives a general and systematic approach to analytic perturbation theory of embedded eigenvalues. The spectral deformation technique originally developed in the theory of dilation analytic potentials in the context of Schrödinger operators is systematized by the use of Mourre theory. The group...
On eigenvectors of multiple eigenvalues obtained in NASTRAN
Pamidi, P. R.; Brown, W. K.
1975-01-01
In the case of nonmultiple eigenvalues, each of the three real eigenvalue extraction methods available in NASTRAN will, for a given type of normalization, give essentially the same eigenvectors, but this is not so in the case of multiple eigenvalues. This discrepancy is explained and illustrated by considering the example of a NASTRAN demonstration problem that has both multiple and nonmultiple eigenvalues.
Institute of Scientific and Technical Information of China (English)
凌莉芸; 凌晨
2016-01-01
For a class of eigenvalue complementarity problem with strictly semi-positive tensors,we study the symbolic features of Pareto-eigenvalue.On this based,we obtain the upper and lower bounds of Pareto-eigenvalue for eigenvalue complementarity problem with strictly semi-positive tensors by using the constant definition and operator definition of strictly semi-positive tensors.%针对一类严格半正张量特征值互补问题，研究了其 Pareto-特征值的符号特征。在此基础上，利用严格半正张量的常量定义和算子定义，得到了严格半正张量特征值互补问题的 Pareto-特征值的上下界估计。
Liu, Tianyu; Du, Xining; Ji, Wei; Xu, X. George; Brown, Forrest B.
2014-06-01
For nuclear reactor analysis such as the neutron eigenvalue calculations, the time consuming Monte Carlo (MC) simulations can be accelerated by using graphics processing units (GPUs). However, traditional MC methods are often history-based, and their performance on GPUs is affected significantly by the thread divergence problem. In this paper we describe the development of a newly designed event-based vectorized MC algorithm for solving the neutron eigenvalue problem. The code was implemented using NVIDIA's Compute Unified Device Architecture (CUDA), and tested on a NVIDIA Tesla M2090 GPU card. We found that although the vectorized MC algorithm greatly reduces the occurrence of thread divergence thus enhancing the warp execution efficiency, the overall simulation speed is roughly ten times slower than the history-based MC code on GPUs. Profiling results suggest that the slow speed is probably due to the memory access latency caused by the large amount of global memory transactions. Possible solutions to improve the code efficiency are discussed.
Some Problems in Using Diffusion Models for New Products.
Bernhardt, Irwin; Mackenzie, Kenneth D.
This paper analyzes some of the problems of using diffusion models to formulate marketing strategies for new products. Though future work in this area appears justified, many unresolved problems limit its application. There is no theory for adoption and diffusion processes; such a theory is outlined in this paper. The present models are too…
QUENCHING PROBLEMS OF DEGENERATE FUNCTIONAL REACTION-DIFFUSION EQUATION
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
This paper is concerned with the quenching problem of a degenerate functional reaction-diffusion equation. The quenching problem and global existence of solution for the reaction-diffusion equation are derived and, some results of the positive steady state solutions for functional elliptic boundary value are also presented.
On the sensitivities of multiple eigenvalues
DEFF Research Database (Denmark)
Gravesen, Jens; Evgrafov, Anton; Nguyen, Dang Manh
2011-01-01
We consider the generalized symmetric eigenvalue problem where matrices depend smoothly on a parameter. It is well known that in general individual eigenvalues, when sorted in accordance with the usual ordering on the real line, do not depend smoothly on the parameter. Nevertheless, symmetric...... a shape of a vibrating membrane with a smallest perimeter and with prescribed four lowest eigenvalues, only two of which have algebraic multiplicity one....
New complex variable meshless method for advection-diffusion problems
Institute of Scientific and Technical Information of China (English)
Wang Jian-Fei; Cheng Yu-Min
2013-01-01
In this paper,an improved complex variable meshless method (ICVMM) for two-dimensional advection-diffusion problems is developed based on improved complex variable moving least-square (ICVMLS) approximation.The equivalent functional of two-dimensional advection-diffusion problems is formed,the variation method is used to obtain the equation system,and the penalty method is employed to impose the essential boundary conditions.The difference method for two-point boundary value problems is used to obtain the discrete equations.Then the corresponding formulas of the ICVMM for advection-diffusion problems are presented.Two numerical examples with different node distributions are used to validate and investigate the accuracy and efficiency of the new method in this paper.It is shown that ICVMM is very effective for advection-diffusion problems,and has a good convergent character,accuracy,and computational efficiency.
Revisiting the Diffusion Problem in a Capillary Tube Geometry
Sullivan, Eric
2012-01-01
The present work revisits the problem of modeling diffusion above a stagnant liquid interface in a capillary tube geometry. In this revisitation we elucidate a misconception found in the classical model proposed by Bird et. al. Furthermore, we propose alternative explanations for thermally forced diffusion and provide a description of natural convection in the absence of forcing terms.
Some Problems in Using Diffusion Models for New Products
Bernhardt, Irwin; Mackenzie, Kenneth D.
1972-01-01
Analyzes some of the problems involved in using diffusion models to formulate marketing strategies for introducing new products. Six models, which remove some of the theoretical and methodological restrictions inherent in current models of the adoption and diffusion process, are presented. (Author/JH)
Mathematics Mechanization in the Eigenvalue Problem of Application%数学机械化方法在特征值问题中的应用
Institute of Scientific and Technical Information of China (English)
白根柱; 乌仁高娃; 冀爱萍
2011-01-01
Mathematics mechanization theory and method will be applied to solve to algebraic eigenvalue problem which was developed by our country mathematician who called WU Wenjun in the 1970 s and the modern mathematics view was reflected in mathematics teaching.It will help student to improve the mathematical thinking level,development innovation consciousness and practical ability.%将我国数学家吴文俊在二十世纪七十年代倡导的并发展起来的数学机械化理论和方法应用到代数特征值问题中,把现代的数学观点反映到数学教学中来,这对于提高学生的数学思维层次,发展创新意识和实践能力会有一定的帮助.
Monotone method for initial value problem for fractional diffusion equation
Institute of Scientific and Technical Information of China (English)
ZHANG Shuqin
2006-01-01
Using the method of upper and lower solutions and its associated monotone iterative, consider the existence and uniqueness of solution of an initial value problem for the nonlinear fractional diffusion equation.
From Sturm-Liouville problems to fractional and anomalous diffusions
D'Ovidio, Mirko
2010-01-01
Some fractional and anomalous diffusions are driven by equations involving fractional derivatives in both time and space. Such diffusions are processes with randomly varying times. In representing the solutions to those diffusions, the explicit laws of certain stable processes turn out to be fundamental. This paper directs one's efforts towards the explicit representation of solutions to fractional and anomalous diffusions related to Sturm-Liouville problems of fractional order associated to fractional power function spaces. Furthermore, we study a new version of the Bochner's subordination rule and we establish some connections between subordination and space-fractional operator
Layer-adapted meshes for reaction-convection-diffusion problems
Linß, Torsten
2010-01-01
This book on numerical methods for singular perturbation problems - in particular, stationary reaction-convection-diffusion problems exhibiting layer behaviour is devoted to the construction and analysis of layer-adapted meshes underlying these numerical methods. A classification and a survey of layer-adapted meshes for reaction-convection-diffusion problems are included. This structured and comprehensive account of current ideas in the numerical analysis for various methods on layer-adapted meshes is addressed to researchers in finite element theory and perturbation problems. Finite differences, finite elements and finite volumes are all covered.
用Lanczos算法进行周期结构固有特性分析的研究%The Analysis of Eigenvalue Problem of Periodic Structures byLanczos Method
Institute of Scientific and Technical Information of China (English)
王翔
2000-01-01
In the FEM dynamic analysis of large flexible spacestructure (LFSS), the solution of frequencies and corresponding modelsis actually a general eigenvalue and eigenvector problem. In thisarticle, Lanczos method was applied to this type of problem. Innumerical examples, the convergency, accuracy-time relation andmulti-root problem of Lanczos method for beam structure were analyzed.Considering the characteristic of the stiffness matrix of beam structure, theauthor also present some ideas to improve Lanczos method byiteration method.
Computing the eigenvalues and eigenvectors of a fuzzy matrix
Directory of Open Access Journals (Sweden)
A. Kumar
2012-08-01
Full Text Available Computation of fuzzy eigenvalues and fuzzy eigenvectors of a fuzzy matrix is a challenging problem. Determining the maximal and minimal symmetric solution can help to find the eigenvalues. So, we try to compute these eigenvalues by determining the maximal and minimal symmetric solution of the fully fuzzy linear system $widetilde{A}widetilde{X}= widetilde{lambda} widetilde{X}.$
1988-04-01
Moreover, if e2 > 61 and we further assume (f.1) f(x,y) is locally Lipschitz continuous in y then inequalities (1.37) and (1.38) are strict. 17 Proof MI...Univ. Math. J. 20 (1971), 983-996.[I 124 18. J. A. Hempel, Superlinear variational boundary value problems and nonuniqueness , Thesis, University of
Matrix-dependent multigrid-homogenization for diffusion problems
Energy Technology Data Exchange (ETDEWEB)
Knapek, S. [Institut fuer Informatik tu Muenchen (Germany)
1996-12-31
We present a method to approximately determine the effective diffusion coefficient on the coarse scale level of problems with strongly varying or discontinuous diffusion coefficients. It is based on techniques used also in multigrid, like Dendy`s matrix-dependent prolongations and the construction of coarse grid operators by means of the Galerkin approximation. In numerical experiments, we compare our multigrid-homogenization method with homogenization, renormalization and averaging approaches.
A Parallel Algorithm for the Convection Diffusion Problem
Institute of Scientific and Technical Information of China (English)
刘晓遇; 赵凯; 陆金甫
2004-01-01
Based on the second-order compact upwind scheme,a group explicit method for solving the two-dimensional time-independent convection-dominated diffusion problem is developed.The stability of the group explicit method is proven strictly.The method has second-order accuracy and good stability.This explicit scheme can be used to solve all Reynolds number convection-dominated diffusion problems.A numerical test using a parallel computer shows high efficiency.The numerical results conform closely to the analytic solution.
Inverse problems of generalized eigenvalue for H-(anti) symmetric mtrices%H-(反)对称矩阵的广义特征值反问题
Institute of Scientific and Technical Information of China (English)
燕列雅; 任学明; 王艳
2012-01-01
讨论H矩阵的性质,给出H-对称矩阵和H-反对称矩阵的结构,证明若x是H-对称矩阵或H-反对称矩阵A -λB的特征向量,则x是H-对称向量或H-反对称向量,或者x可以由H-对称向量及H-反对称向量线性表示,并根据A-λB的特征向量的上述特点,得到H-对称矩阵和H-反对称矩阵的广义特征值反问题AX=BXA解的表达式.%Properties of H-matrices were discussed, the structures of H-symmetric and H-antisymmetric matrices were given, and it was proven that when x was an eigenvector of H-symmetric matrices or H-antisymmetric matrices A-λB, x would be either an H-symmetric vector, or H-antisymmetric vector, or x could be expressed by linear combination of H-symmetric vector with H-antisymmetric vector. Based on a-bove-mentioned feature of eigenvector of A-λB, the expression of solution to inverse problem AX=BXA of generalized eigenvalue of H-symmetric matrices and H-antisymmetric matrices were obtained.
Reaction-diffusion problems in the physics of hot plasmas
Wilhelmsson, H
2000-01-01
The physics of hot plasmas is of great importance for describing many phenomena in the universe and is fundamental for the prospect of future fusion energy production on Earth. Nontrivial results of nonlinear electromagnetic effects in plasmas include the self-organization and self-formation in the plasma of structures compact in time and space. These are the consequences of competing processes of nonlinear interactions and can be best described using reaction-diffusion equations. Reaction-Diffusion Problems in the Physics of Hot Plasmas is focused on paradigmatic problems of a reaction-diffusion type met in many branches of science, concerning in particular the nonlinear interaction of electromagnetic fields with plasmas.
Transmission eigenvalues for operators with constant coefficients
Hitrik, Michael; Ola, Petri; Päivärinta, Lassi
2010-01-01
In this paper we study the interior transmission problem and transmission eigenvalues for multiplicative perturbations of linear partial differential operator of order $\\ge 2$ with constant real coefficients. Under suitable growth conditions on the symbol of the operator and the perturbation, we show the discreteness of the set of transmission eigenvalues and derive sufficient conditions on the existence of transmission eigenvalues. We apply these techniques to the case of the biharmonic operator and the Dirac system. In the hypoelliptic case we present a connection to scattering theory.
HDG schemes for stationary convection-diffusion problems
Dautov, R. Z.; Fedotov, E. M.
2016-11-01
For stationary linear convection-diffusion problems, we construct and study a hybridized scheme of the discontinuous Galerkin method on the basis of an extended mixed statement of the problem. Discrete schemes can be used for the solution of equations degenerating in the leading part and are stated via approximations to the solution of the problem, its gradient, the flow, and the restriction of the solution to the boundaries of elements. For the spaces of finite elements, we represent minimal conditions responsible for the solvability, stability and accuracy of the schemes.
Parallel Symmetric Eigenvalue Problem Solvers
2015-05-01
Jacobi - Davidson, and FEAST), establishing the competitiveness of my methods . Graduate School Form 30 Updated 1/15/2015 PURDUE UNIVERSITY GRADUATE...LOBPCG, Jacobi -Davidson, and FEAST), establishing the competitiveness of our methods . 1 1 INTRODUCTION Many applications in science and engineering give...though SLEPc’s Jacobi -Davidson is the fastest method ; it is roughly twice as fast as TraceMin-Davidson. However, since it uses a block size 90 of 1
Latyshev, A V
2016-01-01
In the present work the second Stokes problem about behaviour of the rarefied gas filling half-space is formulated. A plane limiting half-space makes harmonious fluctuations with variable amplitude in the plane. The amplitude changes on the exponential law. The kinetic equation with model integral of collisions in the form $\\tau$-model is used. The case of diffusion reflexions of gas molecules from a wall is considered. Eigen solutions (continuous modes) of the initial kinetic equation corresponding to the continuous spectrum are searched. Properties of dispersion function are studied. It is investigated the discrete spectrum of the problem consisting of zero of the dispersion functions in the complex plane. It is shown, that number of zero of dispersion function to equally doubled index of problem coefficient. The problem coefficient is understood as the relation of boundary values of dispersion function from above and from below on the real axis. Further are eigen solutions (discrete modes) of the initial k...
Institute of Scientific and Technical Information of China (English)
秦佩华
2012-01-01
Elliptic eigenvalue problems in nonsmooth domain by using of discontinuous galerkin (DG) methods were analyzed. From many numerical results we find that for elliptic eigenvalue problems in nonsmooth domain DG methods provide better approximation than other methods, such as conforming or nonconforming finite element method, and finite element defect correction scheme.%本文针对非光滑区域上椭圆特征值特征值问题利用间断有限元方法(DG)近似.利用大量的数值算例发现,DG方法对非光滑区域(凹角,裂缝等问题)上Laplace特征值问题的近似比协调有限元、非协调元(如C-R元),甚至比有限元校正格式有着更好的效果.
Eigenvalues of singular differential operators by finite difference methods. I.
Baxley, J. V.
1972-01-01
Approximation of the eigenvalues of certain self-adjoint operators defined by a formal differential operator in a Hilbert space. In general, two problems are studied. The first is the problem of defining a suitable Hilbert space operator that has eigenvalues. The second problem concerns the finite difference operators to be used.
Diffusion in fluctuating media: first passage time problem
Energy Technology Data Exchange (ETDEWEB)
Revelli, Jorge A.; Budde, Carlos E.; Wio, Horacio S
2002-12-30
We study the actual and important problem of Mean First Passage Time (MFPT) for diffusion in fluctuating media. We exploit van Kampen's technique of composite stochastic processes, obtaining analytical expressions for the MFPT for a general system, and focus on the two state case where the transitions between the states are modelled introducing both Markovian and non-Markovian processes. The comparison between the analytical and simulations results show an excellent agreement.
Diffuse interface methods for inverse problems: case study for an elliptic Cauchy problem
Burger, Martin; Løseth Elvetun, Ole; Schlottbom, Matthias
2015-12-01
Many inverse problems have to deal with complex, evolving and often not exactly known geometries, e.g. as domains of forward problems modeled by partial differential equations. This makes it desirable to use methods which are robust with respect to perturbed or not well resolved domains, and which allow for efficient discretizations not resolving any fine detail of those geometries. For forward problems in partial differential equations methods based on diffuse interface representations have gained strong attention in the last years, but so far they have not been considered systematically for inverse problems. In this work we introduce a diffuse domain method as a tool for the solution of variational inverse problems. As a particular example we study ECG inversion in further detail. ECG inversion is a linear inverse source problem with boundary measurements governed by an anisotropic diffusion equation, which naturally cries for solutions under changing geometries, namely the beating heart. We formulate a regularization strategy using Tikhonov regularization and, using standard source conditions, we prove convergence rates. A special property of our approach is that not only operator perturbations are introduced by the diffuse domain method, but more important we have to deal with topologies which depend on a parameter \\varepsilon in the diffuse domain method, i.e. we have to deal with \\varepsilon -dependent forward operators and \\varepsilon -dependent norms. In particular the appropriate function spaces for the unknown and the data depend on \\varepsilon . This prevents the application of some standard convergence techniques for inverse problems, in particular interpreting the perturbations as data errors in the original problem does not yield suitable results. We consequently develop a novel approach based on saddle-point problems. The numerical solution of the problem is discussed as well and results for several computational experiments are reported. In
Chuluunbaatar, O.; Gusev, A. A.; Vinitsky, S. I.; Abrashkevich, A. G.
2009-08-01
A FORTRAN 77 program is presented for calculating with the given accuracy eigenvalues, eigenfunctions and their first derivatives with respect to the parameter of the parametric self-adjoined Sturm-Liouville problem with the parametric third type boundary conditions on the finite interval. The program calculates also potential matrix elements - integrals of the eigenfunctions multiplied by their first derivatives with respect to the parameter. Eigenvalues and matrix elements computed by the ODPEVP program can be used for solving the bound state and multi-channel scattering problems for a system of the coupled second-order ordinary differential equations with the help of the KANTBP programs [O. Chuluunbaatar, A.A. Gusev, A.G. Abrashkevich, A. Amaya-Tapia, M.S. Kaschiev, S.Y. Larsen, S.I. Vinitsky, Comput. Phys. Commun. 177 (2007) 649-675; O. Chuluunbaatar, A.A. Gusev, S.I. Vinitsky, A.G. Abrashkevich, Comput. Phys. Commun. 179 (2008) 685-693]. As a test desk, the program is applied to the calculation of the potential matrix elements for an integrable 2D-model of three identical particles on a line with pair zero-range potentials, a 3D-model of a hydrogen atom in a homogeneous magnetic field and a hydrogen atom on a three-dimensional sphere. Program summaryProgram title: ODPEVP Catalogue identifier: AEDV_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEDV_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC license, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 3001 No. of bytes in distributed program, including test data, etc.: 24 195 Distribution format: tar.gz Programming language: FORTRAN 77 Computer: Intel Xeon EM64T, Alpha 21264A, AMD Athlon MP, Pentium IV Xeon, Opteron 248, Intel Pentium IV Operating system: OC Linux, Unix AIX 5.3, SunOS 5.8, Solaris, Windows XP RAM: depends on the number and order of finite
Eigenvector derivatives of repeated eigenvalues using singular value decomposition
Lim, Kyong B.; Juang, Jer-Nan; Ghaemmaghami, Peiman
1989-01-01
An explicit formula is obtained for the first-order eigenvector derivative that corresponds to the eigenvector of a repeated eigenvalue, in the case of the nonself-adjoint eigenvalue problem. This method applies to the class of nondefective problems whose first eigenvalue derivatives of the repeated eigenvalues are distinct. A singular-value decomposition approach is used to compute four requisite bases for eigenspaces, as well as to keep track of the dimensions of state variables and the conditioning of the state equations.
Extremal eigenvalues of measure differential equations with fixed variation
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
In this paper we will study eigenvalues of measure differential equations which are motivated by physical problems when physical quantities are not absolutely continuous.By taking Neumann eigenvalues of measure differential equations as an example,we will show how the extremal problems can be completely solved by exploiting the continuity results of eigenvalues in weak* topology of measures and the Lagrange multiplier rule for nonsmooth functionals.These results can give another explanation for extremal eigenvalues of SturmLiouville operators with integrable potentials.
Simplicity of extremal eigenvalues of the Klein-Gordon equation
Koppen, Mario; Winklmeier, Monika
2010-01-01
We consider the spectral problem associated with the Klein-Gordon equation for unbounded electric potentials. If the spectrum of this problem is contained in two disjoint real intervals and the two inner boundary points are eigenvalues, we show that these extremal eigenvalues are simple and possess strictly positive eigenfunctions. Examples of electric potentials satisfying these assumptions are given.
Directory of Open Access Journals (Sweden)
Akimov Pavel Alekseevich
2012-10-01
Full Text Available The proposed paper covers the operator-related formulation of the eigenvalue problem of analysis of a three-dimensional structure that has piecewise-constant physical and geometrical parameters alongside the so-called basic direction within the framework of a discrete-continual approach (a discrete-continual finite element method, a discrete-continual variation method. Generally, discrete-continual formulations represent contemporary mathematical models that become available for computer implementation. They make it possible for a researcher to consider the boundary effects whenever particular components of the solution represent rapidly varying functions. Another feature of discrete-continual methods is the absence of any limitations imposed on lengths of structures. The three-dimensional problem of elasticity is used as the design model of a structure. In accordance with the so-called method of extended domain, the domain in question is embordered by an extended one of an arbitrary shape. At the stage of numerical implementation, relative key features of discrete-continual methods include convenient mathematical formulas, effective computational patterns and algorithms, simple data processing, etc. The authors present their formulation of the problem in question for an isotropic medium with allowance for supports restrained by elastic elements while standard boundary conditions are also taken into consideration.
Eigenvalue approximation from below using non-conforming finite elements
Institute of Scientific and Technical Information of China (English)
无
2010-01-01
This is a survey article about using non-conforming finite elements in solving eigenvalue problems of elliptic operators,with emphasis on obtaining lower bounds. In addition,this article also contains some new materials for eigenvalue approximations of the Laplace operator,which include:1) the proof of the fact that the non-conforming Crouzeix-Raviart element approximates eigenvalues associated with smooth eigenfunctions from below;2) the proof of the fact that the non-conforming EQ rot1 element approximates eigenvalues from below on polygonal domains that can be decomposed into rectangular elements;3) the explanation of the phenomena that numerical eigenvalues λ 1,h and λ 3,h of the non-conforming Q rot1 element approximate the true eigenvalues from below for the L-shaped domain. Finally,we list several unsolved problems.
EXISTENCE AND UNIQUENESS OF POSITIVE EIGENVALUES FOR CERTAIN EIGENVALUE SYSTEMS
Institute of Scientific and Technical Information of China (English)
XUE Ruying; QIN Yuchun
1999-01-01
In this paper we consider certain eigenvalue systems.Imposing some reasonable hypotheses, we prove that theeigenvalue system has a unique eigenvalue with positiveeigenfunctions, and that the eigenfunction is unique upto a scalar multiple.
A Study on the Problems of EigenValue and Eigenvectors%特征值与特征向量相关问题的研究
Institute of Scientific and Technical Information of China (English)
王新武
2011-01-01
特征值与特征向量是代数研究的中心问题之一，是两个密切相关的概念．在理论和实际应用中，特征值与特征向量都有举足轻重的地位．本文主要是对矩阵特征值与特征向量进行讨论，给出关于特征值与特征向量的相关命题．并对有关特征值与特征向量的题型做了解析以及解题的错误做出相应分析．%Eigenvalue and Eigenvectors are one of key items in Algebra study and closely related concepts. In theory and practice, Eigenvalue and Eigenvectors have important function. The paper discusses on Matrix Eigen value and Eigenvectors, listing and solving related propositions on Matrix Eigenvalue and Eigenvectors, and analyzing the corresponding causes on errors.
Variational methods applied to problems of diffusion and reaction
Strieder, William
1973-01-01
This monograph is an account of some problems involving diffusion or diffusion with simultaneous reaction that can be illuminated by the use of variational principles. It was written during a period that included sabbatical leaves of one of us (W. S. ) at the University of Minnesota and the other (R. A. ) at the University of Cambridge and we are grateful to the Petroleum Research Fund for helping to support the former and the Guggenheim Foundation for making possible the latter. We would also like to thank Stephen Prager for getting us together in the first place and for showing how interesting and useful these methods can be. We have also benefitted from correspondence with Dr. A. M. Arthurs of the University of York and from the counsel of Dr. B. D. Coleman the general editor of this series. Table of Contents Chapter 1. Introduction and Preliminaries . 1. 1. General Survey 1 1. 2. Phenomenological Descriptions of Diffusion and Reaction 2 1. 3. Correlation Functions for Random Suspensions 4 1. 4. Mean Free ...
Eigenvalue ratio detection based on exact moments of smallest and largest eigenvalues
Shakir, Muhammad
2011-01-01
Detection based on eigenvalues of received signal covariance matrix is currently one of the most effective solution for spectrum sensing problem in cognitive radios. However, the results of these schemes always depend on asymptotic assumptions since the close-formed expression of exact eigenvalues ratio distribution is exceptionally complex to compute in practice. In this paper, non-asymptotic spectrum sensing approach to approximate the extreme eigenvalues is introduced. In this context, the Gaussian approximation approach based on exact analytical moments of extreme eigenvalues is presented. In this approach, the extreme eigenvalues are considered as dependent Gaussian random variables such that the joint probability density function (PDF) is approximated by bivariate Gaussian distribution function for any number of cooperating secondary users and received samples. In this context, the definition of Copula is cited to analyze the extent of the dependency between the extreme eigenvalues. Later, the decision threshold based on the ratio of dependent Gaussian extreme eigenvalues is derived. The performance analysis of our newly proposed approach is compared with the already published asymptotic Tracy-Widom approximation approach. © 2011 ICST.
The Least Eigenvalue of Graphs
Institute of Scientific and Technical Information of China (English)
Guidong YU; Yizheng FAN; Yi WANG
2012-01-01
In this paper we investigate the least eigenvalue of a graph whose complement is connected,and present a lower bound for the least eigenvalue of such graph.We also characterize the unique graph whose least eigenvalue attains the second minimum among all graphs of fixed order.
DEFF Research Database (Denmark)
Lindberg, Erik
1997-01-01
In order to obtain insight in the nature of nonlinear oscillators the eigenvalues of the linearized Jacobian of the differential equations describing the oscillator are found and displayed as functions of time. A number of oscillators are studied including Dewey's oscillator (piecewise linear...... with negative resistance), Kennedy's Colpitts-oscillator (with and without chaos) and a new 4'th order oscillator with hyper-chaos....
Aeroelastic stability analysis of wind turbines using an eigenvalue approach
DEFF Research Database (Denmark)
Hansen, M.H.
2004-01-01
. To eliminate the periodic coefficients and avoid using the Floquet Theory, the multi-blade transformation is utilized. From the corresponding eigenvalue problem, the eigenvalues and eigenvectors can be computed at any operation condition to give the aeroelastic modal properties: Natural frequencies, damping...
Numerical pole assignment by eigenvalue Jacobian inversion
Sevaston, George E.
1986-01-01
A numerical procedure for solving the linear pole placement problem is developed which operates by the inversion of an analytically determined eigenvalue Jacobian matrix. Attention is given to convergence characteristics and pathological situations. It is not concluded that the algorithm developed is suitable for computer-aided control system design with particular reference to the scan platform pointing control system for the Galileo spacecraft.
Eigenvalue study of a chaotic resonator
Energy Technology Data Exchange (ETDEWEB)
Banova, Todorka [Technische Universitaet Darmstadt, Institut fuer Theorie Elektromagnetischer Felder (TEMF), Schlossgartenstrasse 8, D-64289 Darmstadt (Germany); Technische Universitaet Darmstadt, Graduate School of Computational Engineering, Dolivostrasse 15, D-64293 Darmstadt (Germany); Ackermann, Wolfgang; Weiland, Thomas [Technische Universitaet Darmstadt, Institut fuer Theorie Elektromagnetischer Felder (TEMF), Schlossgartenstrasse 8, D-64289 Darmstadt (Germany)
2013-07-01
The field of quantum chaos comprises the study of the manifestations of classical chaos in the properties of the corresponding quantum systems. Within this work, we compute the eigenfrequencies that are needed for the level spacing analysis of a microwave resonator with chaotic characteristics. The major challenges posed by our work are: first, the ability of the approaches to tackle the large scale eigenvalue problem and second, the capability to extract many, i.e. order of thousands, eigenfrequencies for the considered cavity. The first proposed approach for an accurate eigenfrequency extraction takes into consideration the evaluated electric field computations in time domain of a superconducting cavity and by means of signal-processing techniques extracts the eigenfrequencies. The second approach is based on the finite element method with curvilinear elements, which transforms the continuous eigenvalue problem to a discrete generalized eigenvalue problem. Afterwards, the Lanczos algorithm is used for the solution of the generalized eigenvalue problem. In the poster, a summary of the applied algorithms, as well as, critical implementation details together with the simulation results are provided.
On the sensitivities of multiple eigenvalues
DEFF Research Database (Denmark)
Gravesen, Jens; Evgrafov, Anton; Nguyen, Dang Manh
2011-01-01
polynomials of a number of eigenvalues, regardless of their multiplicity, which are known to be isolated from the rest depend smoothly on the parameter. We present explicit readily computable expressions for their first derivatives. Finally, we demonstrate the utility of our approach on a problem of finding...
Eigenvalues of the -Laplacian and disconjugacy criteria
Directory of Open Access Journals (Sweden)
Pinasco Juan P
2006-01-01
Full Text Available We derive oscillation and nonoscillation criteria for the one-dimensional -Laplacian in terms of an eigenvalue inequality for a mixed problem. We generalize the results obtained in the linear case by Nehari and Willett, and the proof is based on a Picone-type identity.
Directory of Open Access Journals (Sweden)
R. Sumithra
2012-02-01
Full Text Available The Hydrothermal growth of crystals is mathematically modeled as the onset of double diffusive magnetoconvection in a two-layer system comprising an incompressible two component, electrically conducting fluid saturated porous layer over which lies a layer of the same fluid in the presence a vertical magnetic field. Both the upper boundary of the fluid layer and the lower boundary of the porous layer are rigid and insulating to both heat and mass. At the interface the velocity, shear stress, normal stress, heat, heat flux,mass and mass flux are assumed to be continuous conducive for Darcy-Brinkman model. The resulting eigenvalue problem is solved by regular perturbation technique. The critical Rayleigh number, which is thecriterion for stability of the system is obtained. The effects of different physical parameters on the onset of double diffusive magnetoconvection are investigated in detail which enables to control convection during the growth of crystals in order to obtain pure crystals.
Murthy, D. V.
1989-01-01
This paper considers complex transcendental eigenvalue problems where one is interested in pairs of eigenvalues that are restricted to take real values only. Such eigenvalue problems arise in dynamic stability analysis of nonconservative physical systems, i.e., flutter analysis of aeroelastic systems. Some available solution methods are discussed and a new method is presented. Two computational approaches are described for analytical evaluation of the sensitivities of these eigenvalues when they are dependent on other parameters. The algorithms presented are illustrated through examples.
Distributions of Dirac Operator Eigenvalues
Akemann, G
2004-01-01
The distribution of individual Dirac eigenvalues is derived by relating them to the density and higher eigenvalue correlation functions. The relations are general and hold for any gauge theory coupled to fermions under certain conditions which are stated. As a special case, we give examples of the lowest-lying eigenvalue distributions for QCD-like gauge theories without making use of earlier results based on the relation to Random Matrix Theory.
Organization and diffusion in biological and material fabrication problems
Mangan, Niall Mari
This thesis is composed of two problems. The first is a systems level analysis of the carbon concentrating mechanism in cyanobacteria. The second presents a theoretical analysis of femtosecond laser melting for the purpose of hyperdoping silicon with sulfur. While these systems are very distant, they are both relevant to the development of alternative energy (production of biofuels and methods for fabricating photovoltaics respectively). Both problems are approached through analysis of the underlying diffusion equations. Cyanobacteria are photosynthetic bacteria with a unique carbon concentrating mechanism (CCM) which enhances carbon fixation. A greater understanding of this mechanism would offer new insights into the basic biology and methods for bioengineering more efficient biochemical reactions. The molecular components of the CCM have been well characterized in the last decade, with genetic analysis uncovering both variation and commonalities in CCMs across cyanobacteria strains. Analysis of CCMs on a systems level, however, is based on models formulated prior to the molecular characterization. We present an updated model of the cyanobacteria CCM, and analytic solutions in terms of the various molecular components. The solutions allow us to find the parameter regime (expression levels, catalytic rates, permeability of carboxysome shell) where carbon fixation is maximized and oxygenation is minimized. Saturation of RuBisCO, maximization of the ratio of CO2 to O2, and staying below or at the saturation level for carbonic anhydrase are all needed for maximum efficacy. These constraints limit the parameter regime where the most effective carbon fixation can occur. There is an optimal non-specific carboxysome shell permeability, where trapping of CO2 is maximized, but HCO3 - is not detrimentally restricted. The shell also shields carbonic anhydrase activity and CO2 → HCO3- conversion at the thylakoid and cell membrane from one another. Co-localization of carbonic
Computational Methods for Multi-dimensional Neutron Diffusion Problems
Energy Technology Data Exchange (ETDEWEB)
Song Han
2009-10-15
Lead-cooled fast reactor (LFR) has potential for becoming one of the advanced reactor types in the future. Innovative computational tools for system design and safety analysis on such NPP systems are needed. One of the most popular trends is coupling multi-dimensional neutron kinetics (NK) with thermal-hydraulic (T-H) to enhance the capability of simulation of the NPP systems under abnormal conditions or during rare severe accidents. Therefore, various numerical methods applied in the NK module should be reevaluated to adapt the scheme of coupled code system. In the author's present work a neutronic module for the solution of two dimensional steady-state multigroup diffusion problems in nuclear reactor cores is developed. The module can produce both direct fluxes as well as adjoints, i.e. neutron importances. Different numerical schemes are employed. A standard finite-difference (FD) approach is firstly implemented, mainly to serve as a reference for less computationally challenging schemes, such as transverse-integrated nodal methods (TINM) and boundary element methods (BEM), which are considered in the second part of the work. The validation of the methods proposed is carried out by comparisons of the results for some reference structures. In particular a critical problem for a homogeneous reactor for which an analytical solution exists is considered as a benchmark. The computational module is then applied to a fast spectrum system, having physical characteristics similar to the proposed European Lead-cooled System (ELSY) project. The results show the effectiveness of the numerical techniques presented. The flexibility and the possibility to obtain neutron importances allow the use of the module for parametric studies, design assessments and integral parameter evaluations, as well as for future sensitivity and perturbation analyses and as a shape solver for time-dependent procedures
Dual variational formulas for the first Dirichlet eigenvalue on half-line
Institute of Scientific and Technical Information of China (English)
Chen; Mufa(陈木法); ZHANG; Yuhui(张余辉); ZHAO; Xiaoliang(赵晓亮)
2003-01-01
The aim of the paper is to establish two dual variational formulas for the first Dirichlet eigenvalue of the second order elliptic operators on half-line. Some explicit bounds of the eigenvalue depending only on the coefficients of the operators are presented. Moreover, the corresponding problems in the discrete case and the higher-order eigenvalues in the continuous case are also studied.
On the design derivatives of eigenvalues and eigenvectors for distributed parameter systems
Reiss, R.
1985-01-01
In this paper, analytic expressions are obtained for the design derivatives of eigenvalues and eigenfunctions of self-adjoint linear distributed parameter systems. Explicit treatment of boundary conditions is avoided by casting the eigenvalue equation into integral form. Results are expressed in terms of the linear operators defining the eigenvalue problem, and are therefore quite general. Sufficiency conditions appropriate to structural optimization of eigenvalues are obtained.
Maximization of eigenvalues using topology optimization
DEFF Research Database (Denmark)
Pedersen, Niels Leergaard
2000-01-01
Topology optimization is used to optimize the eigenvalues of plates. The results are intended especially for MicroElectroMechanical Systems (MEMS) but call be seen as more general. The problem is not formulated as a case of reinforcement of an existing structure, so there is a problem related...... to localized modes in low density areas. The topology optimization problem is formulated using the SIMP method. Special attention is paid to a numerical method for removing localized eigenmodes in low density areas. The method is applied to numerical examples of maximizing the first eigenfrequency, One example...... is a practical MEMS application; a probe used in an Atomic Force Microscope (AFM). For the AFM probe the optimization is complicated by a constraint on the stiffness and constraints on higher order eigenvalues....
On the distribution of eigenvalues of non-selfadjoint operators
Demuth, Michael; Katriel, Guy
2008-01-01
We prove quantitative bounds on the eigenvalues of non-selfadjoint bounded and unbounded operators. We use the perturbation determinant to reduce the problem to one of studying the zeroes of a holomorphic function.
Numerical methods for evaluating the derivatives of eigenvalues and eigenvectors
Rudisill, C. S.; Chu, Y.-Y.
1975-01-01
Two numerical methods are presented for computing the derivatives of eigenvalues and eigenvectors which do not require complete solution of the eigenvalue problem if only a few derivatives are sought. The 'iterative' method may be used to find the first derivative of one or all of the eigenvectors together with the second derivative of their eigenvalues in a self-adjoint system. If the left- and right-hand eigenvectors are known, the first derivative of the eigenvector corresponding to the largest eigenvalue and the second derivative of the largest eigenvalue may be obtained for a nonself-adjoint system. The 'algebraic' method may be used to find all orders of the derivatives, provided they exist, without requiring the left-hand eigenvectors.
Eigenvalue based Spectrum Sensing Algorithms for Cognitive Radio
Zeng, Yonghong
2008-01-01
Spectrum sensing is a fundamental component is a cognitive radio. In this paper, we propose new sensing methods based on the eigenvalues of the covariance matrix of signals received at the secondary users. In particular, two sensing algorithms are suggested, one is based on the ratio of the maximum eigenvalue to minimum eigenvalue; the other is based on the ratio of the average eigenvalue to minimum eigenvalue. Using some latest random matrix theories (RMT), we quantify the distributions of these ratios and derive the probabilities of false alarm and probabilities of detection for the proposed algorithms. We also find the thresholds of the methods for a given probability of false alarm. The proposed methods overcome the noise uncertainty problem, and can even perform better than the ideal energy detection when the signals to be detected are highly correlated. The methods can be used for various signal detection applications without requiring the knowledge of signal, channel and noise power. Simulations based ...
Extreme eigenvalues of sample covariance and correlation matrices
DEFF Research Database (Denmark)
Heiny, Johannes
This thesis is concerned with asymptotic properties of the eigenvalues of high-dimensional sample covariance and correlation matrices under an infinite fourth moment of the entries. In the first part, we study the joint distributional convergence of the largest eigenvalues of the sample covariance...... of the problem at hand. We develop a theory for the point process of the normalized eigenvalues of the sample covariance matrix in the case where rows and columns of the data are linearly dependent. Based on the weak convergence of this point process we derive the limit laws of various functionals...... of the eigenvalues. In the second part, we show that the largest and smallest eigenvalues of a highdimensional sample correlation matrix possess almost sure non-random limits if the truncated variance of the entry distribution is “almost slowly varying”, a condition we describe via moment properties of self...
Nonlinear predator-prey singularly perturbed Robin Problems for reaction diffusion systems
Institute of Scientific and Technical Information of China (English)
莫嘉琪; 韩祥临
2003-01-01
The nonlinear predator-prey reaction diffusion systems for singularly perturbed Robin Problems are considered. Under suitable conditions, the theory of differential inequalities can be used to study the asymptotic behavior of the solution for initial boundary value problems.
Institute of Scientific and Technical Information of China (English)
莫嘉琪
2003-01-01
The nonlinear predator-prey singularly perturbed Robin initial boundary value problems for reaction diffusion systems were considered. Under suitable conditions, using theory of differential inequalities the existence and asymptotic behavior of solution for initial boundary value problems were studied.
Institute of Scientific and Technical Information of China (English)
Jia-qi Mo; Wan-tao Lin
2006-01-01
In this paper the singularly perturbed initial boundary value problems for the nonlocal reaction diffusion system are considered. Using the iteration method and the comparison theorem, the existence and its asymptotic behavior of the solution for the problem are studied.
THE CORNER LAYER SOLUTION TO ROBIN PROBLEM FOR REACTION DIFFUSION EQUATION
Institute of Scientific and Technical Information of China (English)
无
2012-01-01
A class of Robin boundary value problem for reaction diffusion equation is considered. Under suitable conditions, using the theory of differential inequalities the existence and asymptotic behavior of the corner layer solution to the initial boundary value problem are studied.
Analysis of Diffusion Problems using Homotopy Perturbation and Variational Iteration Methods
DEFF Research Database (Denmark)
Barari, Amin; Poor, A. Tahmasebi; Jorjani, A.
2010-01-01
In this paper, variational iteration method and homotopy perturbation method are applied to different forms of diffusion equation. The diffusion equations have found wide applications in heat transfer problems, theory of consolidation and many other problems in engineering. The methods proposed t...
An inverse problem for a one-dimensional time-fractional diffusion problem
Jin, Bangti
2012-06-26
We study an inverse problem of recovering a spatially varying potential term in a one-dimensional time-fractional diffusion equation from the flux measurements taken at a single fixed time corresponding to a given set of input sources. The unique identifiability of the potential is shown for two cases, i.e. the flux at one end and the net flux, provided that the set of input sources forms a complete basis in L 2(0, 1). An algorithm of the quasi-Newton type is proposed for the efficient and accurate reconstruction of the coefficient from finite data, and the injectivity of the Jacobian is discussed. Numerical results for both exact and noisy data are presented. © 2012 IOP Publishing Ltd.
Eigenvalue Conditions for Induced Subgraphs
Directory of Open Access Journals (Sweden)
Harant Jochen
2015-05-01
Full Text Available Necessary conditions for an undirected graph G to contain a graph H as induced subgraph involving the smallest ordinary or the largest normalized Laplacian eigenvalue of G are presented.
Lowest Eigenvalues of Random Hamiltonians
Shen, J J; Arima, A; Yoshinaga, N
2008-01-01
In this paper we present results of the lowest eigenvalues of random Hamiltonians for both fermion and boson systems. We show that an empirical formula of evaluating the lowest eigenvalues of random Hamiltonians in terms of energy centroids and widths of eigenvalues are applicable to many different systems (except for $d$ boson systems). We improve the accuracy of the formula by adding moments higher than two. We suggest another new formula to evaluate the lowest eigenvalues for random matrices with large dimensions (20-5000). These empirical formulas are shown to be applicable not only to the evaluation of the lowest energy but also to the evaluation of excited energies of systems under random two-body interactions.
The Laplacian eigenvalues of graphs: a survey
Zhang, Xiao-Dong
2011-01-01
The Laplacian matrix of a simple graph is the difference of the diagonal matrix of vertex degree and the (0,1) adjacency matrix. In the past decades, the Laplacian spectrum has received much more and more attention, since it has been applied to several fields, such as randomized algorithms, combinatorial optimization problems and machine learning. This paper is primarily a survey of various aspects of the eigenvalues of the Laplacian matrix of a graph for the past teens. In addition, some new unpublished results and questions are concluded. Emphasis is given on classifications of the upper and lower bounds for the Laplacian eigenvalues of graphs (including some special graphs, such as trees, bipartite graphs, triangular-free graphs, cubic graphs, etc.) as a function of other graph invariants, such as degree sequence, the average 2-degree, diameter, the maximal independence number, the maximal matching number, vertex connectivity, the domination number, the number of the spanning trees, etc.
The Method of Lines for Ternary Diffusion Problems
Directory of Open Access Journals (Sweden)
Henryk Leszczyński
2014-01-01
Full Text Available The method of lines (MOL for diffusion equations with Neumann boundary conditions is considered. These equations are transformed by a discretization in space variables into systems of ordinary differential equations. The proposed ODEs satisfy the mass conservation law. The stability of solutions of these ODEs with respect to discrete L2 norms and discrete W1,∞ norms is investigated. Numerical examples confirm the parabolic behaviour of this model and very regular dynamics.
A CLASS OF SINGULARLY PERTURBED INITIAL BOUNDARY PROBLEM FOR REACTION DIFFUSION EQUATION
Institute of Scientific and Technical Information of China (English)
Xie Feng
2003-01-01
The singularly perturbed initial boundary value problem for a class of reaction diffusion equation isconsidered. Under appropriate conditions, the existence-uniqueness and the asymptotic behavior of the solu-tion are showed by using the fixed-point theorem.
Singlet states and the estimation of eigenstates and eigenvalues of an unknown Controlled-U gate
Hillery, M; Hillery, Mark; Buzek, Vladimir
2001-01-01
We consider several problems that involve finding the eigenvalues and generating the eigenstates of unknown unitary gates. We first examine Controlled-U gates that act on qubits, and assume that we know the eigenvalues. It is then shown how to use singlet states to produce qubits in the eigenstates of the gate. We then remove the assumption that we know the eigenvalues and show how to both find the eigenvalues and produce qubits in the eigenstates. Finally, we look at the case where the unitary operator acts on qutrits and has eigenvalues of 1 and -1, where the eigenvalue 1 is doubly degenerate. The eigenstates are unknown. We are able to use a singlet state to produce a qutrit in the eigenstate corresponding to the -1 eigenvalue.
An adjoint-based scheme for eigenvalue error improvement
Energy Technology Data Exchange (ETDEWEB)
Merton, S.R.; Smedley-Stevenson, R.P., E-mail: Simon.Merton@awe.co.uk, E-mail: Richard.Smedley-Stevenson@awe.co.uk [AWE plc, Berkshire (United Kingdom); Pain, C.C.; El-Sheikh, A.H.; Buchan, A.G., E-mail: c.pain@imperial.ac.uk, E-mail: a.el-sheikh@imperial.ac.uk, E-mail: andrew.buchan@imperial.ac.uk [Department of Earth Science and Engineering, Imperial College, London (United Kingdom)
2011-07-01
A scheme for improving the accuracy and reducing the error in eigenvalue calculations is presented. Using a rst order Taylor series expansion of both the eigenvalue solution and the residual of the governing equation, an approximation to the error in the eigenvalue is derived. This is done using a convolution of the equation residual and adjoint solution, which is calculated in-line with the primal solution. A defect correction on the solution is then performed in which the approximation to the error is used to apply a correction to the eigenvalue. The method is shown to dramatically improve convergence of the eigenvalue. The equation for the eigenvalue is shown to simplify when certain normalizations are applied to the eigenvector. Two such normalizations are considered; the rst of these is a fission-source type of normalisation and the second is an eigenvector normalisation. Results are demonstrated on a number of demanding elliptic problems using continuous Galerkin weighted nite elements. Moreover, the correction scheme may also be applied to hyperbolic problems and arbitrary discretization. This is not limited to spatial corrections and may be used throughout the phase space of the discrete equation. The applied correction not only improves fidelity of the calculation, it allows assessment of the reliability of numerical schemes to be made and could be used to guide mesh adaption algorithms or to automate mesh generation schemes. (author)
A chaotic model for advertising diffusion problem with competition
Ip, W. H.; Yung, K. L.; Wang, Dingwei
2012-08-01
In this article, the author extends Dawid and Feichtinger's chaotic advertising diffusion model into the duopoly case. A computer simulation system is used to test this enhanced model. Based on the analysis of simulation results, it is found that the best advertising strategy in duopoly is to increase the advertising investment to reach the best Win-Win situation where the oscillation of market portion will not occur. In order to effectively arrive at the best situation, we define a synthetic index and two thresholds. An estimation method for the parameters of the index and thresholds is proposed in this research. We can reach the Win-Win situation by simply selecting the control parameters to make the synthetic index close to the threshold of min-oscillation state. The numerical example and computational results indicated that the proposed chaotic model is useful to describe and analyse advertising diffusion process in duopoly, it is an efficient tool for the selection and optimisation of advertising strategy.
Institute of Scientific and Technical Information of China (English)
张勇; 林皋; 胡志强
2012-01-01
提出比例边界等几何方法（scaled boundary isogeometric analysis，SBIGA），并用以求解波导本征值问题．在比例边界等几何坐标变换的基础上，利用加权余量法将控制偏微分方程进行离散处理，半弱化为关于边界控制点变量的二阶常微分方程，即TE波或TM波波导的比例边界等几何分析的频域方程以及波导动刚度方程，同时利用连分式求解波导动刚度矩阵．通过引入辅助变量进一步得出波导本征方程．该方法只需在求解域的边界上进行等几何离散，使问题降低一维，计算工作量大为节约，并且由于边界的等几何离散，使得解的精度更高，进一步节省求解自由度．以矩形和L形波导的本征问题分析为例，通过与解析解和其他数值方法比较，结果表明该方法具有精度高、计算工作量小的优点．%Scaled boundary isogeometric analysis （SBIGA） is approved and applied for waveguide eigenvalue problem. Based on scaled boundary isogeometric transformation, the governing partial differential equations （PDEs） for waveguide eigenvalue problem are semi-weakened to a set of 2nd order ordinary differential equations （ODEs） by weighted residual methods, and transformed to a set of 1st order ODEs about the dynamic stiffness matrix in wavenumber domain. Approximating the dynamic stiffness matrix in the continued fraction expression and introducing auxiliary variables, the ODEs are finally and thus the cutoff wavenumber of waveguide is obtained. exported to algebraic general eigenvalue equations, The main property of SBIGA is that the governing PDEs are isogeometricly discretized on domain boundary, which reduces the spatial dimension by one and analytical feature in the radial direction like traditional SBFEM, additionally, boundary is exactly discretized as its geometry design. The numerical examples, including rectangular and L-shaped waveguides, are presented and
Sturm-Liouville eigenvalue characterizations
Directory of Open Access Journals (Sweden)
Paul B. Bailey
2012-07-01
Full Text Available We study the relationship between the eigenvalues of separated self-adjoint boundary conditions and coupled self-adjoint conditions. Given an arbitrary real coupled boundary condition determined by a coupling matrix K we construct a one parameter family of separated conditions and show that all the eigenvalues for K and -K are extrema of the eigencurves of this family. This characterization makes it possible to use the well known Prufer transformation which has been used very successfully, both theoretically and numerically, for separated conditions, also in the coupled case. In particular, this characterization makes it possible to compute the eigenvalues for any real coupled self-adjoint boundary condition using any code which works for separated conditions.
Analytical computation of the eigenvalues and eigenvectors in DT-MRI.
Hasan, K M; Basser, P J; Parker, D L; Alexander, A L
2001-09-01
In this paper a noniterative algorithm to be used for the analytical determination of the sorted eigenvalues and corresponding orthonormalized eigenvectors obtained by diffusion tensor magnetic resonance imaging (DT-MRI) is described. The algorithm uses the three invariants of the raw water spin self-diffusion tensor represented by a 3 x 3 positive definite matrix and certain math functions that do not require iteration. The implementation requires a positive definite mask to preserve the physical meaning of the eigenvalues. This algorithm can increase the speed of eigenvalue/eigenvector calculations by a factor of 5-40 over standard iterative Jacobi or singular-value decomposition techniques. This approach may accelerate the computation of eigenvalues, eigenvalue-dependent metrics, and eigenvectors especially when having high-resolution measurements with large numbers of slices and large fields of view.
Institute of Scientific and Technical Information of China (English)
尹凤; 黄光鑫
2012-01-01
Let R∈Cm×m" andSeCm×n be two nontrivial involutions, i.e. , R=R-1≠±Im and S = S-1≠ ± /zl. A matrix A∈Cm×n is called (R, S)-symmetric matrix if A is satisfactory to RAS = A. This paper first gives the solvable conditions and the general expressions of the (R, S) -symmetric solutions for the left and right inverse eigenvalue problem. Then, the corresponding best approximation problem to the ieft and right inverse eigenvalue problem is also solved over (R, S)-symmetric matrix solution.%令R∈Cm×m和S∈Cn×n是2个非平凡卷积矩阵,即R=R-1≠±Im且S=S-1≠±In.如果一个矩阵A∈Cm×n满足RAS-A,则矩阵A称为(R,S)对称矩阵.本文首先分别给出了左右逆特征值问题的(R,S)对称矩阵解的可解条件和一般表达式；然后,给出了左右逆特征值问题相应的最佳逼近问题的(R,S)对称矩阵解.
On the eigenvalues of a "dumb-bell with a thin handle"
Gadyl'shin, R. R.
2005-04-01
We consider the Neumann boundary-value problem of finding the small-parameter asymptotics of the eigenvalues and eigenfunctions for the Laplace operator in a singularly perturbed domain consisting of two bounded domains joined by a thin "handle". The small parameter is the diameter of the cross-section of the handle. We show that as the small parameter tends to zero these eigenvalues converge either to the eigenvalues corresponding to the domains joined or to the eigenvalues of the Dirichlet problem for the Sturm-Liouville operator on the segment to which the thin handle contracts. The main results of this paper are the complete power small-parameter asymptotics of the eigenvalues and the corresponding eigenfunctions and explicit formulae for the first terms of the asymptotics. We consider critical cases generated by the choice of the place where the thin "handle" is joined to the domains, as well as by the multiplicity of the eigenvalues corresponding to the domains joined.
A Series Solution of the Cauchy Problem for Turing Reaction-diffusion Model
Directory of Open Access Journals (Sweden)
L. Päivärinta
2011-12-01
Full Text Available In this paper, the series pattern solution of the Cauchy problem for Turing reaction-diffusion model is obtained by using the homotopy analysis method (HAM. Turing reaction-diffusion model is nonlinear reaction-diffusion system which usually has power-law nonlinearities or may be rewritten in the form of power-law nonlinearities. Using the HAM, it is possible to find the exact solution or an approximate solution of the problem. This technique provides a series of functions which converges rapidly to the exact solution of the problem. The efficiency of the approach will be shown by applying the procedure on two problems. Furthermore, the so-called homotopy-Pade technique (HPT is applied to enlarge the convergence region and rate of solution series given by the HAM.
Li, Xianping
2010-01-01
Heterogeneous anisotropic diffusion problems arise in the various areas of science and engineering including plasma physics, petroleum engineering, and image processing. Standard numerical methods can produce spurious oscillations when they are used to solve those problems. A common approach to avoid this difficulty is to design a proper numerical scheme and/or a proper mesh so that the numerical solution validates the discrete counterpart (DMP) of the maximum principle satisfied by the continuous solution. A well known mesh condition for the DMP satisfaction by the linear finite element solution of isotropic diffusion problems is the non-obtuse angle condition that requires the dihedral angles of mesh elements to be non-obtuse. In this paper, a generalization of the condition, the so-called anisotropic non-obtuse angle condition, is developed for the finite element solution of heterogeneous anisotropic diffusion problems. The new condition is essentially the same as the existing one except that the dihedral ...
Energy Technology Data Exchange (ETDEWEB)
Lehoucq, Richard B.; Salinger, Andrew G.
1999-08-01
We present an approach for determining the linear stability of steady states of PDEs on massively parallel computers. Linearizing the transient behavior around a steady state leads to a generalized eigenvalue problem. The eigenvalues with largest real part are calculated using Arnoldi's iteration driven by a novel implementation of the Cayley transformation to recast the problem as an ordinary eigenvalue problem. The Cayley transformation requires the solution of a linear system at each Arnoldi iteration, which must be done iteratively for the algorithm to scale with problem size. A representative model problem of 3D incompressible flow and heat transfer in a rotating disk reactor is used to analyze the effect of algorithmic parameters on the performance of the eigenvalue algorithm. Successful calculations of leading eigenvalues for matrix systems of order up to 4 million were performed, identifying the critical Grashof number for a Hopf bifurcation.
Mimetic Discretization of Vector-valued Diffusion Problems
DEFF Research Database (Denmark)
Olesen, Kennet
relations are based on universal physical laws and if these are not represented correctly by the numerical scheme, the wrong physics are being solved for. General schemes like the Finite Difference Method (FDM) and the Finite Element Method (FEM) approximate the solution of all the involved Partial...... the following attractive properties: - There exist a natural link between geometrical objects (points, lines, surfaces and volumes) and physical quantities through integration. - The integral of differential operators, defined through the exterior derivative, allows relations between geometrical object...... of vector- and covector-valued differential forms. Special considerations are taken to maintain the intrinsic nature of the operators. Several solutions to relevant 2D problems are shown to document the preserving nature of the method and the attractive convergence rates obtained when using spectral...
The second boundary value problem for equations of viscoelastic diffusion in polymers
Vorotnikov, Dmitry A
2009-01-01
The classical approach to diffusion processes is based on Fick's law that the flux is proportional to the concentration gradient. Various phenomena occurring during propagation of penetrating liquids in polymers show that this type of diffusion exhibits anomalous behavior and contradicts the just mentioned law. However, they can be explained in the framework of non-Fickian diffusion theories based on viscoelasticity of polymers. Initial-boundary value problems for viscoelastic diffusion equations have been studied by several authors. Most of the studies are devoted to the Dirichlet BVP (the concentration is given on the boundary of the domain). In this chapter we study the second BVP, i.e. when the normal component of the concentration flux is prescribed on the boundary, which is more realistic in many physical situations. We establish existence of weak solutions to this problem. We suggest some conditions on the coefficients and boundary data under which all the solutions tend to the homogeneous state as tim...
Integral transform methodology for convection-diffusion problems in petroleum reservoir engineering
Energy Technology Data Exchange (ETDEWEB)
Almeida, A.R. [PETROBRAS, Rio de Janeiro, RJ (Brazil); Cotta, R.M. [Universidade Federal, Rio de Janeiro, RJ (Brazil)
1995-12-01
The convection-diffusion equation is present in the formulation of many petroleum reservoir engineering problems. A representative example, the tracer injection problem, is solved analytically here, through the generalised integral transform technique so as to illustrate the usefulness of this approach, for this class of problems. Classical assumptions, such as steady-state single phase flow and unit mobility ratio, are adopted. Comparisons with alternative analytical (when available) or numerical (finite difference) solutions are performed and benchmark results are established. (author)
A multigroup radiation diffusion test problem: Comparison of code results with analytic solution
Energy Technology Data Exchange (ETDEWEB)
Shestakov, A I; Harte, J A; Bolstad, J H; Offner, S R
2006-12-21
We consider a 1D, slab-symmetric test problem for the multigroup radiation diffusion and matter energy balance equations. The test simulates diffusion of energy from a hot central region. Opacities vary with the cube of the frequency and radiation emission is given by a Wien spectrum. We compare results from two LLNL codes, Raptor and Lasnex, with tabular data that define the analytic solution.
Generalized eigenvalue based spectrum sensing
Shakir, Muhammad
2012-01-01
Spectrum sensing is one of the fundamental components in cognitive radio networks. In this chapter, a generalized spectrum sensing framework which is referred to as Generalized Mean Detector (GMD) has been introduced. In this context, we generalize the detectors based on the eigenvalues of the received signal covariance matrix and transform the eigenvalue based spectrum sensing detectors namely: (i) the Eigenvalue Ratio Detector (ERD) and two newly proposed detectors which are referred to as (ii) the GEometric Mean Detector (GEMD) and (iii) the ARithmetic Mean Detector (ARMD) into an unified framework of generalize spectrum sensing. The foundation of the proposed framework is based on the calculation of exact analytical moments of the random variables of the decision threshold of the respective detectors. The decision threshold has been calculated in a closed form which is based on the approximation of Cumulative Distribution Functions (CDFs) of the respective test statistics. In this context, we exchange the analytical moments of the two random variables of the respective test statistics with the moments of the Gaussian (or Gamma) distribution function. The performance of the eigenvalue based detectors is compared with the several traditional detectors including the energy detector (ED) to validate the importance of the eigenvalue based detectors and the performance of the GEMD and the ARMD particularly in realistic wireless cognitive radio network. Analytical and simulation results show that the newly proposed detectors yields considerable performance advantage in realistic spectrum sensing scenarios. Moreover, the presented results based on proposed approximation approaches are in perfect agreement with the empirical results. © 2012 Springer Science+Business Media Dordrecht.
THE EIGENVALUE PERTURBATION BOUND FOR ARBITRARY MATRICES
Institute of Scientific and Technical Information of China (English)
Wen Li; Jian-xin Chen
2006-01-01
In this paper we present some new absolute and relative perturbation bounds for the eigenvalue for arbitrary matrices, which improves some recent results. The eigenvalue inclusion region is also discussed.
Two regularization methods for solving a Riesz-Feller space-fractional backward diffusion problem
Zheng, G. H.; Wei, T.
2010-11-01
In this paper, a backward diffusion problem for a space-fractional diffusion equation (SFDE) in a strip is investigated. Such a problem is obtained from the classical diffusion equation in which the second-order space derivative is replaced with a Riesz-Feller derivative of order β in (0, 2]. We show that such a problem is severely ill-posed and further propose a new regularization method and apply a spectral regularization method to solve it based on the solution given by the Fourier method. Convergence estimates are presented under a priori bound assumptions for the exact solution. Finally, numerical examples are given to show that the proposed numerical methods are effective.
On a Fourth-order Eigenvalue Problem
Institute of Scientific and Technical Information of China (English)
无
2000-01-01
@@ We consider the existence of positive solutions for the equation d4y/dx4-λf(x,y(x))=0,(1) with one of the following sets of boundary value conditions y(0)=y(1)=y"(0)=y"(1)=0,(2) y(0)=y′(1)=y"(0)=y"′(1)=0.(3)
Methods of intermediate problems for eigenvalues
Weinstein, Alexander
1972-01-01
In this book, we study theoretical and practical aspects of computing methods for mathematical modelling of nonlinear systems. A number of computing techniques are considered, such as methods of operator approximation with any given accuracy; operator interpolation techniques including a non-Lagrange interpolation; methods of system representation subject to constraints associated with concepts of causality, memory and stationarity; methods of system representation with an accuracy that is the best within a given class of models; methods of covariance matrix estimation;methods for low-rank mat
Polar-symmetric problem of elastic diffusion for isotropic multi-component plane
Zemskov, A. V.; Tarlakovskii, D. V.
2016-11-01
The paper considers a polar-symmetric problem of finding a stress strain condition of a plane influenced by non-stationary volume elastic diffusion disturbances. The mathematical model is based on a connected system of equations of elastic diffusion in a polar coordinate system. The solution of the problem is sought in an integral for and presented in the form of convolutions of Green's function with the right side of equation of motion and mass transfer. Laplace time and Hankel's radial coordinate transformations are used to find the Green's functions. The inverse Laplace transform is done analytically by residue. The inverse Hankel's transform is done numerically by quadrature formulas.
Directory of Open Access Journals (Sweden)
Medet Nursultanov
2017-01-01
Full Text Available We find asymptotic formulas for the eigenvalues of the Sturm-Liouville operator on the finite interval, with potential having a strong negative singularity at one endpoint. This is the case of limit circle in H. Weyl sense. We establish that, unlike the case of an infinite interval, the asymptotics for positive eigenvalues does not depend on the potential and it is the same as in the regular case. The asymptotics of the negative eigenvalues may depend on the potential quite strongly, however there are always asymptotically fewer negative eigenvalues than positive ones. By unknown reasons this type of problems had not been studied previously.
Institute of Scientific and Technical Information of China (English)
焦宝宝
2016-01-01
在116 Sn原子核壳模型结构下，利用广义辛弱数截断多体空间得到的哈密顿矩阵是一个大型的实对称非正交归一基稀疏矩阵，因此求解大型矩阵的能量特征值和能量特征向量是原子核物理上的一个重要问题。为此，利用重正交化Lanczos法与Cholesky分解法和Elementary transformation法相结合的方法，实现了用内存较小的计算机求解大型实对称非正交归一基稀疏矩阵的特征值和特征向量。用这种方法计算小型矩阵得到的特征值和精确值符合得较好，且运用这个方法计算了116Sn壳模型截断后的大型非正交归一基稀疏矩阵的能量特征值，得到的原子核低态能量与实验测量能量相吻合，计算结果表明Lanczos法在Matlab编程和大型壳模型计算中的精确性和可行性。此方法也有助于求解一些中质核或者重核的低态能量，同时也有利于用内存稍大的计算机求解更大的非正交归一基矩阵的特征值问题。%Using shell model to calculate the nuclear systems in a large model space is an important method in the field of nuclear physics. On the basis of the nuclear shell model, a large symmetric non-orthonormal sparse Hamiltonian matrix is generated when adopting the generalized seniority method to truncate the many-body space. Calculating the energy eigenvalues and energy eigenvectors of the large symmetric non-orthonormal sparse Hamiltonian matrix is of indispensable steps before energies of nucleus are further calculated. In the mean time, some low-lying energy eigenvalues are always the focus of attention on the occasion of large scale shell model calculation. In this paper, by combining reorthogonalization Lanczos method with Cholesky decomposition method and Elementary transformation method, converting the generalized eigenvalue problems into the standard eigenvalue problems, and transforming the large standard eigenvalue problems into the small standard
h-p finite element method for the Lambda modes problem in hexagonal geometries
Energy Technology Data Exchange (ETDEWEB)
Fayez, R.; Vidal-Ferrandiz, A.; Ginestar, D.; Verdu, G.
2014-07-01
Most of the simulation codes of a nuclear power reactor use the multigroup neutron diffusion equation to describe the neutron distribution inside the reactor core. to study the stationary state of a reactor, the reactor criticality is forced in artificial way leading to a generalized different eigenvalue problem, known as the Lambda modes equation, which is solved to obtain the dominant eigenvalues of the reactor and their corresponding eigenfunctions. (Author)
Eigenvalue pinching on spinc manifolds
Roos, Saskia
2017-02-01
We derive various pinching results for small Dirac eigenvalues using the classification of spinc and spin manifolds admitting nontrivial Killing spinors. For this, we introduce a notion of convergence for spinc manifolds which involves a general study on convergence of Riemannian manifolds with a principal S1-bundle. We also analyze the relation between the regularity of the Riemannian metric and the regularity of the curvature of the associated principal S1-bundle on spinc manifolds with Killing spinors.
Institute of Scientific and Technical Information of China (English)
Jingsun Yao; Jiaqi Mo
2005-01-01
The nonlinear nonlocal singularly perturbed initial boundary value problems for reaction diffusion equations with a boundary perturbation is considered. Under suitable conditions, the outer solution of the original problem is obtained. Using the stretched variable, the composing expansion method and the expanding theory of power series the initial layer is constructed. And then using the theory of differential inequalities the asymptotic behavior of solution for the initial boundary value problems is studied. Finally the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation are discussed.
On strongly degenerate convection-diffusion Problems Modeling sedimentation-consolidation Processes
Energy Technology Data Exchange (ETDEWEB)
Buerger, R.; Evje, S.; Karlsen, S. Hvistendahl
1999-10-01
This report investigates initial-boundary value problems for a quasilinear strongly degenerate convection-diffusion equation with a discontinuous diffusion coefficient. These problems come from the mathematical modelling of certain sedimentation-consolidation processes. Existence of entropy solutions belonging to BV is shown by the vanishing viscosity method. The existence proof for one of the models includes a new regularity result for the integrated diffusion coefficient. New uniqueness proofs for entropy solutions are also presented. These proofs rely on a recent extension to second order equations of Kruzkov`s method of `doubling of the variables`. The application to a sedimentation-consolidation model is illustrated by two numerical examples. 25 refs., 2 figs.
Finite volume element method for analysis of unsteady reaction-diffusion problems
Institute of Scientific and Technical Information of China (English)
Sutthisak Phongthanapanich; Pramote Dechaumphai
2009-01-01
A finite volume element method is developed for analyzing unsteady scalar reaction--diffusion problems in two dimensions. The method combines the concepts that are employed in the finite volume and the finite element method together. The finite volume method is used to discretize the unsteady reaction--diffusion equation, while the finite element method is applied to estimate the gradient quantities at cell faces. Robustness and efficiency of the combined method have been evaluated on uniform rectangular grids by using available numerical solutions of the two-dimensional reaction-diffusion problems. The numerical solutions demonstrate that the combined method is stable and can provide accurate solution without spurious oscillation along the highgradient boundary layers.
A two-phase free boundary problem for a nonlinear diffusion-convection equation
Energy Technology Data Exchange (ETDEWEB)
De Lillo, S; Lupo, G [Dipartimento di Matematica e Informatica, Universita degli Studi di Perugia, Via Vanvitelli 1, 06123 Perugia (Italy)], E-mail: silvana.delillo@pg.infn.it
2008-04-11
A two-phase free boundary problem associated with a diffusion-convection equation is considered. The problem is reduced to a system of nonlinear integral equations, which admits a unique solution for small times. The system admits an explicit two-component solution corresponding to a two-component shock wave of the Burgers equation. The stability of such a solution is also discussed.
Derivatives of eigenvalues and eigenvectors of a general complex matrix
Murthy, Durbha V.; Haftka, Raphael T.
1988-01-01
A survey of methods for sensitivity analysis of the algebraic eigenvalue problem for non-Hermitian matrices is presented. In addition, a modification of one method based on a better normalizing condition is proposed. Methods are classified as Direct or Adjoint and are evaluated for efficiency. Operation counts are presented in terms of matrix size, number of design variables and number of eigenvalues and eigenvectors of interest. The effect of the sparsity of the matrix and its derivatives is also considered, and typical solution times are given. General guidelines are established for the selection of the most efficient method.
A subspace preconditioning algorithm for eigenvector/eigenvalue computation
Energy Technology Data Exchange (ETDEWEB)
Bramble, J.H.; Knyazev, A.V.; Pasciak, J.E.
1996-12-31
We consider the problem of computing a modest number of the smallest eigenvalues along with orthogonal bases for the corresponding eigen-spaces of a symmetric positive definite matrix. In our applications, the dimension of a matrix is large and the cost of its inverting is prohibitive. In this paper, we shall develop an effective parallelizable technique for computing these eigenvalues and eigenvectors utilizing subspace iteration and preconditioning. Estimates will be provided which show that the preconditioned method converges linearly and uniformly in the matrix dimension when used with a uniform preconditioner under the assumption that the approximating subspace is close enough to the span of desired eigenvectors.
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
In this paper, we study the semi-discrete mortar upwind finite volume element method with the Crouzeix-Raviart element for the parabolic convection diffusion problems.It is proved that the semi-discrete mortar upwind finite volume element approximations derived are convergent in the H1- and L2-norms.
The Induced Dimension Reduction method applied to convection-diffusion-reaction problems
Astudillo, R.; Van Gijzen, M.B.
2016-01-01
Discretization of (linearized) convection-diffusion-reaction problems yields a large and sparse non symmetric linear system of equations, Ax = b. (1) In this work, we compare the computational behavior of the Induced Dimension Reduction method (IDR(s)) [10], with other short-recurrences Krylov met
Why does renewable energy diffuse so slowly? A review of innovation system problems
Negro, S.O.; Alkemade, F.; Hekkert, M.P.
2012-01-01
In this paper we present a literature review of studies that have analysed the troublesome trajectory of different renewable energy technologies (RETs) development and diffusion in different, mainly European countries. We present an overview of typical systemic problems in the development of
Effects of upwinding on the solution of a 1-D advection-diffusion problem
Energy Technology Data Exchange (ETDEWEB)
Sahai, V.
1991-12-01
A one-dimensional advection-diffusion problem whose solution is known was solved using TOPAZ2D. Two numerical upwinding techniques were used to damp out the numerical oscillations that occur. Comparisons between the exact and numerical solution were made.
A non-local non-autonomous diffusion problem: linear and sublinear cases
Figueiredo-Sousa, Tarcyana S.; Morales-Rodrigo, Cristian; Suárez, Antonio
2017-10-01
In this work we investigate an elliptic problem with a non-local non-autonomous diffusion coefficient. Mainly, we use bifurcation arguments to obtain existence of positive solutions. The structure of the set of positive solutions depends strongly on the balance between the non-local and the reaction terms.
Why does renewable energy diffuse so slowly? A review of innovation system problems
Negro, S.O.; Alkemade, F.; Hekkert, M.P.
2012-01-01
In this paper we present a literature review of studies that have analysed the troublesome trajectory of different renewable energy technologies (RETs) development and diffusion in different, mainly European countries. We present an overview of typical systemic problems in the development of innovat
Characterizing and approximating eigenvalue sets of symmetric interval matrices
DEFF Research Database (Denmark)
Hladík, Milan; Daney, David; Tsigaridas, Elias
2011-01-01
We consider the eigenvalue problem for the case where the input matrix is symmetric and its entries are perturbed, with perturbations belonging to some given intervals. We present a characterization of some of the exact boundary points, which allows us to introduce an inner approximation algorith...
"Universal" inequalities for the eigenvalues of the biharmonic operator
Ilias, Said
2010-01-01
In this paper, we establish universal inequalities for eigenvalues of the clamped plate problem on compact submanifolds of Euclidean spaces, of spheres and of real, complex and quaternionic projective spaces. We also prove similar results for the biharmonic operator on domains of Riemannian manifolds admitting spherical eigenmaps (this includes the compact homogeneous Riemannian spaces) and nally on domains of the hyperbolic space.
New algorithms for the symmetric tridiagonal eigenvalue computation
Energy Technology Data Exchange (ETDEWEB)
Pan, V. [City Univ. of New York, Bronx, NY (United States)]|[International Computer Sciences Institute, Berkeley, CA (United States)
1994-12-31
The author presents new algorithms that accelerate the bisection method for the symmetric eigenvalue problem. The algorithms rely on some new techniques, which include acceleration of Newton`s iteration and can also be further applied to acceleration of some other iterative processes, in particular, of iterative algorithms for approximating polynomial zeros.
Vedelago, J.; Quiroga, A.; Valente, M.
2014-10-01
Diffusion of ferric ions in ferrous sulfate (Fricke) gels represents one of the main drawbacks of some radiation detectors, such as Fricke gel dosimeters. In practice, this disadvantage can be overcome by prompt dosimeter analysis, and constraining strongly the time between irradiation and analysis, implementing special dedicated protocols aimed at minimizing signal blurring due to diffusion effects. This work presents a novel analytic modeling and numerical calculation approach of diffusion coefficients in Fricke gel radiation sensitive materials. Samples are optically analyzed by means of visible light transmission measurements by capturing images with a charge-coupled device camera provided with a monochromatic filter corresponding to the XO-infused Fricke solution absorbance peak. Dose distributions in Fricke gels are suitably delivered by assessing specific initial conditions further studied by periodical sample image acquisitions. Diffusion coefficient calculations were performed using a set of computational algorithms based on inverse problem formulation. Although 1D approaches to the diffusion equation might provide estimations of the diffusion coefficient, it should be calculated in the 2D framework due to the intrinsic bi-dimensional characteristics of Fricke gel layers here considered as radiation dosimeters. Thus a suitable 2D diffusion model capable of determining diffusion coefficients was developed by fitting the obtained algorithm numerical solutions with the corresponding experimental data. Comparisons were performed by introducing an appropriate functional in order to analyze both experimental and numerical values. Solutions to the second-order diffusion equation are calculated in the framework of a dedicated method that incorporates finite element method. Moreover, optimized solutions can be attained by gradient-type minimization algorithms. Knowledge about diffusion coefficient for a Fricke gel radiation detector is helpful in accounting for
Many Sparse Cuts via Higher Eigenvalues
Louis, Anand; Tetali, Prasad; Vempala, Santosh
2011-01-01
Cheeger's fundamental inequality states that any edge-weighted graph has a vertex subset $S$ such that its expansion (a.k.a. conductance) is bounded as follows: \\[ \\phi(S) \\defeq \\frac{w(S,\\bar{S})}{\\min \\set{w(S), w(\\bar{S})}} \\leq 2\\sqrt{\\lambda_2} \\] where $w$ is the total edge weight of a subset or a cut and $\\lambda_2$ is the second smallest eigenvalue of the normalized Laplacian of the graph. Here we prove the following natural generalization: for any integer $k \\in [n]$, there exist $ck$ disjoint subsets $S_1, ..., S_{ck}$, such that \\[ \\max_i \\phi(S_i) \\leq C \\sqrt{\\lambda_{k} \\log k} \\] where $\\lambda_i$ is the $i^{th}$ smallest eigenvalue of the normalized Laplacian and $c0$ are suitable absolute constants. Our proof is via a polynomial-time algorithm to find such subsets, consisting of a spectral projection and a randomized rounding. As a consequence, we get the same upper bound for the small set expansion problem, namely for any $k$, there is a subset $S$ whose weight is at most a $\\bigO(1/k)$ fra...
Perturbation of eigenvalues of preconditioned Navier-Stokes operators
Energy Technology Data Exchange (ETDEWEB)
Elman, H.C. [Univ. of Maryland, College Park, MD (United States)
1996-12-31
We study the sensitivity of algebraic eigenvalue problems associated with matrices arising from linearization and discretization of the steady-state Navier-Stokes equations. In particular, for several choices of preconditioners applied to the system of discrete equations, we derive upper bounds on perturbations of eigenvalues as functions of the viscosity and discretization mesh size. The bounds suggest that the sensitivity of the eigenvalues is at worst linear in the inverse of the viscosity and quadratic in the inverse of the mesh size, and that scaling can be used to decrease the sensitivity in some cases. Experimental results supplement these results and confirm the relatively mild dependence on viscosity. They also indicate a dependence on the mesh size of magnitude smaller than the analysis suggests.
Directory of Open Access Journals (Sweden)
Maria Malejki
2007-01-01
Full Text Available We investigate the problem of approximation of eigenvalues of some self-adjoint operator in the Hilbert space \\(l^2(\\mathbb{N}\\ by eigenvalues of suitably chosen principal finite submatrices of an infinite Jacobi matrix that defines the operator considered. We assume the Jacobi operator is bounded from below with compact resolvent. In our research we estimate the asymptotics (with \\(n\\to \\infty\\ of the joint error of approximation for the first \\(n\\ eigenvalues and eigenvectors of the operator by the eigenvalues and eigenvectors of the finite submatrix of order \\(n \\times n\\. The method applied in our research is based on the Rayleigh-Ritz method and Volkmer's results included in [H. Volkmer, Error Estimates for Rayleigh-Ritz Approximations of Eigenvalues and Eigenfunctions of the Mathieu and Spheroidal Wave Equation, Constr. Approx. 20 (2004, 39-54]. We extend the method to cover a class of infinite symmetric Jacobi matrices with three diagonals satisfying some polynomial growth estimates.
A balancing domain decomposition method by constraints for advection-diffusion problems
Energy Technology Data Exchange (ETDEWEB)
Tu, Xuemin; Li, Jing
2008-12-10
The balancing domain decomposition methods by constraints are extended to solving nonsymmetric, positive definite linear systems resulting from the finite element discretization of advection-diffusion equations. A pre-conditioned GMRES iteration is used to solve a Schur complement system of equations for the subdomain interface variables. In the preconditioning step of each iteration, a partially sub-assembled finite element problem is solved. A convergence rate estimate for the GMRES iteration is established, under the condition that the diameters of subdomains are small enough. It is independent of the number of subdomains and grows only slowly with the subdomain problem size. Numerical experiments for several two-dimensional advection-diffusion problems illustrate the fast convergence of the proposed algorithm.
Asymptotic solution for a class of weakly nonlinear singularly perturbed reaction diffusion problem
Institute of Scientific and Technical Information of China (English)
TANG Rong-rong
2009-01-01
Under appropriate conditions, with the perturbation method and the theory of differential inequalities, a class of weakly nonlinear singularly perturbed reaction diffusion problem is considered. The existence of solution of the original problem is proved by constructing the auxiliary functions. The uniformly valid asymptotic expansions of the solution for arbitrary mth order approximation are obtained through constructing the formal solutions of the original problem, expanding the nonlinear terms to the power in small parameter e and comparing the coefficient for the same powers of ε. Finally, an example is provided, resulting in the error of O(ε2).
A Symmetric Characteristic Finite Volume Element Scheme for Nonlinear Convection-Diffusion Problems
Institute of Scientific and Technical Information of China (English)
Min Yang; Yi-rang Yuan
2008-01-01
In this paper, we implement alternating direction strategy and construct a symmetric FVE scheme for nonlinear convection-diffusion problems. Comparing to general FVE methods, our method has two advantages. First, the coefficient matrices of the discrete schemes will be symmetric even for nonlinear problems.Second, since the solution of the algebraic equations at each time step can be inverted into the solution of several one-dimensional problems, the amount of computation work is smaller. We prove the optimal H1-norm error estimates of order O(△t2 + h) and present some numerical examples at the end of the paper.
Free boundary value problems for a class of generalized diffusion equation
Institute of Scientific and Technical Information of China (English)
无
2002-01-01
The transport behavior of free boundary value problems for a class of generalized diffusion equations was studied. Suitable similarity transformations were used to convert the problems into a class of singular nonlinear two-point boundary value problems and similarity solutions were numerical presented for different representations of heat conduction function, convection function, heat flux function, and power law parameters by utilizing the shooting technique. The results revealed the flux transfer mechanism and the character as well as the effects of parameters on the solutions.
Numerical solution of a diffusion problem by exponentially fitted finite difference methods.
D'Ambrosio, Raffaele; Paternoster, Beatrice
2014-01-01
This paper is focused on the accurate and efficient solution of partial differential differential equations modelling a diffusion problem by means of exponentially fitted finite difference numerical methods. After constructing and analysing special purpose finite differences for the approximation of second order partial derivatives, we employed them in the numerical solution of a diffusion equation with mixed boundary conditions. Numerical experiments reveal that a special purpose integration, both in space and in time, is more accurate and efficient than that gained by employing a general purpose solver.
Indian Academy of Sciences (India)
S Nayak; S Chakraverty
2015-10-01
In this paper, neutron diffusion equation of a triangular homogeneous bare reactor with uncertain parameters has been investigated. Here the involved parameters viz. geometry of the reactor, diffusion coefficient and absorption coefficient, etc. are uncertain and these are considered as fuzzy. Fuzzy values are handled through limit method which was defined for interval computations. The concept of fuzziness is hybridised with traditional finite element method to propose fuzzy finite element method. The proposed fuzzy finite element method has been used to obtain the uncertain eigenvalues of the said problem. Further these uncertain eigenvalues are compared with the traditional finite element method in special cases.
Energy Technology Data Exchange (ETDEWEB)
Perfetti, C.; Martin, W. [Univ. of Michigan, Dept. of Nuclear Engineering and Radiological Sciences, 2355 Bonisteel Boulevard, Ann Arbor, MI 48109-2104 (United States); Rearden, B.; Williams, M. [Oak Ridge National Laboratory, Reactor and Nuclear Systems Div., Bldg. 5700, P.O. Box 2008, Oak Ridge, TN 37831-6170 (United States)
2012-07-01
This study introduced two new approaches for calculating the F*(r) importance weighting function for Contributon and CLUTCH eigenvalue sensitivity coefficient calculations, and compared them in terms of accuracy and applicability. The necessary levels of F*(r) mesh refinement and mesh convergence for obtaining accurate eigenvalue sensitivity coefficients were determined for two preliminary problems through two parametric studies, and the results of these studies suggest that a sufficiently accurate F*(r) mesh for calculating eigenvalue sensitivity coefficients can be obtained for these problems with only a small increase in problem runtime. (authors)
Multi-input partial eigenvalue assignment for high order control systems with time delay
Zhang, Lei
2016-05-01
In this paper, we consider the partial eigenvalue assignment problem for high order control systems with time delay. Ram et al. (2011) [1] have shown that a hybrid method can be used to solve partial quadratic eigenvalue assignment problem of single-input vibratory system. Based on this theory, a rather simple algorithm for solving multi-input partial eigenvalue assignment for high order control systems with time delay is proposed. Our method can assign the expected eigenvalues and keep the no spillover property. The solution can be implemented with only partial information of the eigenvalues and the corresponding eigenvectors of the matrix polynomial. Numerical examples are given to illustrate the efficiency of our approach.
Li, Tiexiang; Huang, Tsung-Ming; Lin, Wen-Wei; Wang, Jenn-Nan
2017-03-01
We propose an efficient eigensolver for computing densely distributed spectra of the two-dimensional transmission eigenvalue problem (TEP), which is derived from Maxwell’s equations with Tellegen media and the transverse magnetic mode. The governing equations, when discretized by the standard piecewise linear finite element method, give rise to a large-scale quadratic eigenvalue problem (QEP). Our numerical simulation shows that half of the positive eigenvalues of the QEP are densely distributed in some interval near the origin. The quadratic Jacobi-Davidson method with a so-called non-equivalence deflation technique is proposed to compute the dense spectrum of the QEP. Extensive numerical simulations show that our proposed method processes the convergence efficiently, even when it needs to compute more than 5000 desired eigenpairs. Numerical results also illustrate that the computed eigenvalue curves can be approximated by nonlinear functions, which can be applied to estimate the denseness of the eigenvalues for the TEP.
Eigenvalue distributions of Wilson loops
Energy Technology Data Exchange (ETDEWEB)
Lohmayer, Robert
2010-07-01
In the first part of this thesis, we focus on the distribution of the eigenvalues of the unitary Wilson loop matrix in the two-dimensional case at arbitrary finite N. To characterize the distribution of the eigenvalues, we introduce three density functions (the ''symmetric'', the ''antisymmetric'', and the ''true'' eigenvalue density) which differ at finite N but possess the same infinite-N limit, exhibiting the Durhuus-Olesen phase transition. Using expansions of determinants and inverse determinants in characters of totally symmetric or totally antisymmetric representations of SU(N), the densities at finite N can be expressed in terms of simple sums involving only dimensions and quadratic Casimir invariants of certain irreducible representations of SU(N), allowing for a numerical computation of the densities at arbitrary N to any desired accuracy. We find that the true eigenvalue density, adding N oscillations to the monotonic symmetric density, is in some sense intermediate between the symmetric and the antisymmetric density, which in turn is given by a sum of N delta peaks located at the zeros of the average of the characteristic polynomial. Furthermore, we show that the dependence on N can be made explicit by deriving integral representations for the resolvents associated to the three eigenvalue densities. Using saddle-point approximations, we confirm that all three densities reduce to the Durhuus-Olesen result in the infinite-N limit. In the second part, we study an exponential form of the multiplicative random complex matrix model introduced by Gudowska-Nowak et al. Varying a parameter which can be identified with the area of the Wilson loop in the unitary case, the region of non-vanishing eigenvalue density of the N-dimensional complex product matrix undergoes a topological change at a transition point in the infinite-N limit. We study the transition by a detailed analysis of the average of the
Eigenvalue dynamics for multimatrix models
de Mello Koch, Robert; Gossman, David; Nkumane, Lwazi; Tribelhorn, Laila
2017-07-01
By performing explicit computations of correlation functions, we find evidence that there is a sector of the two matrix model defined by the S U (2 ) sector of N =4 super Yang-Mills theory that can be reduced to eigenvalue dynamics. There is an interesting generalization of the usual Van der Monde determinant that plays a role. The observables we study are the Bogomol'nyi-Prasad-Sommerfield operators of the S U (2 ) sector and include traces of products of both matrices, which are genuine multimatrix observables. These operators are associated with supergravity solutions of string theory.
Higher dimensional nonclassical eigenvalue asymptotics
Camus, Brice; Rautenberg, Nils
2015-02-01
In this article, we extend Simon's construction and results [B. Simon, J. Funct. Anal. 53(1), 84-98 (1983)] for leading order eigenvalue asymptotics to n-dimensional Schrödinger operators with non-confining potentials given by Hn α = - Δ + ∏ i = 1 n |x i| α i on ℝn (n > 2), α ≔ ( α 1 , … , α n ) ∈ ( R+ ∗ ) n . We apply the results to also derive the leading order spectral asymptotics in the case of the Dirichlet Laplacian -ΔD on domains Ωn α = { x ∈ R n : ∏ j = 1 n }x j| /α j α n < 1 } .
Eigenvalue Dynamics for Multimatrix Models
Koch, Robert de Mello; Nkumane, Lwazi; Tribelhorn, Laila
2016-01-01
By performing explicit computations of correlation functions, we find evidence that there is a sector of the two matrix model defined by the $SU(2)$ sector of ${\\cal N}=4$ super Yang-Mills theory, that can be reduced to eigenvalue dynamics. There is an interesting generalization of the usual Van der Monde determinant that plays a role. The observables we study are the BPS operators of the $SU(2)$ sector and include traces of products of both matrices, which are genuine multi matrix observables. These operators are associated to supergravity solutions of string theory.
Tensor eigenvalues and entanglement of symmetric states
Bohnet-Waldraff, F.; Braun, D.; Giraud, O.
2016-10-01
Tensor eigenvalues and eigenvectors have been introduced in the recent mathematical literature as a generalization of the usual matrix eigenvalues and eigenvectors. We apply this formalism to a tensor that describes a multipartite symmetric state or a spin state, and we investigate to what extent the corresponding tensor eigenvalues contain information about the multipartite entanglement (or, equivalently, the quantumness) of the state. This extends previous results connecting entanglement to spectral properties related to the state. We show that if the smallest tensor eigenvalue is negative, the state is detected as entangled. While for spin-1 states the positivity of the smallest tensor eigenvalue is equivalent to separability, we show that for higher values of the angular momentum there is a correlation between entanglement and the value of the smallest tensor eigenvalue.
Preconditioned iterations to calculate extreme eigenvalues
Energy Technology Data Exchange (ETDEWEB)
Brand, C.W.; Petrova, S. [Institut fuer Angewandte Mathematik, Leoben (Austria)
1994-12-31
Common iterative algorithms to calculate a few extreme eigenvalues of a large, sparse matrix are Lanczos methods or power iterations. They converge at a rate proportional to the separation of the extreme eigenvalues from the rest of the spectrum. Appropriate preconditioning improves the separation of the eigenvalues. Davidson`s method and its generalizations exploit this fact. The authors examine a preconditioned iteration that resembles a truncated version of Davidson`s method with a different preconditioning strategy.
Approximations of the operator exponential in a periodic diffusion problem with drift
Energy Technology Data Exchange (ETDEWEB)
Pastukhova, Svetlana E [Moscow State Institute of Radio-Engineering, Electronics and Automation (Technical University), Moscow (Russian Federation)
2013-02-28
A Cauchy problem for a parabolic diffusion equation with 1-periodic coefficients containing first order terms is studied. For the corresponding semigroup we construct approximations in the L{sup 2}-operator norm on sections t=const of order O(t{sup -m/2}) as t{yields}{infinity} for m=1 or m=2. The spectral method based on the Bloch representation of an operator with periodic coefficients is used. Bibliography: 25 titles.
On the hierarchical risk-averse control problems for diffusion processes
Befekadu, Getachew K.; Veremyev, Alexander; Pasiliao, Eduardo L.
2016-01-01
In this paper, we consider a risk-averse control problem for diffusion processes, in which there is a partition of the admissible control strategy into two decision-making groups (namely, the {\\it leader} and {\\it follower}) with different cost functionals and risk-averse satisfactions. Our approach, based on a hierarchical optimization framework, requires that a certain level of risk-averse satisfaction be achieved for the {\\it leader} as a priority over that of the {\\it follower's} risk-ave...
Moving-boundary problems for the time-fractional diffusion equation
Directory of Open Access Journals (Sweden)
Sabrina D. Roscani
2017-02-01
Full Text Available We consider a one-dimensional moving-boundary problem for the time-fractional diffusion equation. The time-fractional derivative of order $\\alpha\\in (0,1$ is taken in the sense of Caputo. We study the asymptotic behaivor, as t tends to infinity, of a general solution by using a fractional weak maximum principle. Also, we give some particular exact solutions in terms of Wright functions.
Directory of Open Access Journals (Sweden)
Xiang-Chao Shi
2016-02-01
Full Text Available The fractional reaction diffusion equation is one of the popularly used fractional partial differential equations in recent years. The fast Adomian decomposition method is used to obtain the solution of the Cauchy problem. Also, the analytical scheme is extended to the fractional one where the Taylor series is employed. In comparison with the classical Adomian decomposition method, the ratio of the convergence is increased. The method is more reliable for the fractional partial differential equations.
Transport dissipative particle dynamics model for mesoscopic advection-diffusion-reaction problems
Li, Zhen; Yazdani, Alireza; Tartakovsky, Alexandre; Karniadakis, George Em
2015-01-01
We present a transport dissipative particle dynamics (tDPD) model for simulating mesoscopic problems involving advection-diffusion-reaction (ADR) processes, along with a methodology for implementation of the correct Dirichlet and Neumann boundary conditions in tDPD simulations. tDPD is an extension of the classic dissipative particle dynamics (DPD) framework with extra variables for describing the evolution of concentration fields. The transport of concentration is modeled by a Fickian flux a...
Institute of Scientific and Technical Information of China (English)
Ningning YAN; Zhaojie ZHOU
2008-01-01
In this paper,we study a posteriori error estimates of the edge stabilization Galerkin method for the constrained optimal control problem governed by convection-dominated diffusion equations.The residual-type a posteriori error estimators yield both upper and lower bounds for control u measured in L2-norm and for state y and costate p measured in energy norm.Two numerical examples are presented to illustrate the effectiveness of the error estimators provided in this paper.
A free boundary problem of a diffusive SIRS model with nonlinear incidence
Cao, Jia-Feng; Li, Wan-Tong; Wang, Jie; Yang, Fei-Ying
2017-04-01
This paper is concerned with the spreading (persistence) and vanishing (extinction) of a disease which is characterized by a diffusive SIRS model with a bilinear incidence rate and free boundary. Through discussing the dynamics of a free boundary problem of an SIRS model, the spreading of a disease is described. We get the sufficient conditions which ensure the disease spreading or vanishing. In addition, the estimate of the expanding speed is also given when the free boundaries extend to the whole R.
Parabolic inverse convection-diffusion-reaction problem solved using an adaptive parametrization
Deolmi, Giulia
2011-01-01
This paper investigates the solution of a parabolic inverse problem based upon the convection-diffusion-reaction equation, which can be used to estimate both water and air pollution. We will consider both known and unknown source location: while in the first case the problem is solved using a projected damped Gauss-Newton, in the second one it is ill-posed and an adaptive parametrization with time localization will be adopted to regularize it. To solve the optimization loop a model reduction technique (Proper Orthogonal Decomposition) is used.
The Second Stokes Problem with Specular - Diffusive Boundary Conditions in Kinetic Theory
Akimova, V A; Yushkanov, A A
2012-01-01
The second Stokes problem with specular - diffusive boundary conditions of the kinetic theory is considered. The new method of the decision of the boundary problems of the kinetic theory is applied. The method allows to receive the decision with any degree of accuracy. At the basis of a method lays the idea of representation of a boundary condition on distribution function in the form of a source in the kinetic equation. By means of integrals Fourier the kinetic equation with a source is reduced to the integral equation of Fredholm type of the second kind. The decision is received in the form of Neumann's series.
Bai, Zheng-Jian; Yang, Jin-Ku; Datta, Biswa Nath
2016-12-01
In this paper, we consider the robust partial quadratic eigenvalue assignment problem in vibration by active feedback control. Based on the receptance measurements and the system matrices, we propose an optimization method for the robust and minimum norm partial quadratic eigenvalue assignment problem. We provide a new cost function and the closed-loop eigenvalue sensitivity and the feedback norms can be minimized simultaneously. Our method is also extended to the case of time delay between measurements of state and actuation of control. Numerical tests demonstrate the effectiveness of our method.
Radial point collocation method (RPCM) for solving convection-diffusion problems
Institute of Scientific and Technical Information of China (English)
LIU Xin
2006-01-01
In this paper, Radial point collocation method (RPCM), a kind ofmeshfree method, is applied to solve convectiondiffusion problem. The main feature of this approach is to use the interpolation schemes in local supported domains based on radial basis functions. As a result, this method is local and hence the system matrix is banded which is very attractive for practical engineering problems. In the numerical examination, RPCM is applied to solve non-linear convection-diffusion 2D Burgers equations. The results obtained by RPCM demonstrate the accuracy and efficiency of the proposed method for solving transient fluid dynamic problems. A fictitious point scheme is adopted to improve the solution accuracy while Neumann boundary conditions exist. The meshfree feature of the present method is very attractive in solving computational fluid problems.
Chen, Meng-Huo
2015-03-18
Summary: A two-grid convergence analysis based on the paper [Algebraic analysis of aggregation-based multigrid, by A. Napov and Y. Notay, Numer. Lin. Alg. Appl. 18 (2011), pp. 539-564] is derived for various aggregation schemes applied to a finite element discretization of a rotated anisotropic diffusion equation. As expected, it is shown that the best aggregation scheme is one in which aggregates are aligned with the anisotropy. In practice, however, this is not what automatic aggregation procedures do. We suggest approaches for determining appropriate aggregates based on eigenvectors associated with small eigenvalues of a block splitting matrix or based on minimizing a quantity related to the spectral radius of the iteration matrix. © 2015 John Wiley & Sons, Ltd.
A variational eigenvalue solver on a quantum processor
Peruzzo, Alberto; Shadbolt, Peter; Yung, Man-Hong; Zhou, Xiao-Qi; Love, Peter J; Aspuru-Guzik, Alán; O'Brien, Jeremy L
2013-01-01
Quantum computers promise to efficiently solve important problems that are intractable on a conventional computer. For quantum systems, where the dimension of the problem space grows exponentially, finding the eigenvalues of certain operators is one such intractable problem and remains a fundamental challenge. The quantum phase estimation algorithm can efficiently find the eigenvalue of a given eigenvector but requires fully coherent evolution. We present an alternative approach that greatly reduces the requirements for coherent evolution and we combine this method with a new approach to state preparation based on ans\\"atze and classical optimization. We have implemented the algorithm by combining a small-scale photonic quantum processor with a conventional computer. We experimentally demonstrate the feasibility of this approach with an example from quantum chemistry: calculating the ground state molecular energy for He-H+, to within chemical accuracy. The proposed approach, by drastically reducing the cohere...
On the eigenvalue-eigenvector method for solution of the stationary discrete matrix Riccati equation
DEFF Research Database (Denmark)
Michelsen, Michael Locht
1979-01-01
The purpose of this correspondence is to point out that certain numerical problems encountered in the solution of the stationary discrete matrix Riccati equation by the eigenvalue-eigenvector method of Vanghan [1] can be avoided by a simple reformulation....
The Lower Bounds for Eigenvalues of Elliptic Operators --By Nonconforming Finite Element Methods
Hu, Jun; Lin, Qun
2011-01-01
Finding eigenvalues of partial differential operators is important in the mathematical science. Since the exact eigenvalues are almost impossible, many papers and books investigate their bounds from above and below. It is well known that the variational principle (including the conforming finite element methods) provides the upper bounds, while there are no general theories to provide the lower bounds. The aim of our paper is to introduce a new systematic method that can produce the lower bounds for eigenvalues. The main idea is to use the nonconforming finite element methods. However, the numerics from the literature demonstrate that some nonconforming elements produce upper bounds of eigenvalues though some other nonconforming elements yield lower bounds. The general herein conclusion is that if the local approximation property of the nonconforming finite element space $V_h$ is better than the global continuity property of $V_h$, the corresponding method of the eigenvalue problem will produce the lower boun...
Stratified source-sampling techniques for Monte Carlo eigenvalue analysis.
Energy Technology Data Exchange (ETDEWEB)
Mohamed, A.
1998-07-10
In 1995, at a conference on criticality safety, a special session was devoted to the Monte Carlo ''Eigenvalue of the World'' problem. Argonne presented a paper, at that session, in which the anomalies originally observed in that problem were reproduced in a much simplified model-problem configuration, and removed by a version of stratified source-sampling. In this paper, stratified source-sampling techniques are generalized and applied to three different Eigenvalue of the World configurations which take into account real-world statistical noise sources not included in the model problem, but which differ in the amount of neutronic coupling among the constituents of each configuration. It is concluded that, in Monte Carlo eigenvalue analysis of loosely-coupled arrays, the use of stratified source-sampling reduces the probability of encountering an anomalous result over that if conventional source-sampling methods are used. However, this gain in reliability is substantially less than that observed in the model-problem results.
Active partial eigenvalue assignment for friction-induced vibration using receptance method
Liang, Y.; Ouyang, H. J.; Yamaura, H.
2016-09-01
Generally, a mechanical system always has symmetric system matrices. Nevertheless, when some non-conservative forces are included, such as friction and aerodynamic force, the symmetry of the stiffness matrix or damping matrix or both violated. Moreover, such an asymmetric system is prone to dynamic instability. Distinct from the eigenvalue assignment for symmetric systems to reassign their natural frequencies, the main purpose of eigenvalue assignment for asymmetric systems is to shift the unstable eigenvalues to the stable region. In this research, only the unstable eigenvalues and eigenvalues which are close to the imaginary axis of the complex eigenvalue plane are assigned due to their predominant influence on the response of the system. The remaining eigenvalues remain unchanged. The state-feedback control gains are obtained by solving the constrained linear least-squares problems in which the linear system matrices are deduced based on the receptance method and the constraint is derived from the unobservability condition. The numerical simulation results demonstrate that the proposed method is capable of partially assigning those targeted eigenvalues of the system for stabilisation.
Matrix methods for bare resonator eigenvalue analysis.
Latham, W P; Dente, G C
1980-05-15
Bare resonator eigenvalues have traditionally been calculated using Fox and Li iterative techniques or the Prony method presented by Siegman and Miller. A theoretical framework for bare resonator eigenvalue analysis is presented. Several new methods are given and compared with the Prony method.
An eigenvalue study of the MLC circuit
DEFF Research Database (Denmark)
Lindberg, Erik; Murali, K.
1998-01-01
The MLC (Murali-Lakshmanan-Chua) circuit is the simplest non-autonomous chaotic circuit. Insight in the behaviour of the circuit is obtained by means of a study of the eigenvalues of the linearized Jacobian of the nonlinear differential equations. The trajectories of the eigenvalues as functions...
LARGEST EIGENVALUE OF A UNICYCLIC MIXED GRAPH
Institute of Scientific and Technical Information of China (English)
FanYizheng
2004-01-01
The graphs which maximize and minimize respectively the largest eigenvalue over all unicyclic mixed graphs U on n vertices are determined. The unicyclic mixed graphs U with the largest eigenvalue λ1 (U)=n or λ1 (U)∈ (n ,n+1] are characterized.
Solution verification, goal-oriented adaptive methods for stochastic advection–diffusion problems
Almeida, Regina C.
2010-08-01
A goal-oriented analysis of linear, stochastic advection-diffusion models is presented which provides both a method for solution verification as well as a basis for improving results through adaptation of both the mesh and the way random variables are approximated. A class of model problems with random coefficients and source terms is cast in a variational setting. Specific quantities of interest are specified which are also random variables. A stochastic adjoint problem associated with the quantities of interest is formulated and a posteriori error estimates are derived. These are used to guide an adaptive algorithm which adjusts the sparse probabilistic grid so as to control the approximation error. Numerical examples are given to demonstrate the methodology for a specific model problem. © 2010 Elsevier B.V.
MOVING BOUNDARY PROBLEM FOR DIFFUSION RELEASE OF DRUG FROM A CYLINDER POLYMERIC MATRIX
Institute of Scientific and Technical Information of China (English)
谭文长; 吴望一; 严宗毅; 温功碧
2001-01-01
An approximate analytical solution of moving boundary problem for diffusion release of drug from a cylinder polymeric matrix was obtained by use of refined integral method. The release kinetics has been analyzed for non-erodible matrices with perfect sink condition. The formulas of the moving boundary and the fractional drug release were given.The moving boundary and the fractional drug release have been calculated at various drug loading levels, and the calculated results were in good agreement with those of experiments.The comparison of the moving boundary in spherical, cylinder, planar matrices has been completed. An approximate formula for estimating the available release time was presented.These results are useful for the clinic experiments. This investigation provides a new theoretical tool for studying the diffusion release of drug from a cylinder polymeric matrix and designing the controlled released drug.
Directory of Open Access Journals (Sweden)
Sumit Gupta
2015-09-01
Full Text Available The aim of this paper was to present a user friendly numerical algorithm based on homotopy perturbation transform method for solving various linear and nonlinear convection-diffusion problems arising in physical phenomena where particles, energy, or other physical quantities are transferred inside a physical system due to two processes: diffusion and convection. The homotopy perturbation transform method is a combined form of the homotopy perturbation method and Laplace transform method. The nonlinear terms can be easily obtained by the use of He’s polynomials. The technique presents an accurate methodology to solve many types of partial differential equations The approximate solutions obtained by proposed scheme in a wide range of the problem’s domain were compared with those results obtained from the actual solutions. The comparison shows a precise agreement between the results.
A Novel Characteristic Expanded Mixed Method for Reaction-Convection-Diffusion Problems
Directory of Open Access Journals (Sweden)
Yang Liu
2013-01-01
Full Text Available A novel characteristic expanded mixed finite element method is proposed and analyzed for reaction-convection-diffusion problems. The diffusion term ∇·(a(x,t∇u is discretized by the novel expanded mixed method, whose gradient belongs to the square integrable space instead of the classical H(div;Ω space and the hyperbolic part d(x(∂u/∂t+c(x,t·∇u is handled by the characteristic method. For a priori error estimates, some important lemmas based on the novel expanded mixed projection are introduced. The fully discrete error estimates based on backward Euler scheme are obtained. Moreover, the optimal a priori error estimates in L2- and H1-norms for the scalar unknown u and a priori error estimates in (L22-norm for its gradient λ and its flux σ (the coefficients times the negative gradient are derived. Finally, a numerical example is provided to verify our theoretical results.
Modelling of free boundary problems for phase change with diffuse interfaces
Directory of Open Access Journals (Sweden)
Sanyal Dipayan
2005-01-01
Full Text Available We present a continuum thermodynamical framework for simulating multiphase Stefan problem. For alloy solidification, which is marked by a diffuse interface called the mushy zone, we present a phase filed like formalism which comprises a set of macroscopic conservation equations with an order parameter which can account for the solid, liquid, and the mushy zones with the help of a phase function defined on the basis of the liquid fraction, the Gibbs relation, and the phase diagram with local approximations. Using the above formalism for alloy solidification, the width of the diffuse interface (mushy zone was computed rather accurately for iron-carbon and ammonium chloride-water binary alloys and validated against experimental data from literature.
Yamilov, A; Sarma, R; Cao, H
2015-01-01
The universal bimodal distribution of transmission eigenvalues in lossless diffusive systems un- derpins such celebrated phenomena as universal conductance fluctuations, quantum shot noise in condensed matter physics and enhanced transmission in optics and acoustics. Here, we show that in the presence of absorption, density of the transmission eigenvalues depends on the confinement geometry of scattering media. Furthermore, in an asymmetric waveguide, densities of the reflection and absorption eigenvalues also depend of the side from which the waves are incident. With increas- ing absorpotion, the density of absorption eigenvalues transforms from single-peak to double-peak function. Our findings open a new avenue for coherent control of wave transmission, reflection and absorption in random media.
ANALYTICAL RELATIONS BETWEEN EIGENVALUES OF CIRCULAR PLATE BASED ON VARIOUS PLATE THEORIES
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
Based on the mathematical similarity of the axisymmetric eigenvalue problems of a circular plate between the classical plate theory(CPT), the first-order shear deformation plate theory(FPT) and the Reddy's third-order shear deformation plate theory(RPT), analytical relations between the eigenvalues of circular plate based on various plate theories are investigated. In the present paper, the eigenvalue problem is transformed to solve an algebra equation. Analytical relationships that are expressed explicitly between various theories are presented. Therefore, from these relationships one can easily obtain the exact RPT and FPT solutions of critical buckling load and natural frequencyfor a circular plate with CPT solutions. The relationships are useful for engineering application, and can be used to check the validity, convergence and accuracy of numerical results for the eigenvalue problem of plates.
A variational eigenvalue solver on a photonic quantum processor.
Peruzzo, Alberto; McClean, Jarrod; Shadbolt, Peter; Yung, Man-Hong; Zhou, Xiao-Qi; Love, Peter J; Aspuru-Guzik, Alán; O'Brien, Jeremy L
2014-07-23
Quantum computers promise to efficiently solve important problems that are intractable on a conventional computer. For quantum systems, where the physical dimension grows exponentially, finding the eigenvalues of certain operators is one such intractable problem and remains a fundamental challenge. The quantum phase estimation algorithm efficiently finds the eigenvalue of a given eigenvector but requires fully coherent evolution. Here we present an alternative approach that greatly reduces the requirements for coherent evolution and combine this method with a new approach to state preparation based on ansätze and classical optimization. We implement the algorithm by combining a highly reconfigurable photonic quantum processor with a conventional computer. We experimentally demonstrate the feasibility of this approach with an example from quantum chemistry--calculating the ground-state molecular energy for He-H(+). The proposed approach drastically reduces the coherence time requirements, enhancing the potential of quantum resources available today and in the near future.
Verification of high-order mixed FEM solution of transient Magnetic diffusion problems
Energy Technology Data Exchange (ETDEWEB)
Rieben, R; White, D A
2005-05-12
We develop and present high order mixed finite element discretizations of the time dependent electromagnetic diffusion equations for solving eddy current problems on 3D unstructured grids. The discretizations are based on high order H(grad), H(curl) and H(div) conforming finite element spaces combined with an implicit and unconditionally stable generalized Crank-Nicholson time differencing method. We develop three separate electromagnetic diffusion formulations, namely the E (electric field), H (magnetic field) and the A-{phi} (potential) formulations. For each formulation, we also provide a consistent procedure for computing the secondary variables F (current flux density) and B (magnetic flux density), as these fields are required for the computation of electromagnetic force and heating terms. We verify the error convergence properties of each formulation via a series of numerical experiments on canonical problems with known analytic solutions. The key result is that the different formulations are equally accurate, even for the secondary variables J and B, and hence the choice of which formulation to use depends mostly upon relevance of the Natural and Essential boundary conditions to the problem of interest. In addition, we highlight issues with numerical verification of finite element methods which can lead to false conclusions on the accuracy of the methods.
Energy Technology Data Exchange (ETDEWEB)
Svyatskiy, Daniil [Los Alamos National Laboratory; Shashkov, Mikhail [Los Alamos National Laboratory; Kuzmin, D [DORTMUND UNIV
2008-01-01
A new approach to the design of constrained finite element approximations to second-order elliptic problems is introduced. This approach guarantees that the finite element solution satisfies the discrete maximum principle (DMP). To enforce these monotonicity constrains the sufficient conditions for elements of the stiffness matrix are formulated. An algebraic splitting of the stiffness matrix is employed to separate the contributions of diffusive and antidiffusive numerical fluxes, respectively. In order to prevent the formation of spurious undershoots and overshoots, a symmetric slope limiter is designed for the antidiffusive part. The corresponding upper and lower bounds are defined using an estimate of the steepest gradient in terms of the maximum and minimum solution values at surrounding nodes. The recovery of nodal gradients is performed by means of a lumped-mass L{sub 2} projection. The proposed slope limiting strategy preserves the consistency of the underlying discrete problem and the structure of the stiffness matrix (symmetry, zero row and column sums). A positivity-preserving defect correction scheme is devised for the nonlinear algebraic system to be solved. Numerical results and a grid convergence study are presented for a number of anisotropic diffusion problems in two space dimensions.
Directory of Open Access Journals (Sweden)
Thái Anh Nhan
2016-01-01
Full Text Available A one-dimensional linear convection-diffusion problem with a perturbation parameter ɛ multiplying the highest derivative is considered. The problem is solved numerically by using the standard upwind scheme on special layer-adapted meshes. It is proved that the numerical solution is ɛ-uniform accurate in the maximum norm. This is done by a new proof technique in which the discrete system is preconditioned in order to enable the use of the principle where “ɛ-uniform stability plus ɛ-uniform consistency implies ɛ-uniform convergence.” Without preconditioning, this principle cannot be applied to convection-diffusion problems because the consistency error is not uniform in ɛ. At the same time, the condition number of the discrete system becomes independent of ɛ due to the same preconditioner; otherwise, the condition number of the discrete system before preconditioning increases when ɛ tends to 0. We obtained such results in an earlier paper, but only for the standard Shishkin mesh. In a nontrivial generalization, we show here that the same proof techniques can be applied to the whole class of Shishkin-type meshes.
EDGE SINGULARITY OF BONDED PIEZOELECTRIC MATERIALS WITH REPEATED EIGENVALUES
Institute of Scientific and Technical Information of China (English)
王效贵; 许金泉
2001-01-01
In piezoelectric problems, the form of the general solution is dependent on the eigenvalues of the material. The singular stress field and electrical displacement field near the interface edge were deduced in this study. The results showed that the stress field and the electrical displacement field have the same singularity; and that the singularity depends not only on the mechanical properties and shape of the interface edge, but also on the piezoelectric properties of the composite material.
Local-instantaneous filtering in the integral transform solution of nonlinear diffusion problems
Macêdo, E. N.; Cotta, R. M.; Orlande, H. R. B.
A novel filtering strategy is proposed to be utilized in conjunction with the Generalized Integral Transform Technique (GITT), in the solution of nonlinear diffusion problems. The aim is to optimize convergence enhancement, yielding computationally efficient eigenfunction expansions. The proposed filters include space and time dependence, extracted from linearized versions of the original partial differential system. The scheme automatically updates the filter along the time integration march, as the required truncation orders for the user requested accuracy begin to exceed a prescribed maximum system size. A fully nonlinear heat conduction example is selected to illustrate the computational performance of the filtering strategy, against the classical single-filter solution behavior.
[Outstanding problems of normal and pathological morphology of the diffuse endocrine system].
Iaglov, V V; Iaglova, N V
2011-01-01
The diffuse endocrine system (DES)--a mosaic-cellular endoepithelial gland--is the biggest part of the human endocrine system. Scientists used to consider cells of DES as neuroectodermal. According to modem data cells of DES are different cytogenetic types because they develop from the different embryonic blastophyllum. So that any hormone-active tumors originated from DES of the digestive, respiratory and urogenital system shouldn't be considered as neuroendocrinal tumors. The basic problems of DES morphology and pathology are the creation of scientifically substantiated histogenetic classification of DES tumors.
Exact and approximate interior corner problem in neutron diffusion by integral transform methods
Energy Technology Data Exchange (ETDEWEB)
Bareiss, E.H.; Chang, K.S.J.; Constatinescu, D.A.
1976-09-01
The mathematical solution of the neutron diffusion equation exhibits singularities in its derivatives at material corners. A mathematical treatment of the nature of these singularities and its impact on coarse network approximation methods in computational work is presented. The mathematical behavior is deduced from Green's functions, based on a generalized theory for two space dimensions, and the resulting systems of integral equations, as well as from the Kontorovich--Lebedev Transform. The effect on numerical calculations is demonstrated for finite difference and finite element methods for a two-region corner problem.
Energy Technology Data Exchange (ETDEWEB)
Perfetti, Christopher M [ORNL; Martin, William R [University of Michigan; Rearden, Bradley T [ORNL; Williams, Mark L [ORNL
2012-01-01
This study introduced three approaches for calculating the importance weighting function for Contributon and CLUTCH eigenvalue sensitivity coefficient calculations, and compared them in terms of accuracy and applicability. The necessary levels of mesh refinement and mesh convergence for obtaining accurate eigenvalue sensitivity coefficients were determined through two parametric studies, and the results of these studies suggest that a sufficiently-accurate mesh for calculating eigenvalue sensitivity coefficients can be obtained for the Contributon and CLUTCH methods with only a small increase in problem runtime.
Bifurcations of Eigenvalues of Gyroscopic Systems with Parameters Near Stability Boundaries
DEFF Research Database (Denmark)
Seyranian, Alexander P.; Kliem, Wolfhard
1999-01-01
. It is shown that the bifurcation (splitting) of double eigenvalues is closely related to the stability, flutter and divergence boundaries in the parameter space. Normal vectors to these boundaries are derived using only information at a boundary point: eigenvalues, eigenvectors and generalized eigenvectors......The paper deals with stability problems of linear gyroscopic systems with finite or infinite degrees of freedom, where the system matrices or operators depend smoothly on several real parameters. Explicit formulae for the behavior of eigenvalues under a change of parameters are obtained...
Eigenvalues, inequalities and ergodic theory
Institute of Scientific and Technical Information of China (English)
无
2000-01-01
This paper surveys the main results obtained during the period 1992-1999 on three aspects mentioned in the title. The first result is a new and general variational formula for the lower bound of spectral gap (i.e. the first non-trivial eigenvalue) of elliptic operators in Euclidean space, Laplacian on Riemannian manifolds or Markov chains (§1). Here, a probabilistic method -coupling method is adopted. The new formula is a dual of the classical variational formula. The last formula is actually equivalent to Poincaré inequality. To which, there are closely related logarithmic Sobolev inequality, Nash inequality, Liggett inequality and so on. These inequalities are treated in a unified way by using Cheeger's method which comes from Riemannian geometry. This consists of §2. The results on these two aspects are mainly completed by the author joint with F. Y. Wang. Furthermore, a diagram of the inequalities and the traditional three types of ergodicity is presented (§3). The diagram extends the ergodic theory of Markov processes. The details of the methods used in the paper will be explained in a subsequent paper under the same title.
Transport dissipative particle dynamics model for mesoscopic advection- diffusion-reaction problems
Energy Technology Data Exchange (ETDEWEB)
Zhen, Li; Yazdani, Alireza; Tartakovsky, Alexandre M.; Karniadakis, George E.
2015-07-07
We present a transport dissipative particle dynamics (tDPD) model for simulating mesoscopic problems involving advection-diffusion-reaction (ADR) processes, along with a methodology for implementation of the correct Dirichlet and Neumann boundary conditions in tDPD simulations. tDPD is an extension of the classic DPD framework with extra variables for describing the evolution of concentration fields. The transport of concentration is modeled by a Fickian flux and a random flux between particles, and an analytical formula is proposed to relate the mesoscopic concentration friction to the effective diffusion coefficient. To validate the present tDPD model and the boundary conditions, we perform three tDPD simulations of one-dimensional diffusion with different boundary conditions, and the results show excellent agreement with the theoretical solutions. We also performed two-dimensional simulations of ADR systems and the tDPD simulations agree well with the results obtained by the spectral element method. Finally, we present an application of the tDPD model to the dynamic process of blood coagulation involving 25 reacting species in order to demonstrate the potential of tDPD in simulating biological dynamics at the mesoscale. We find that the tDPD solution of this comprehensive 25-species coagulation model is only twice as computationally expensive as the DPD simulation of the hydrodynamics only, which is a significant advantage over available continuum solvers.
Numerical simulations of the moving contact line problem using a diffuse-interface model
Afzaal, Muhammad; Sibley, David; Duncan, Andrew; Yatsyshin, Petr; Duran-Olivencia, Miguel A.; Nold, Andreas; Savva, Nikos; Schmuck, Markus; Kalliadasis, Serafim
2015-11-01
Moving contact lines are a ubiquitous phenomenon both in nature and in many modern technologies. One prevalent way of numerically tackling the problem is with diffuse-interface (phase-field) models, where the classical sharp-interface model of continuum mechanics is relaxed to one with a finite thickness fluid-fluid interface, capturing physics from mesoscopic lengthscales. The present work is devoted to the study of the contact line between two fluids confined by two parallel plates, i.e. a dynamically moving meniscus. Our approach is based on a coupled Navier-Stokes/Cahn-Hilliard model. This system of partial differential equations allows a tractable numerical solution to be computed, capturing diffusive and advective effects in a prototypical case study in a finite-element framework. Particular attention is paid to the static and dynamic contact angle of the meniscus advancing or receding between the plates. The results obtained from our approach are compared to the classical sharp-interface model to elicit the importance of considering diffusion and associated effects. We acknowledge financial support from European Research Council via Advanced Grant No. 247031.
Eigenvalue Expansion Approach to Study Bio-Heat Equation
Khanday, M. A.; Nazir, Khalid
2016-07-01
A mathematical model based on Pennes bio-heat equation was formulated to estimate temperature profiles at peripheral regions of human body. The heat processes due to diffusion, perfusion and metabolic pathways were considered to establish the second-order partial differential equation together with initial and boundary conditions. The model was solved using eigenvalue method and the numerical values of the physiological parameters were used to understand the thermal disturbance on the biological tissues. The results were illustrated at atmospheric temperatures TA = 10∘C and 20∘C.
Edge fluctuations of eigenvalues of Wigner matrices
Döring, Hanna
2012-01-01
We establish a moderate deviation principle (MDP) for the number of eigenvalues of a Wigner matrix in an interval close to the edge of the spectrum. Moreover we prove a MDP for the $i$th largest eigenvalue close to the edge. The proof relies on fine asymptotics of the variance of the eigenvalue counting function of GUE matrices due to Gustavsson. The extension to large families of Wigner matrices is based on the Tao and Vu Four Moment Theorem. Possible extensions to other random matrix ensembles are commented.
Eigenvalue Decomposition-Based Modified Newton Algorithm
Directory of Open Access Journals (Sweden)
Wen-jun Wang
2013-01-01
Full Text Available When the Hessian matrix is not positive, the Newton direction may not be the descending direction. A new method named eigenvalue decomposition-based modified Newton algorithm is presented, which first takes the eigenvalue decomposition of the Hessian matrix, then replaces the negative eigenvalues with their absolute values, and finally reconstructs the Hessian matrix and modifies the searching direction. The new searching direction is always the descending direction. The convergence of the algorithm is proven and the conclusion on convergence rate is presented qualitatively. Finally, a numerical experiment is given for comparing the convergence domains of the modified algorithm and the classical algorithm.
A high-order discontinuous Galerkin method for unsteady advection-diffusion problems
Borker, Raunak; Farhat, Charbel; Tezaur, Radek
2017-03-01
A high-order discontinuous Galerkin method with Lagrange multipliers is presented for the solution of unsteady advection-diffusion problems in the high Péclet number regime. It operates directly on the second-order form of the governing equation and does not require any stabilization. Its spatial basis functions are chosen among the free-space solutions of the homogeneous form of the partial differential equation obtained after time-discretization. It also features Lagrange multipliers for enforcing a weak continuity of the approximated solution across the element interface boundaries. This leads to a system of differential-algebraic equations which are time-integrated by an implicit family of schemes. The numerical stability of these schemes and the well-posedness of the overall discretization method are supported by a theoretical analysis. The performance of this method is demonstrated for various high Péclet number constant-coefficient model flow problems.
Institute of Scientific and Technical Information of China (English)
Yuefang Wang; Lihua Huang; Xuetao Liu; Keren Wang
2005-01-01
The Hamiltonian dynamics is adopted to solve the eigenvalue problem for transverse vibrations of axially moving strings. With the explicit Hamiltonian function the canonical equation of the free vibration is derived. Non-singular modal functions are obtained through a linear, symplectic eigenvalue analysis, and the symplectic-type orthogonality conditions of modes are derived. Stability of the transverse motion is examined by means of analyzing the eigenvalues and their bifurcation, especially for strings transporting with the critical speed. It is pointed out that the motion of the string does not possess divergence instability at the critical speed due to the weak interaction between eigenvalue pairs. The expansion theorem is applied with the non-singular modal functions to solve the displacement response to free and forced vibrations. It is demonstrated that the modal functions can be used as the base functions for solving linear and nonlinear vibration problems.
Energy Technology Data Exchange (ETDEWEB)
Couto, Nozimar do
2003-07-01
Diffusion theory is traditionally applied to nuclear reactor global calculations. Based on the good results generated by the one-dimensional spectral nodal diffusion (SND) method for benchmark problems, we offer the SND method for nuclear reactor global calculations in X,Y geometry. In this method, the continuity equation and Flick law are transverse integrated in each spatial direction leading to a system of two 'one-dimensional' equations coupled by the transverse leakage terms. We then apply the SND method to numerically solve this system with constant approximations for the transverse leakage terms. We perform a spectral analysis to determine the local general solution of each 'one-dimensional' nodal equation with flat approximation for the transverse leakages. We used special auxiliary equations with parameters that are to be determined in order to preserve the analytical general solutions in the numerical algorithm. By considering continuity conditions at the node interfaces and appropriate boundary conditions, we obtain a solvable system of discretized equations involving the node-edge average scalar fluxes at each estimate of the dominant eigenvalue (k{sub eff}) in the outer power iterations. As we considered approximations to the transverse leakages, the SND method is not free of spatial truncation errors. Nevertheless, it generated good results for the typical model problems that we considered. (author)
Arbitrary eigenvalue assignments for linear time-varying multivariable control systems
Nguyen, Charles C.
1987-01-01
The problem of eigenvalue assignments for a class of linear time-varying multivariable systems is considered. Using matrix operators and canonical transformations, it is shown that a time-varying system that is 'lexicography-fixedly controllable' can be made via state feedback to be equivalent to a time-invariant system whose eigenvalues are arbitrarily assignable. A simple algorithm for the design of the state feedback is provided.
Energy Technology Data Exchange (ETDEWEB)
Perfetti, Christopher M [ORNL; Martin, William R [University of Michigan; Rearden, Bradley T [ORNL; Williams, Mark L [ORNL
2012-01-01
Three methods for calculating continuous-energy eigenvalue sensitivity coefficients were developed and implemented into the SHIFT Monte Carlo code within the Scale code package. The methods were used for several simple test problems and were evaluated in terms of speed, accuracy, efficiency, and memory requirements. A promising new method for calculating eigenvalue sensitivity coefficients, known as the CLUTCH method, was developed and produced accurate sensitivity coefficients with figures of merit that were several orders of magnitude larger than those from existing methods.
Eigenvalue spectra of a $\\mathcal{PT}$ -symmetric coupled quartic potential in two dimensions
Indian Academy of Sciences (India)
Fakir Chand; Savita; S C Mishra
2010-10-01
The Schrödinger equation was solved for a generalized $\\mathcal{PT}$-symmetric quartic potential in two dimensions. It was found that, under a suitable ansatz for the wave function, the system possessed real and discrete energy eigenvalues. Analytic expressions for the energy eigenvalues and the eigenfunctions for the first four states were obtained. Some constraining relations among the wave function parameters rendered the problem quasi-solvable.
The Upper Bounds of Arbitrary Eigenvalues for Uniformly Elliptic Operators with Higher Orders
Institute of Scientific and Technical Information of China (English)
Gao Jia; Xiao-ping Yang
2006-01-01
Let Ω(∪)Rm (m≥2) be a bounded domain with piecewise smooth and Lipschitz boundary (e)Ω. Let t and r be two nonnegative integers with t ≥ r + 1. In this paper, we consider the variable-coefficient eigenvalue problems with uniformly elliptic differential operators on the left-hand side and (-△)r on the right-hand side.Some upper bounds of the arbitrary eigenvalue are obtained, and several known results are generalized.
Eigenvalues of non-selfadjoint operators: A comparison of two approaches
Demuth, Michael; Hansmann, Marcel; Katriel, Guy
2012-01-01
The central problem we consider is the distribution of eigenvalues of closed linear operators which are not selfadjoint, with a focus on those operators which are obtained as perturbations of selfadjoint linear operators. Two methods are explained and elaborated. One approach uses complex analysis to study a holomorphic function whose zeros can be identified with the eigenvalues of the linear operator. The second method is an operator theoretic approach involving the numerical range. General ...
Eigenvalues and expansion of bipartite graphs
DEFF Research Database (Denmark)
Høholdt, Tom; Janwa, Heeralal
2012-01-01
We prove lower bounds on the largest and second largest eigenvalue of the adjacency matrix of bipartite graphs and give necessary and sufficient conditions for equality. We give several examples of classes that are optimal with respect to the bouns. We prove that BIBD-graphs are characterized by ...... by their eigenvalues. Finally we present a new bound on the expansion coefficient of (c,d)-regular bipartite graphs and compare that with aclassical bound....
Partial eigenvalue assignment and its stability in a time delayed system
Singh, Kumar V.; Dey, Rajeeb; Datta, Biswa N.
2014-01-01
Active vibration control strategy is an effective way to control dangerous vibrations in a structure, caused by resonance and to manipulate the dynamics of vibrational response. Implementation of this strategy requires real-time computations of two feedback control matrices such that a small amount of eigenvalues of the associated quadratic matrix pencil are replaced by suitably chosen ones while the remaining large number of eigenvalues and eigenvectors remain unchanged ensuring the no spill-over. This mathematical problem is referred to as the Quadratic Partial Eigenvalue Assignment problem. The greatest challenge there is to solve the problems using the knowledge of only a small number of eigenvalues and eigenvectors that are computable using state-of-the-art techniques. This paper generalizes the earlier work on partial assignment to constant time-delay systems. Furthermore, a posterior stability analysis is carried out to identify the ranges of the time-delay that maintains the closed-loop assignment while keeping the stability of the infinite number of eigenvalues for the time-delayed systems. The practical features of the proposed methods are that it is implemented in the second-order setting itself using only those small number of eigenvalues and the eigenvectors that are to be assigned and the no spill-over is established by means of mathematical results. The results of our numerical experiments support the validity of our proposed methods.
Anharmonic oscillators in the complex plane, $\\mathcal{PT}$-symmetry, and real eigenvalues
Shin, Kwang C
2010-01-01
For integers $m\\geq 3$ and $1\\leq\\ell\\leq m-1$, we study the eigenvalue problems $-u^{\\prime\\prime}(z)+[(-1)^{\\ell}(iz)^m-P(iz)]u(z)=\\lambda u(z)$ with the boundary conditions that $u(z)$ decays to zero as $z$ tends to infinity along the rays $\\arg z=-\\frac{\\pi}{2}\\pm \\frac{(\\ell+1)\\pi}{m+2}$ in the complex plane, where $P$ is a polynomial of degree at most $m-1$. We provide asymptotic expansions of the eigenvalues $\\lambda_{n}$. Then we show that if the eigenvalue problem is $\\mathcal{PT}$-symmetric, then the eigenvalues are all real and positive with at most finitely many exceptions. Moreover, we show that when $\\gcd(m,\\ell)=1$, the eigenvalue problem has infinitely many real eigenvalues if and only if its translation or itself is $\\mathcal{PT}$-symmetric. Also, we will prove some other interesting direct and inverse spectral results.
Kvashnin, A Yu; Yushkanov, A A
2012-01-01
The classical Kramers problem of the kinetic theory is solved. The Kramers problem about isothermal sliding for quantum Fermi gases is considered. Quantum gases with the velocity - dependent collision frequency are considered. Specular - diffusive boundary conditions are applied. Dependence of isothermal sliding on the resulted chemical potential is found out.
Transport dissipative particle dynamics model for mesoscopic advection-diffusion-reaction problems
Li, Zhen; Yazdani, Alireza; Tartakovsky, Alexandre; Karniadakis, George Em
2015-07-01
We present a transport dissipative particle dynamics (tDPD) model for simulating mesoscopic problems involving advection-diffusion-reaction (ADR) processes, along with a methodology for implementation of the correct Dirichlet and Neumann boundary conditions in tDPD simulations. tDPD is an extension of the classic dissipative particle dynamics (DPD) framework with extra variables for describing the evolution of concentration fields. The transport of concentration is modeled by a Fickian flux and a random flux between tDPD particles, and the advection is implicitly considered by the movements of these Lagrangian particles. An analytical formula is proposed to relate the tDPD parameters to the effective diffusion coefficient. To validate the present tDPD model and the boundary conditions, we perform three tDPD simulations of one-dimensional diffusion with different boundary conditions, and the results show excellent agreement with the theoretical solutions. We also performed two-dimensional simulations of ADR systems and the tDPD simulations agree well with the results obtained by the spectral element method. Finally, we present an application of the tDPD model to the dynamic process of blood coagulation involving 25 reacting species in order to demonstrate the potential of tDPD in simulating biological dynamics at the mesoscale. We find that the tDPD solution of this comprehensive 25-species coagulation model is only twice as computationally expensive as the conventional DPD simulation of the hydrodynamics only, which is a significant advantage over available continuum solvers.
Optimization of the principal eigenvalue of the one-dimensional Schrodinger operator
Directory of Open Access Journals (Sweden)
Ryan I. Fernandes
2008-04-01
Full Text Available In this paper we consider two optimization problems related to the principal eigenvalue of the one dimensional Schrodinger operator. These optimization problems are formulated relative to the rearrangement of a fixed function. We show that both problems have unique solutions, and each of these solutions is a fixed point of an appropriate function.
Computable eigenvalue bounds for rank-k perturbations
Brandts, J.H.; Reis da Silva, R.
2010-01-01
We investigate lower bounds for the eigenvalues of perturbations of matrices. In the footsteps of Weyl and Ipsen & Nadler, we develop approximating matrices whose eigenvalues are lower bounds for the eigenvalues of the perturbed matrix. The number of available eigenvalues and eigenvectors of the ori
A second eigenvalue bound for the Dirichlet Schrodinger equation wtih a radially symmetric potential
Directory of Open Access Journals (Sweden)
Craig Haile
2000-01-01
Full Text Available We study the time-independent Schrodinger equation with radially symmetric potential $k|x|^alpha$, $k ge 0$, $k in mathbb{R}, alpha ge 2$ on a bounded domain $Omega$ in $mathbb{R}^n$, $(n ge 2$ with Dirichlet boundary conditions. In particular, we compare the eigenvalue $lambda_2(Omega$ of the operator $-Delta + k |x|^alpha $ on $Omega$ with the eigenvalue $lambda_2(S_1$ of the same operator $-Delta +kr^alpha$ on a ball $S_1$, where $S_1$ has radius such that the first eigenvalues are the same ($lambda_1(Omega = lambda_1(S_1$. The main result is to show $lambda_2(Omega le lambda_2(S_1$. We also give an extension of the main result to the case of a more general elliptic eigenvalue problem on a bounded domain $Omega$ with Dirichlet boundary conditions.
Luchko, Yuri; Mainardi, Francesco
2013-06-01
In this paper, the one-dimensional time-fractional diffusion-wave equation with the Caputo fractional derivative of order α, 1 ≤ α ≤ 2 and with constant coefficients is revisited. It is known that the diffusion and the wave equations behave quite differently regarding their response to a localized disturbance. Whereas the diffusion equation describes a process where a disturbance spreads infinitely fast, the propagation speed of the disturbance is a constant for the wave equation. We show that the time-fractional diffusion-wave equation interpolates between these two different responses and investigate the behavior of its fundamental solution for the signalling problem in detail. In particular, the maximum location, the maximum value, and the propagation velocity of the maximum point of the fundamental solution for the signalling problem are described analytically and calculated numerically.
Superaccurate finite element eigenvalues via a Rayleigh quotient correction
Fried, Isaac; Leong, Kaiwen
2005-11-01
The consistent finite element formulation of the vibration problem generates upper bounds on the corresponding exact eigenvalues but requires the solution of the highly expensive general algebraic eigenproblem Kx=λMx with a global matrix M that is of the same sparsity pattern as the global stiffness K. The lumped, diagonal, mass matrix finite element formulation is no longer variationally correct but results in a simplified algebraic eigenproblem of comparable accuracy. We may write the mass matrix as a linear matrix function, M(γ)=M1+γM2, of parameter γ such that M(γ=1) is the (diagonal) lumped mass matrix and M(γ=0) is the consistent mass matrix. It has been shown that an optimal γ exists between these two states which results in superaccurate eigenvalues. What detracts from the appeal of this approach is that the superior accuracy thus achieved comes at the hefty price of having to solve the still general algebraic eigenproblem with a nondiagonal mass matrix. In this note we show that the same superior accuracy can be had by first computing an eigenvector u from Ku=λDu, in which D=M1+M2 is the lumped, diagonal, mass matrix, and then obtaining the corresponding, superaccurate, eigenvalue from the Rayleigh quotient R[u]=uTKu/uTM(γ)u, M(γ)=M1+γM2 for an optimal γ.
Bai, Zheng-Jian; Datta, Biswa Nath; Wang, Jinwei
2010-04-01
The partial quadratic eigenvalue assignment problem (PQEVAP) concerns reassigning a few undesired eigenvalues of a quadratic matrix pencil to suitably chosen locations and keeping the other large number of eigenvalues and eigenvectors unchanged (no spill-over). The problem naturally arises in controlling dangerous vibrations in structures by means of active feedback control design. For practical viability, the design must be robust, which requires that the norms of the feedback matrices and the condition number of the closed-loop eigenvectors are as small as possible. The problem of computing feedback matrices that satisfy the above two practical requirements is known as the Robust Partial Quadratic Eigenvalue Assignment Problem (RPQEVAP). In this paper, we formulate the RPQEVAP as an unconstrained minimization problem with the cost function involving the condition number of the closed-loop eigenvector matrix and two feedback norms. Since only a small number of eigenvalues of the open-loop quadratic pencil are computable using the state-of-the-art matrix computational techniques and/or measurable in a vibration laboratory, it is imperative that the problem is solved using these small number of eigenvalues and the corresponding eigenvectors. To this end, a class of the feedback matrices are obtained in parametric form, parameterized by a single parametric matrix, and the cost function and the required gradient formulas for the optimization problem are developed in terms of the small number of eigenvalues that are reassigned and their corresponding eigenvectors. The problem is solved directly in quadratic setting without transforming it to a standard first-order control problem and most importantly, the significant "no spill-over property" of the closed-loop eigenvalues and eigenvectors is established by means of a mathematical result. These features make the proposed method practically applicable even for very large structures. Results on numerical experiments show
Eigenvalue Separation in Some Random Matrix Models
Bassler, Kevin E; Frankel, Norman E
2008-01-01
The eigenvalue density for members of the Gaussian orthogonal and unitary ensembles follows the Wigner semi-circle law. If the Gaussian entries are all shifted by a constant amount c/Sqrt(2N), where N is the size of the matrix, in the large N limit a single eigenvalue will separate from the support of the Wigner semi-circle provided c > 1. In this study, using an asymptotic analysis of the secular equation for the eigenvalue condition, we compare this effect to analogous effects occurring in general variance Wishart matrices and matrices from the shifted mean chiral ensemble. We undertake an analogous comparative study of eigenvalue separation properties when the size of the matrices are fixed and c goes to infinity, and higher rank analogues of this setting. This is done using exact expressions for eigenvalue probability densities in terms of generalized hypergeometric functions, and using the interpretation of the latter as a Green function in the Dyson Brownian motion model. For the shifted mean Gaussian u...
Cell-centered nonlinear finite-volume methods for the heterogeneous anisotropic diffusion problem
Terekhov, Kirill M.; Mallison, Bradley T.; Tchelepi, Hamdi A.
2017-02-01
We present two new cell-centered nonlinear finite-volume methods for the heterogeneous, anisotropic diffusion problem. The schemes split the interfacial flux into harmonic and transversal components. Specifically, linear combinations of the transversal vector and the co-normal are used that lead to significant improvements in terms of the mesh-locking effects. The harmonic component of the flux is represented using a conventional monotone two-point flux approximation; the component along the parameterized direction is treated nonlinearly to satisfy either positivity of the solution as in [29], or the discrete maximum principle as in [9]. In order to make the method purely cell-centered, we derive a homogenization function that allows for seamless interpolation in the presence of heterogeneity following a strategy similar to [46]. The performance of the new schemes is compared with existing multi-point flux approximation methods [3,5]. The robustness of the scheme with respect to the mesh-locking problem is demonstrated using several challenging test cases.
Institute of Scientific and Technical Information of China (English)
张俊; 朱克超; 范馨月
2016-01-01
讨论三维 Schrödinger 方程特征值问题的 Wilson 元离散及二网格离散方案,得到相应的误差估计结果。数值实验结果表明,Schrödinger 方程的 Wilson 元下逼近于准确特征值。%This paper discusses the nonconforming finite element for eigenvalueproblem of Schrödinger equation.Based on the two-scale scheme,this paper establish two-scale scheme for Wilson nonconforming element of eigenvalue problem of Schrödinger equation.It is proved the resulting solution of the scheme still maintainsan asymptotically optimal order of accuracy and numerical experiments areprovided to support theoretical conclusions.
Institute of Scientific and Technical Information of China (English)
马龙; 刘杰
2011-01-01
This paper discusses the residual type a posterior error estimate of EQ1^rot element for the Possion eigenvalue problem. The posterior error estimator is obtained by using the properties in nonconforming EQ1^rot element space and the unified framework of Carstensen et al. （ in SIAM. J. Numer. Anal. 2007,45 （1）：68- 82）. The results of numerical experiments confirm the posterior error estimator is reliable and effective.%把Carstensen和Jun Hu（见SIAM JNumer Anal，2007，107：473—502）建立的残差型误差指示子，移植到特征值问题EQ1^not元近似。给出了Matlab程序，计算结果说明该指示子是有效和可靠的。
Directory of Open Access Journals (Sweden)
D. Goos
2015-01-01
Full Text Available We consider the time-fractional derivative in the Caputo sense of order α∈(0, 1. Taking into account the asymptotic behavior and the existence of bounds for the Mainardi and the Wright function in R+, two different initial-boundary-value problems for the time-fractional diffusion equation on the real positive semiaxis are solved. Moreover, the limit when α↗1 of the respective solutions is analyzed, recovering the solutions of the classical boundary-value problems when α = 1, and the fractional diffusion equation becomes the heat equation.
Energy Technology Data Exchange (ETDEWEB)
Betzler, Benjamin R., E-mail: betzlerbr@ornl.gov [Department of Nuclear Engineering and Radiological Sciences, University of Michigan, 2355 Bonisteel Boulevard, Ann Arbor, MI 48109 (United States); Kiedrowski, Brian C., E-mail: bckiedro@umich.edu [Department of Nuclear Engineering and Radiological Sciences, University of Michigan, 2355 Bonisteel Boulevard, Ann Arbor, MI 48109 (United States); Brown, Forrest B., E-mail: fbrown@lanl.gov [Los Alamos National Laboratory, P.O. Box 1663, MS A143, Los Alamos, NM 87545 (United States); Martin, William R., E-mail: wrm@umich.edu [Department of Nuclear Engineering and Radiological Sciences, University of Michigan, 2355 Bonisteel Boulevard, Ann Arbor, MI 48109 (United States)
2015-12-15
Highlights: • A transition rate matrix method for calculating α-eigenvalues is formulated. • Verification of this method is performed using multigroup infinite-medium problems. • Applications to continuous-energy media examine the slowing down of neutrons. • The effect of the α-eigenvalue spectrum on the short-time flux behavior is discussed. - Abstract: The time-dependent behavior of the energy spectrum in neutron transport was investigated with a formulation, based on continuous-time Markov processes, for computing α eigenvalues and eigenvectors in an infinite medium. For this, a research Monte Carlo code called “TORTE” (To Obtain Real Time Eigenvalues) was created and used to estimate elements of a transition rate matrix. TORTE is capable of using both multigroup and continuous-energy nuclear data, and verification was performed. Eigenvalue spectra for infinite homogeneous mixtures were obtained, and an eigenfunction expansion was used to investigate transient behavior of the neutron energy spectrum.
Capiński, Maciej J.; Gidea, Marian; de la Llave, Rafael
2017-01-01
We present a diffusion mechanism for time-dependent perturbations of autonomous Hamiltonian systems introduced in Gidea (2014 arXiv:1405.0866). This mechanism is based on shadowing of pseudo-orbits generated by two dynamics: an ‘outer dynamics’, given by homoclinic trajectories to a normally hyperbolic invariant manifold, and an ‘inner dynamics’, given by the restriction to that manifold. On the inner dynamics the only assumption is that it preserves area. Unlike other approaches, Gidea (2014 arXiv:1405.0866) does not rely on the KAM theory and/or Aubry-Mather theory to establish the existence of diffusion. Moreover, it does not require to check twist conditions or non-degeneracy conditions near resonances. The conditions are explicit and can be checked by finite precision calculations in concrete systems (roughly, they amount to checking that Melnikov-type integrals do not vanish and that some manifolds are transversal). As an application, we study the planar elliptic restricted three-body problem. We present a rigorous theorem that shows that if some concrete calculations yield a non zero value, then for any sufficiently small, positive value of the eccentricity of the orbits of the main bodies, there are orbits of the infinitesimal body that exhibit a change of energy that is bigger than some fixed number, which is independent of the eccentricity. We verify numerically these calculations for values of the masses close to that of the Jupiter/Sun system. The numerical calculations are not completely rigorous, because we ignore issues of round-off error and do not estimate the truncations, but they are not delicate at all by the standard of numerical analysis. (Standard tests indicate that we get 7 or 8 figures of accuracy where 1 would be enough.) The code of these verifications is available. We hope that some full computer assisted proofs will be obtained in the near future since there are packages (CAPD) designed for problems of this type.
Block Locally Optimal Preconditioned Eigenvalue Xolvers (BLOPEX) in hypre and PETSc
Knyazev, A V; Lashuk, I; Ovtchinnikov, E E
2007-01-01
We describe our software package Block Locally Optimal Preconditioned Eigenvalue Xolvers (BLOPEX) publicly released recently. BLOPEX is available as a stand-alone serial library, as an external package to PETSc (``Portable, Extensible Toolkit for Scientific Computation'', a general purpose suite of tools for the scalable solution of partial differential equations and related problems developed by Argonne National Laboratory), and is also built into {\\it hypre} (``High Performance Preconditioners'', scalable linear solvers package developed by Lawrence Livermore National Laboratory). The present BLOPEX release includes only one solver--the Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) method for symmetric eigenvalue problems. {\\it hypre} provides users with advanced high-quality parallel preconditioners for linear systems, in particular, with domain decomposition and multigrid preconditioners. With BLOPEX, the same preconditioners can now be efficiently used for symmetric eigenvalue problems...
Shakir, Muhammad
2011-12-01
In this paper, we introduce a new detector referred to as Geometric mean detector (GEMD) which is based on the ratio of the largest eigenvalue to the Geometric mean of the eigenvalues for collaborative spectrum sensing. The decision threshold has been derived by employing Gaussian approximation approach. In this approach, the two random variables, i.e. The largest eigenvalue and the Geometric mean of the eigenvalues are considered as independent Gaussian random variables such that their cumulative distribution functions (CDFs) are approximated by a univariate Gaussian distribution function for any number of cooperating secondary users and received samples. The approximation approach is based on the calculation of exact analytical moments of the largest eigenvalue and the Geometric mean of the eigenvalues of the received covariance matrix. The decision threshold has been calculated by exploiting the CDF of the ratio of two Gaussian distributed random variables. In this context, we exchange the analytical moments of the two random variables with the moments of the Gaussian distribution function. The performance of the detector is compared with the performance of the energy detector and eigenvalue ratio detector. Analytical and simulation results show that our newly proposed detector yields considerable performance advantage in realistic spectrum sensing scenarios. Moreover, our results based on proposed approximation approach are in perfect agreement with the empirical results. © 2011 IEEE.
Selection Theory of Dendritic Growth with Anisotropic Diffusion
Directory of Open Access Journals (Sweden)
Martin von Kurnatowski
2015-01-01
Full Text Available Dendritic patterns frequently arise when a crystal grows into its own undercooled melt. Latent heat released at the two-phase boundary is removed by some transport mechanism, and often the problem can be described by a simple diffusion model. Its analytic solution is based on a perturbation expansion about the case without capillary effects. The length scale of the pattern is determined by anisotropic surface tension, which provides the mechanism for stabilizing the dendrite. In the case of liquid crystals, diffusion can be anisotropic too. Growth is faster in the direction of less efficient heat transport (inverted growth. Any physical solution should include this feature. A simple spatial rescaling is used to reduce the bulk equation in 2D to the case of isotropic diffusion. Subsequently, an eigenvalue problem for the growth mode results from the interface conditions. The eigenvalue is calculated numerically and the selection problem of dendritic growth with anisotropic diffusion is solved. The length scale is predicted and a quantitative description of the inverted growth phenomenon is given. It is found that anisotropic diffusion cannot take the stabilizing role of anisotropic surface tension.
Deswal, Sunita; Kalkal, Kapil Kumar; Sheoran, Sandeep Singh
2016-09-01
A mathematical model of fractional order two-temperature generalized thermoelasticity with diffusion and initial stress is proposed to analyze the transient wave phenomenon in an infinite thermoelastic half-space. The governing equations are derived in cylindrical coordinates for a two dimensional axi-symmetric problem. The analytical solution is procured by employing the Laplace and Hankel transforms for time and space variables respectively. The solutions are investigated in detail for a time dependent heat source. By using numerical inversion method of integral transforms, we obtain the solutions for displacement, stress, temperature and diffusion fields in physical domain. Computations are carried out for copper material and displayed graphically. The effect of fractional order parameter, two-temperature parameter, diffusion, initial stress and time on the different thermoelastic and diffusion fields is analyzed on the basis of analytical and numerical results. Some special cases have also been deduced from the present investigation.
A comparison between the fission matrix method, the diffusion model and the transport model
Energy Technology Data Exchange (ETDEWEB)
Dehaye, B.; Hugot, F. X.; Diop, C. M. [Commissariat a l' Energie Atomique et aux Energies Alternatives, Direction de l' Energie Nucleaire, Departement de Modelisation des Systemes et Structures, CEA DEN/DM2S, PC 57, F-91191 Gif-sur-Yvette cedex (France)
2013-07-01
The fission matrix method may be used to solve the critical eigenvalue problem in a Monte Carlo simulation. This method gives us access to the different eigenvalues and eigenvectors of the transport or fission operator. We propose to compare the results obtained via the fission matrix method with those of the diffusion model, and an approximated transport model. To do so, we choose to analyse the mono-kinetic and continuous energy cases for a Godiva-inspired critical sphere. The first five eigenvalues are computed with TRIPOLI-4{sup R} and compared to the theoretical ones. An extension of the notion of the extrapolation distance is proposed for the modes other than the fundamental one. (authors)
Yu, Zhiyong
2016-01-01
In this paper, we investigate infinite horizon jump-diffusion forward-backward stochastic differential equations under some monotonicity conditions. We establish an existence and uniqueness theorem, two stability results and a comparison theorem for solutions to such kind of equations. Then the theoretical results are applied to study a kind of infinite horizon backward stochastic linear-quadratic optimal control problems, and then differential game problems. The unique optimal controls for t...
2D BEM modeling of a singular thermal diffusion free boundary problem with phase change
Nikolayev, Vadim
2016-01-01
We report a 2D Boundary Element Method (BEM) modeling of the thermal diffusion-controlled growth of a vapor bubble attached to a heating surface during saturated pool boiling. The transient heat conduction problem is solved in a liquid that surrounds a bubble with a free boundary and in a semi-infinite solid heater. The heat generated homogeneously in the heater causes evaporation, i. e. the bubble growth. A singularity exists at the point of the triple (liquid-vapor-solid) contact. At high system pressure the bubble is assumed to grow slowly, its shape being defined by the surface tension and the vapor recoil force, a force coming from the liquid evaporating into the bubble. It is shown that at some typical time the dry spot under the bubble begins to grow rapidly under the action of the vapor recoil. Such a bubble can eventually spread into a vapor film that can separate the liquid from the heater, thus triggering the boiling crisis (Critical Heat Flux phenomenon).
DEVELOPMENT OF AN ITERATIVE METHOD TO SOLVE THE DIFFUSION PROBLEM IN THE PREDESIGN STEP
Abouali Sanchez, Said
2014-01-01
One of the most important and difficult steps in the design process is the predesign. In this step, the main goal is to obtain the geometry characteristics and some other features to solve the problem we want to solve. Thus, reducing the duration of this phase is one of the challenging objectives in the actual engineering design process. As regards our study, following the energy deposition calculations with the means of the FLUKA code, a heat diffusion code is usually needed to calculate the evolution in time of the temperature distributions as well the associated stresses on the object under study. As a preliminary optimization step in the object design process, in this project we are proposing thedevelopment of a new method implemented in a code which allows us to obtain the solution for this equation in a complex geometry by using the information given by the FLUKA code for a simple geometry. By this way, we will be able to reduce the predesign phase duration.
Directory of Open Access Journals (Sweden)
Pratibha Joshi
2014-12-01
Full Text Available In this paper, we have achieved high order solution of a three dimensional nonlinear diffusive-convective problem using modified variational iteration method. The efficiency of this approach has been shown by solving two examples. All computational work has been performed in MATHEMATICA.
A simple method for complex eigenvalues
Energy Technology Data Exchange (ETDEWEB)
Killingbeck, John P [Mathematics Department, University of Hull, Hull HU6 7RX (United Kingdom); Grosjean, Alain [Laboratoire d' Astrophysique de l' Observatoire de Besancon, CNRS, UMR 6091, 41 bis Avenue de l' Observatoire, BP1615, 25010 Besancon Cedex (France); Jolicard, Georges [Laboratoire d' Astrophysique de l' Observatoire de Besancon, CNRS, UMR 6091, 41 bis Avenue de l' Observatoire, BP1615, 25010 Besancon Cedex (France)
2004-11-05
A simple iterative method is described for finding the eigenvalues of a general square complex matrix. Several numerical examples involving complex symmetric matrices are treated. In particular, it is found that a naive matrix calculation without complex rotation produces resonant state energies in accord with those given by the recently introduced naive complex hypervirial perturbation theory. (letter to the editor)
Controlling chaos to solutions with complex eigenvalues.
Kwon, Oh-Jong; Lee, Hoyun
2003-02-01
We derive formulas for parameter and variable perturbations to control chaos using linearized dynamics. They are available irrespective of the dimension of the system, the number of perturbed parameters or variables, and the kinds of eigenvalues of the linearized dynamics. We illustrate this using the two coupled Duffing oscillators and the two coupled standard maps.
Analysis of eigenvalue correction applied to biometrics
Hendrikse, A.J.; Veldhuis, Raymond N.J.; Spreeuwers, Lieuwe Jan; Bazen, A.M.
Eigenvalue estimation plays an important role in biometrics. However, if the number of samples is limited, estimates are significantly biased. In this article we analyse the influence of this bias on the error rates of PCA/LDA based verification systems, using both synthetic data with realistic
Second order perturbation theory for embedded eigenvalues
DEFF Research Database (Denmark)
Faupin, Jeremy; Møller, Jacob Schach; Skibsted, Erik
2011-01-01
We study second order perturbation theory for embedded eigenvalues of an abstract class of self-adjoint operators. Using an extension of the Mourre theory, under assumptions on the regularity of bound states with respect to a conjugate operator, we prove upper semicontinuity of the point spectrum...
Perturbation of eigenvalues embedded at a threshold
DEFF Research Database (Denmark)
Jensen, Arne; Melgaard, Michael
2002-01-01
Results are obtained on perturbation of eigenvalues and half-bound states (zero-resonances) embedded at a threshold. The results are obtained in a two-channel framework for small off-diagonal perturbations. The results are based on given asymptotic expansions of the component Hamiltonians....
Spectral Analysis of Diffusions with Jump Boundary
Kolb, Martin
2011-01-01
In this paper we consider one-dimensional diffusions with constant coefficients in a finite interval with jump boundary and a certain deterministic jump distribution. We use coupling methods in order to identify the spectral gap in the case of a large drift and prove that that there is a threshold drift above which the bottom of the spectrum no longer depends on the drift. As a Corollary to our result we are able to answer two questions concerning elliptic eigenvalue problems with non-local boundary conditions formulated previously by Iddo Ben-Ari and Ross Pinsky.
A finite element algorithm for high-lying eigenvalues with Neumann and Dirichlet boundary conditions
Báez, G.; Méndez-Sánchez, R. A.; Leyvraz, F.; Seligman, T. H.
2014-01-01
We present a finite element algorithm that computes eigenvalues and eigenfunctions of the Laplace operator for two-dimensional problems with homogeneous Neumann or Dirichlet boundary conditions, or combinations of either for different parts of the boundary. We use an inverse power plus Gauss-Seidel algorithm to solve the generalized eigenvalue problem. For Neumann boundary conditions the method is much more efficient than the equivalent finite difference algorithm. We checked the algorithm by comparing the cumulative level density of the spectrum obtained numerically with the theoretical prediction given by the Weyl formula. We found a systematic deviation due to the discretization, not to the algorithm itself.
Using Generalized Annotated Programs to Solve Social Network Diffusion Optimization Problems
2013-01-01
0000000.0000000 1. INTRODUCTION There is a rapid proliferation of different types of graph data in the world today. These include social network data ( FaceBook ...et al. 2008] (diffusion of photos in Flickr) and [Sun et al. 2009] (diffusion of bookmarks in FaceBook ) both look at diffusion process in social...a similar property (search term) will be interested in the same advertised item. In fact, [Cha et al. 2008] explicitly pre-processed their Flickr
Institute of Scientific and Technical Information of China (English)
何楚宁
2001-01-01
This paper considers the following problem:Given X,B∈Rm×n, find A∈SE∩Rmn such that AX = B.where SE= {A∈Rm×n l AE-F ‖ =min, E,F∈Rm×k}, Rmn= {A∈Rm×n|YTAY≤0, A Y∈Rm×1}, ‖.‖ is the Frobenius norm.The necessary and sufficient condition for the problem having a solution is studied. The expressions for general solutions of the problem are also given.
Universal inequalities for the eigenvalues of a power of the Laplace operator
Ilias, Said
2010-01-01
In this paper, we obtain a new abstract formula relating eigenvalues of a self-adjoint operator to two families of symmetric and skew-symmetric operators and their commutators. This formula generalizes earlier ones obtained by Harrell, Stubbe, Hook, Ashbaugh, Hermi, Levitin and Parnovski. We also show how one can use this abstract formulation both for giving dierent and simpler proofs for all the known results obtained for the eigenvalues of a power of the Laplace operator (i.e. the Dirichlet Laplacian, the clamped plate problem for the bilaplacian and more generally for the polyharmonic problem on a bounded Euclidean domain) and to obtain new ones. In a last paragraph, we derive new bounds for eigenvalues of any power of the Kohn Laplacian on the Heisenberg group.
Existence and uniqueness of positive eigenfunctions for certain eigenvalue systems
Xue, Ru-Ying; Yang, Yi-Min
2004-01-01
The existence and uniqueness of eigenvalues and positive eigenfunctions for some quasilinear elliptic systems are considered. Some necessary and sufficient conditions which guarantee the existence and uniqueness of eigenvalues and positive eigenfunctions are given.
Generalization of Samuelson’s inequality and location of eigenvalues
Indian Academy of Sciences (India)
R Sharma; R Saini
2015-02-01
We prove a generalization of Samuelson’s inequality for higher order central moments. Bounds for the eigenvalues are obtained when a given complex × matrix has real eigenvalues. Likewise, we discuss bounds for the roots of polynomial equations.
Travelling fronts of the CO oxidation on Pd(111) with coverage-dependent diffusion
Energy Technology Data Exchange (ETDEWEB)
Cisternas, Jaime, E-mail: jecisternas@miuandes.cl [Facultad de Ingeniería y Ciencias Aplicadas, Universidad de los Andes, Monseñor Alvaro del Portillo 12455, Las Condes, Santiago (Chile); Karpitschka, Stefan [Physics of Fluids, University of Twente, Drienerlolaan 5, 7522 NB Enschede (Netherlands); Wehner, Stefan [Institut für Integrierte Naturwissenschaften - Physik, Universität Koblenz-Landau, 56070 Koblenz (Germany)
2014-10-28
In this work, we study a surface reaction on Pd(111) crystals under ultra-high-vacuum conditions that can be modeled by two coupled reaction-diffusion equations. In the bistable regime, the reaction exhibits travelling fronts that can be observed experimentally using photo electron emission microscopy. The spatial profile of the fronts reveals a coverage-dependent diffusivity for one of the species. We propose a method to solve the nonlinear eigenvalue problem and compute the direction and the speed of the fronts based on a geometrical construction in phase-space. This method successfully captures the dependence of the speed on control parameters and diffusivities.
A program to solve a solute diffusion problem with segregation at a moving interface
Bakker, M.; Hoonhout, D.
1981-01-01
The one-dimensional transient diffusion of glucose, inulin and dextran into adult bovine knee articular cartilage was determined for transport times of 1, 5, 15 and 60 min, and 4, 12, 24 and 48 h. The apparent diffusion coefficient and apparent interface partition coefficient were calculated from th
Nitsche, Ludwig C.; Nitsche, Johannes M.; Brenner, Howard
1988-01-01
The sedimentation and diffusion of a nonneutrally buoyant Brownian particle in vertical fluid-filled cylinder of finite length which is instantaneously inverted at regular intervals are investigated analytically. A one-dimensional convective-diffusive equation is derived to describe the temporal and spatial evolution of the probability density; a periodicity condition is formulated; the applicability of Fredholm theory is established; and the parameter-space regions are determined within which the existence and uniqueness of solutions are guaranteed. Numerical results for sample problems are presented graphically and briefly characterized.
On the Eigenvalue Two and Matching Number of a Tree
Institute of Scientific and Technical Information of China (English)
Yi-zheng Fan
2004-01-01
In [6], Guo and Tan have shown that 2 is a Laplacian eigenvalue of any tree with perfect matchings.For trees without perfect matchings, we study whether 2 is one of its Laplacian eigenvalues. If the matchingnumber is 1 or 2, the answer is negative; otherwise, there exists a tree with that matching number which has (hasnot) the eigenvalue 2. In particular, we determine all trees with matching number 3 which has the eigenvalue2.
Front propagation in cellular flows for fast reaction and small diffusivity
Tzella, Alexandra
2014-01-01
We investigate the influence of fluid flows on the propagation of chemical fronts arising in FKPP type models. For the cellular flows we consider, the front propagation speed can be determined numerically by solving an eigenvalue problem; this is however difficult for small molecular diffusivity and fast reaction, i.e., when the P\\'eclet (Pe) and Damk\\"ohler (Da) numbers are large. Here, we employ a WKB approach to obtain the front speed for a broad range of Pe,Da$\\gg 1$ in terms of a periodic path -- an instanton -- that minimizes a certain functional, and to derive closed-form results for Da$\\ll$Pe and for Da$\\gg$Pe. Our theoretical predictions are compared with (i) numerical solutions of the eigenvalue problem and (ii) simulations of the advection--diffusion--reaction equation.
Decomposition of spectral density in individual eigenvalue contributions
Energy Technology Data Exchange (ETDEWEB)
Bohigas, O; Pato, M P, E-mail: mpato@if.usp.b [CNRS, Universite Paris-Sud, UMR8626, LPTMS, Orsay Cedex, F-91405 (France)
2010-09-10
The eigenvalue densities of two random matrix ensembles, the Wigner Gaussian matrices and the Wishart covariant matrices, are decomposed in the contributions of each individual eigenvalue distribution. It is shown that the fluctuations of all eigenvalues, for medium matrix sizes, are described with a good precision by nearly normal distributions.
Bicyclic graphs with exactly two main signless Laplacian eigenvalues
Huang, He
2012-01-01
A signless Laplacian eigenvalue of a graph $G$ is called a main signless Laplacian eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this paper, all connected bicyclic graphs with exactly two main eigenvalues are determined.
Directory of Open Access Journals (Sweden)
Baoyan Li
2003-09-01
Full Text Available We study the hp version of three families of Eulerian-Lagrangian mixed discontinuous finite element (MDFE methods for the numerical solution of advection-diffusion problems. These methods are based on a space-time mixed formulation of the advection-diffusion problems. In space, they use discontinuous finite elements, and in time they approximately follow the Lagrangian flow paths (i.e., the hyperbolic part of the problems. Boundary conditions are incorporated in a natural and mass conservative manner. In fact, these methods are locally conservative. The analysis of this paper focuses on advection-diffusion problems in one space dimension. Error estimates are explicitly obtained in the grid size h, the polynomial degree p, and the solution regularity; arbitrary space grids and polynomial degree are allowed. These estimates are asymptotically optimal in both h and p for some of these methods. Numerical results to show convergence rates in h and p of the Eulerian-Lagrangian MDFE methods are presented. They are in a good agreement with the theory.
Marek, A; Blum, V; Johanni, R; Havu, V; Lang, B; Auckenthaler, T; Heinecke, A; Bungartz, H-J; Lederer, H
2014-05-28
Obtaining the eigenvalues and eigenvectors of large matrices is a key problem in electronic structure theory and many other areas of computational science. The computational effort formally scales as O(N(3)) with the size of the investigated problem, N (e.g. the electron count in electronic structure theory), and thus often defines the system size limit that practical calculations cannot overcome. In many cases, more than just a small fraction of the possible eigenvalue/eigenvector pairs is needed, so that iterative solution strategies that focus only on a few eigenvalues become ineffective. Likewise, it is not always desirable or practical to circumvent the eigenvalue solution entirely. We here review some current developments regarding dense eigenvalue solvers and then focus on the Eigenvalue soLvers for Petascale Applications (ELPA) library, which facilitates the efficient algebraic solution of symmetric and Hermitian eigenvalue problems for dense matrices that have real-valued and complex-valued matrix entries, respectively, on parallel computer platforms. ELPA addresses standard as well as generalized eigenvalue problems, relying on the well documented matrix layout of the Scalable Linear Algebra PACKage (ScaLAPACK) library but replacing all actual parallel solution steps with subroutines of its own. For these steps, ELPA significantly outperforms the corresponding ScaLAPACK routines and proprietary libraries that implement the ScaLAPACK interface (e.g. Intel's MKL). The most time-critical step is the reduction of the matrix to tridiagonal form and the corresponding backtransformation of the eigenvectors. ELPA offers both a one-step tridiagonalization (successive Householder transformations) and a two-step transformation that is more efficient especially towards larger matrices and larger numbers of CPU cores. ELPA is based on the MPI standard, with an early hybrid MPI-OpenMPI implementation available as well. Scalability beyond 10,000 CPU cores for problem
Eigenvalue conjecture and colored Alexander polynomials
Mironov, A
2016-01-01
We connect two important conjectures in the theory of knot polynomials. The first one is the property Al_R(q) = Al_{[1]}(q^{|R|}) for all single hook Young diagrams R, which is known to hold for all knots. The second conjecture claims that all the mixing matrices U_{i} in the relation {\\cal R}_i = U_i{\\cal R}_1U_i^{-1} between the i-th and the first generators {\\cal R}_i of the braid group are universally expressible through the eigenvalues of {\\cal R}_1. Since the above property of Alexander polynomials is very well tested, this relation provides a new support to the eigenvalue conjecture, especially for i>2, when its direct check by evaluation of the Racah matrices and their convolutions is technically difficult.
Fuller, Nathaniel J
2016-01-01
Obtaining a detailed understanding of the physical interactions between a cell and its environment often requires information about the flow of fluid surrounding the cell. Cells must be able to effectively absorb and discard material in order to survive. Strategies for nutrient acquisition and toxin disposal, which have been evolutionarily selected for their efficacy, should reflect knowledge of the physics underlying this mass transport problem. Motivated by these considerations, in this paper we consider a two-dimensional advection-diffusion problem at small Reynolds number and large P\\'eclet number. We discuss the problem of mass transport for a circular cell in a uniform far-field flow. We approach the problem numerically, and also analytically through a rescaling of the concentration boundary layer. A biophysically motivated first-passage problem for the absorption of material by the cell demonstrates quantitative agreement between the numerical and analytical approaches.
Phase-space diffusion in turbulent plasmas: The random acceleration problem revisited
DEFF Research Database (Denmark)
Pécseli, H.L.; Trulsen, J.
1991-01-01
Phase-space diffusion of test particles in turbulent plasmas is studied by an approach based on a conditional statistical analysis of fluctuating electrostatic fields. Analytical relations between relevant conditional averages and higher-order correlations, , and triple...
A novel schedule for solving the two-dimensional diffusion problem in fractal heat transfer
Directory of Open Access Journals (Sweden)
Xu Shu
2015-01-01
Full Text Available In this work, the local fractional variational iteration method is employed to obtain approximate analytical solution of the two-dimensional diffusion equation in fractal heat transfer with help of local fractional derivative and integral operators.
Eigenvalue translation method for mode calculations.
Gerck, E; Cruz, C H
1979-05-01
A new method is described for the first few modes calculations in a interferometer that has several advantages over the Allmat subroutine, the Prony method, and the Fox and Li method. In the illustrative results shown for some cases it can be seen that the eigenvalue translation method is typically 100-fold times faster than the usual Fox and Li method and ten times faster than Allmat.
Upper Bounds for the Laplacian Graph Eigenvalues
Institute of Scientific and Technical Information of China (English)
Jiong Sheng LI; Yong Liang PAN
2004-01-01
We first apply non-negative matrix theory to the matrix K = D + A, where D and A are the degree-diagonal and adjacency matrices of a graph G, respectively, to establish a relation on the largest Laplacian eigenvalue λ1 (G) of G and the spectral radius ρ(K) of K. And then by using this relation we present two upper bounds for λ1 (G) and determine the extremal graphs which achieve the upper bounds.
Eigenvalue variance bounds for covariance matrices
Dallaporta, Sandrine
2013-01-01
This work is concerned with finite range bounds on the variance of individual eigenvalues of random covariance matrices, both in the bulk and at the edge of the spectrum. In a preceding paper, the author established analogous results for Wigner matrices and stated the results for covariance matrices. They are proved in the present paper. Relying on the LUE example, which needs to be investigated first, the main bounds are extended to complex covariance matrices by means of the Tao, Vu and Wan...
Jia, Junxiong; Peng, Jigen; Yang, Jiaqing
2017-04-01
In this paper, we focus on a space-time fractional diffusion equation with the generalized Caputo's fractional derivative operator and a general space nonlocal operator (with the fractional Laplace operator as a special case). A weak Harnack's inequality has been established by using a special test function and some properties of the space nonlocal operator. Based on the weak Harnack's inequality, a strong maximum principle has been obtained which is an important characterization of fractional parabolic equations. With these tools, we establish a uniqueness result of an inverse source problem on the determination of the temporal component of the inhomogeneous term, which seems to be the first theoretical result of the inverse problem for such a general fractional diffusion model.
Energy Technology Data Exchange (ETDEWEB)
Perfetti, C.; Martin, W. [Univ. of Michigan, Dept. of Nuclear Engineering and Radiological Sciences, 2355 Bonisteel Boulevard, Ann Arbor, MI 48109-2104 (United States); Rearden, B.; Williams, M. [Oak Ridge National Laboratory, Reactor and Nuclear Systems Div., Bldg. 5700, P.O. Box 2008, Oak Ridge, TN 37831-6170 (United States)
2012-07-01
Three methods for calculating continuous-energy eigenvalue sensitivity coefficients were developed and implemented into the Shift Monte Carlo code within the SCALE code package. The methods were used for two small-scale test problems and were evaluated in terms of speed, accuracy, efficiency, and memory requirements. A promising new method for calculating eigenvalue sensitivity coefficients, known as the CLUTCH method, was developed and produced accurate sensitivity coefficients with figures of merit that were several orders of magnitude larger than those from existing methods. (authors)
Bai, Zheng-Jian; Wan, Qiu-Yue
2017-05-01
In this paper, we consider the partial quadratic eigenvalue assignment problem (PQEAP) in vibration by active feedback control. Based on the receptance measurements and system matrices, we propose a constructive method for solving PQEAP, where we only need to solve a small linear system and only a few undesired open-loop eigenvalues with associated eigenvectors are needed. Our method is designed for both single-input and multiple-input vibration controls of vibrating structures. The real form of our method is also presented. Numerical tests show that our method is effective for constructing a solution to PQEAP with both single-input and multiple-input vibration controls.
Real eigenvalue analysis in NASTRAN by the tridiagonal reduction (FEER) method
Newman, M.; Flanagen, P. F.; Rogers, J. L., Jr.
1976-01-01
Implementation of the tridiagonal reduction method for real eigenvalue extraction in structural vibration and buckling problems is described. The basic concepts underlying the method are summarized and special features, such as the computation of error bounds and default modes of operation are discussed. In addition, the new user information and error messages and optional diagnostic output relating to the tridiagonal reduction method are presented. Some numerical results and initial experiences relating to usage in the NASTRAN environment are provided, including comparisons with other existing NASTRAN eigenvalue methods.
Lq-perturbations of leading coefficients of elliptic operators: Asymptotics of eigenvalues
Directory of Open Access Journals (Sweden)
Vladimir Kozlov
2006-01-01
Full Text Available We consider eigenvalues of elliptic boundary value problems, written in variational form, when the leading coefficients are perturbed by terms which are small in some integral sense. We obtain asymptotic formulae. The main specific of these formulae is that the leading term is different from that in the corresponding formulae when the perturbation is small in L∞-norm.
EXACT CONTROLLABILITY FOR FIRST ORDER QUASILINEAR HYPERBOLIC SYSTEMS WITH ZERO EIGENVALUES
Institute of Scientific and Technical Information of China (English)
LI TATSIEN; YU LIXIN
2003-01-01
For a class of mixed initial-boundary value problem for general quasilinear hyperbolic sys-tems with zero eigenvalues, the authors establish the local exact controllability with boundarycontrols acting on one end or on two ends and internal controls acting on a part of equationsin the system.
Directory of Open Access Journals (Sweden)
Boubakari Ibrahimou
2013-01-01
maximal monotone with and . Using the topological degree theory developed by Kartsatos and Quarcoo we study the eigenvalue problem where the operator is a single-valued of class . The existence of continuous branches of eigenvectors of infinite length then could be easily extended to the case where the operator is multivalued and is investigated.
Expansion by eigenvectors in case of simple eigenvalues of singular differential operator
Directory of Open Access Journals (Sweden)
O. V. Makhnei
2011-06-01
Full Text Available The asymptotic formulas with large values of parameter for solutions of singular differential equation allow us to estimate Green's function of the boundary-value problem. With the help of this estimation the expansion of singular dierential operator by eigenvectors in the case of simple eigenvalues is constructed.
Convergence of eigenvalues for a highly non-self-adjoint differential operator
Davies, E B
2008-01-01
In this paper we study a family of operators dependent on a small parameter $\\epsilon > 0$, which arise in a problem in fluid mechanics. We show that the spectra of these operators converge to N as $\\epsilon \\to 0$, even though, for fixed $\\epsilon > 0$, the eigenvalue asymptotics are quadratic.
Liang, Yao; Yamaura, Hiroshi; Ouyang, Huajiang
2017-06-01
As friction couples tangential and lateral degrees-of-freedom of a structure at contact interfaces, the resulting asymmetric dynamic system is prone to dynamic instability. Using state-feedback control, such a frictional asymmetric system can be stabilized through assigning the system desirable eigenvalues; but uncertainties in system parameters can cause assigned eigenvalues to deviate from desired locations and thus stability may be lost. This study presents a robust stabilization method that assigns both desirable eigenvalues and their sensitivities and thus render assigned eigenvalues stable and insensitive to perturbations in uncertain contact parameters (the friction coefficient, contact damping, and contact stiffness). This method utilizes receptances of the corresponding symmetric part of the asymmetric system. The optimal control input location is first determined by minimizing the Frobenius norm of the normalized eigen-sensitivity matrix. The normalized eigen-sensitivities indicate that the friction coefficient and contact stiffness intrinsically have similar crucial effects on the stability of the system. To demonstrate the application of the proposed control method, the eigen-sensitivities with respect to only the friction coefficient are assigned. A constrained over-determined least-squares problem is solved to assign both required eigenvalues and eigen-sensitivities. Numerical examples validate the effectiveness of the proposed robust control scheme by Monte Carlo simulations.
Energy Technology Data Exchange (ETDEWEB)
Dawes, Alan Sidney [Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Malone, Christopher M. [Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Shashkov, Mikhail Jurievich [Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
2016-07-07
In this report a number of new verification test problems for multimaterial diffusion will be shown. Using them we will show that homogenization of multimaterial cells in either Arbitrary Lagrangian Eulerian (ALE) or Eulerian simulations can lead to errors in the energy flow at the interfaces. Results will be presented that show that significant improvements and predictive capability can be gained by using either a surrogate supermesh, such as Thin Mesh in FLAG, or the emerging method based on Static Condensation.
A Decentralized Eigenvalue Computation Method for Spectrum Sensing Based on Average Consensus
Mohammadi, Jafar; Limmer, Steffen; Stańczak, Sławomir
2016-07-01
This paper considers eigenvalue estimation for the decentralized inference problem for spectrum sensing. We propose a decentralized eigenvalue computation algorithm based on the power method, which is referred to as generalized power method GPM; it is capable of estimating the eigenvalues of a given covariance matrix under certain conditions. Furthermore, we have developed a decentralized implementation of GPM by splitting the iterative operations into local and global computation tasks. The global tasks require data exchange to be performed among the nodes. For this task, we apply an average consensus algorithm to efficiently perform the global computations. As a special case, we consider a structured graph that is a tree with clusters of nodes at its leaves. For an accelerated distributed implementation, we propose to use computation over multiple access channel (CoMAC) as a building block of the algorithm. Numerical simulations are provided to illustrate the performance of the two algorithms.
Gorpas, Dimitris; Andersson-Engels, Stefan
2012-03-01
The solution of the forward problem in fluorescence molecular imaging is among the most important premises for the successful confrontation of the inverse reconstruction problem. To date, the most typical approach has been the application of the diffusion approximation as the forward model. This model is basically a first order angular approximation for the radiative transfer equation, and thus it presents certain limitations. The scope of this manuscript is to present the dual coupled radiative transfer equation and diffusion approximation model for the solution of the forward problem in fluorescence molecular imaging. The integro-differential equations of its weak formalism were solved via the finite elements method. Algorithmic blocks with cubature rules and analytical solutions of the multiple integrals have been constructed for the solution. Furthermore, specialized mapping matrices have been developed to assembly the finite elements matrix. As a radiative transfer equation based model, the integration over the angular discretization was implemented analytically, while quadrature rules were applied whenever required. Finally, this model was evaluated on numerous virtual phantoms and its relative accuracy, with respect to the radiative transfer equation, was over 95%, when the widely applied diffusion approximation presented almost 85% corresponding relative accuracy for the fluorescence emission.
Simple deterministic dynamical systems with fractal diffusion coefficients
Klages, R
1999-01-01
We analyze a simple model of deterministic diffusion. The model consists of a one-dimensional periodic array of scatterers in which point particles move from cell to cell as defined by a piecewise linear map. The microscopic chaotic scattering process of the map can be changed by a control parameter. This induces a parameter dependence for the macroscopic diffusion coefficient. We calculate the diffusion coefficent and the largest eigenmodes of the system by using Markov partitions and by solving the eigenvalue problems of respective topological transition matrices. For different boundary conditions we find that the largest eigenmodes of the map match to the ones of the simple phenomenological diffusion equation. Our main result is that the difffusion coefficient exhibits a fractal structure by varying the system parameter. To understand the origin of this fractal structure, we give qualitative and quantitative arguments. These arguments relate the sequence of oscillations in the strength of the parameter-dep...
Delshams, Amadeu; Gidea, Marian; Roldan, Pablo
2016-11-01
We consider the spatial circular restricted three-body problem, on the motion of an infinitesimal body under the gravity of Sun and Earth. This can be described by a 3-degree of freedom Hamiltonian system. We fix an energy level close to that of the collinear libration point L1, located between Sun and Earth. Near L1 there exists a normally hyperbolic invariant manifold, diffeomorphic to a 3-sphere. For an orbit confined to this 3-sphere, the amplitude of the motion relative to the ecliptic (the plane of the orbits of Sun and Earth) can vary only slightly. We show that we can obtain new orbits whose amplitude of motion relative to the ecliptic changes significantly, by following orbits of the flow restricted to the 3-sphere alternatively with homoclinic orbits that turn around the Earth. We provide an abstract theorem for the existence of such 'diffusing' orbits, and numerical evidence that the premises of the theorem are satisfied in the three-body problem considered here. We provide an explicit construction of diffusing orbits. The geometric mechanism underlying this construction is reminiscent of the Arnold diffusion problem for Hamiltonian systems. Our argument, however, does not involve transition chains of tori as in the classical example of Arnold. We exploit mostly the 'outer dynamics' along homoclinic orbits, and use very little information on the 'inner dynamics' restricted to the 3-sphere. As a possible application to astrodynamics, diffusing orbits as above can be used to design low cost maneuvers to change the inclination of an orbit of a satellite near L1 from a nearly-planar orbit to a tilted orbit with respect to the ecliptic. We explore different energy levels, and estimate the largest orbital inclination that can be achieved through our construction.
Huang, Tsung-Ming; Lin, Wen-Wei; Wang, Weichung
2016-10-01
We study how to efficiently solve the eigenvalue problems in computing band structure of three-dimensional dispersive metallic photonic crystals with face-centered cubic lattices based on the lossless Drude model. The discretized Maxwell equations result in large-scale standard eigenvalue problems whose spectrum contains many zero and cluster eigenvalues, both prevent existed eigenvalue solver from being efficient. To tackle this computational difficulties, we propose a hybrid Jacobi-Davidson method (hHybrid) that integrates harmonic Rayleigh-Ritz extraction, a new and hybrid way to compute the correction vectors, and a FFT-based preconditioner. Intensive numerical experiments show that the hHybrid outperforms existed eigenvalue solvers in terms of timing and convergence behaviors.
Chushnyakova, M. V.; Gontchar, I. I.
2013-01-01
We attempt to make some progress in the problem of the apparently large diffuseness of the Woods-Saxon strong nucleus-nucleus interaction potential (SnnP) needed to fit a large number of precision fusion excitation functions. This problem has been formulated in Newton [Phys. Lett. B10.1016/j.physletb.2004.02.052 586, 219 (2004);Phys. Rev. C10.1103/PhysRevC.70.024605 70, 024605 (2004)]. We applied the classical dissipative trajectory model to describe the data on fusion (capture) of 16O with 92Zr, 144Sm, and 208Pb. No fluctuations or dynamical deformations of the interacting nuclei are accounted for. The friction force is supposed to be proportional to the squared derivative of the SnnP (the surface friction model). The SnnP is calculated within the framework of the double-folding model with the density-dependent M3Y NN forces. This potential is known to possess rather small diffuseness in contradistinction to what is required by the data analysis in Newton [Phys. Lett. B10.1016/j.physletb.2004.02.052 586, 219 (2004);Phys. Rev. C10.1103/PhysRevC.70.024605 70, 024605 (2004)]. Varying slightly the strength of radial friction (universally for all three reactions) and the diffuseness of the charge density of 208Pb we have obtained satisfactory agreement of the calculated excitation functions with the data.
Maximal imaginery eigenvalues in optimal systems
Directory of Open Access Journals (Sweden)
David Di Ruscio
1991-07-01
Full Text Available In this note we present equations that uniquely determine the maximum possible imaginary value of the closed loop eigenvalues in an LQ-optimal system, irrespective of how the state weight matrix is chosen, provided a real symmetric solution of the algebraic Riccati equation exists. In addition, the corresponding state weight matrix and the solution to the algebraic Riccati equation are derived for a class of linear systems. A fundamental lemma for the existence of a real symmetric solution to the algebraic Riccati equation is derived for this class of linear systems.
ON GRAPHS WITH THREE DISTINCT LAPLACIAN EIGENVALUES
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
In this paper, an equivalent condition of a graph G with t (2 ≤ t ≤ n) distinct Laplacian eigenvalues is established. By applying this condition to t = 3, if G is regular (necessarily be strongly regular), an equivalent condition of G being Laplacian integral is given. Also for the case of t = 3, if G is non-regular, it is found that G has diameter 2 and girth at most 5 if G is not a tree. Graph G is characterized in the case of its being triangle-free, bipartite and pentagon-free. In both cases, G is Laplacian integral.
Random matrices, Frobenius eigenvalues, and monodromy
Katz, Nicholas M
1998-01-01
The main topic of this book is the deep relation between the spacings between zeros of zeta and L-functions and spacings between eigenvalues of random elements of large compact classical groups. This relation, the Montgomery-Odlyzko law, is shown to hold for wide classes of zeta and L-functions over finite fields. The book draws on and gives accessible accounts of many disparate areas of mathematics, from algebraic geometry, moduli spaces, monodromy, equidistribution, and the Weil conjectures, to probability theory on the compact classical groups in the limit as their dimension goes to infinit
Remarks on extreme eigenvalues of Toeplitz matrices
Directory of Open Access Journals (Sweden)
Mohsen Pourahmadi
1988-01-01
Full Text Available Let f be a nonnegative integrable function on [−π,π], Tn(f the (n+1×(n+1 Toeplitz matrix associated with f and λ1,n its smallest eigenvalue. It is shown that the convergence of λ1,n to minf(0 can be exponentially fast even when f does not satisfy the smoothness condition of Kac, Murdoch and Szegö (1953. Also a lower bound for λ1,n corresponding to a large class of functions which do not satisfy this smoothness condition is provided.
Shetty, Anil N; Chiang, Sharon; Maletic-Savatic, Mirjana; Kasprian, Gregor; Vannucci, Marina; Lee, Wesley
2014-01-01
In this article, we discuss the theoretical background for diffusion weighted imaging and diffusion tensor imaging. Molecular diffusion is a random process involving thermal Brownian motion. In biological tissues, the underlying microstructures restrict the diffusion of water molecules, making diffusion directionally dependent. Water diffusion in tissue is mathematically characterized by the diffusion tensor, the elements of which contain information about the magnitude and direction of diffusion and is a function of the coordinate system. Thus, it is possible to generate contrast in tissue based primarily on diffusion effects. Expressing diffusion in terms of the measured diffusion coefficient (eigenvalue) in any one direction can lead to errors. Nowhere is this more evident than in white matter, due to the preferential orientation of myelin fibers. The directional dependency is removed by diagonalization of the diffusion tensor, which then yields a set of three eigenvalues and eigenvectors, representing the magnitude and direction of the three orthogonal axes of the diffusion ellipsoid, respectively. For example, the eigenvalue corresponding to the eigenvector along the long axis of the fiber corresponds qualitatively to diffusion with least restriction. Determination of the principal values of the diffusion tensor and various anisotropic indices provides structural information. We review the use of diffusion measurements using the modified Stejskal-Tanner diffusion equation. The anisotropy is analyzed by decomposing the diffusion tensor based on symmetrical properties describing the geometry of diffusion tensor. We further describe diffusion tensor properties in visualizing fiber tract organization of the human brain.
SHETTY, ANIL N.; CHIANG, SHARON; MALETIC-SAVATIC, MIRJANA; KASPRIAN, GREGOR; VANNUCCI, MARINA; LEE, WESLEY
2016-01-01
In this article, we discuss the theoretical background for diffusion weighted imaging and diffusion tensor imaging. Molecular diffusion is a random process involving thermal Brownian motion. In biological tissues, the underlying microstructures restrict the diffusion of water molecules, making diffusion directionally dependent. Water diffusion in tissue is mathematically characterized by the diffusion tensor, the elements of which contain information about the magnitude and direction of diffusion and is a function of the coordinate system. Thus, it is possible to generate contrast in tissue based primarily on diffusion effects. Expressing diffusion in terms of the measured diffusion coefficient (eigenvalue) in any one direction can lead to errors. Nowhere is this more evident than in white matter, due to the preferential orientation of myelin fibers. The directional dependency is removed by diagonalization of the diffusion tensor, which then yields a set of three eigenvalues and eigenvectors, representing the magnitude and direction of the three orthogonal axes of the diffusion ellipsoid, respectively. For example, the eigenvalue corresponding to the eigenvector along the long axis of the fiber corresponds qualitatively to diffusion with least restriction. Determination of the principal values of the diffusion tensor and various anisotropic indices provides structural information. We review the use of diffusion measurements using the modified Stejskal–Tanner diffusion equation. The anisotropy is analyzed by decomposing the diffusion tensor based on symmetrical properties describing the geometry of diffusion tensor. We further describe diffusion tensor properties in visualizing fiber tract organization of the human brain. PMID:27441031
Ivanisenko, P V
2012-01-01
The Kramers problem for quantum fermi-gases with specular - diffuse boundary conditions of the kinetic theory is considered. On an example of Kramers problem the new generalised method of a source of the decision of the boundary problems from the kinetic theory is developed. The method allows to receive the decision with any degree of accuracy. At the basis of a method lays the idea of representation of a boundary condition on distribution function in the form of a source in the kinetic equation. By means of integrals Fourier the kinetic equation with a source is reduced to the integral equation of Fredholm type of the second kind. The decision is received in the form of Neumann's series.
Bedrikova, E A
2012-01-01
The Kramers problem for quantum Bose-gases with specular-diffuse boundary conditions of the kinetic theory is considered. On an example of Kramers' problem the new generalized method of a source of the decision of the boundary problems from the kinetic theory is developed. The method allows to receive the decision with any degree of accuracy. At the basis of a method lays the idea of representation of a boundary condition on distribution function in the form of a source in the kinetic equation. By means of integrals Fourier the kinetic equation with a source is reduced to the integral equation of Fredholm type of the second kind. The decision is received in the form of Neumann's series.
Johnson, Philip; Johnsen, Eric
2016-11-01
The Discontinuous Galerkin (DG) numerical method, while well-suited for hyperbolic PDE systems such as the Euler equations, is not naturally competitive for convection-diffusion systems, such as the Navier-Stokes equations. Where the DG weak form of the Euler equations depends only on the field variables for calculation of numerical fluxes, the traditional form of the Navier-Stokes equations requires calculation of the gradients of field variables for flux calculations. It is this latter task for which the standard DG discretization is ill-suited, and several approaches have been proposed to treat the issue. The most popular strategy for handling diffusion is the "mixed" approach, where the solution gradient is constructed from the primal as an auxiliary. We designed a new mixed approach, called Gradient-Recovery DG; it uses the Recovery concept of Van Leer & Nomura with the mixed approach to produce a scheme with excellent stability, high accuracy, and unambiguous implementation when compared to typical mixed approach concepts. In addition to describing the scheme, we will perform analysis with comparison to other DG approaches for diffusion. Gas dynamics examples will be presented to demonstrate the scheme's capabilities.
Einstein Spacetimes with Constant Weyl Eigenvalues
Barnes, Alan
2014-01-01
Einstein spacetimes (that is vacuum spacetimes possibly with a non-zero cosmological constant {\\Lambda}) with constant non-zero Weyl eigenvalus are considered. For type Petrov II & D this assumption allows one to prove that the non-repeated eigenvalue necessarily has the value 2{\\Lambda}/3 and it turns out that the only possible spacetimes are some Kundt-waves considered by Lewandowski which are type II and a Robinson-Bertotti solution of type D. For Petrov type I the only solution turns out to be a homogeneous pure vacuum solution found long ago by Petrov using group theoretic methods. These results can be summarised by the statement that the only vacuum spacetimes with constant Weyl eigenvalues are either homogeneous or are Kundt space- times. This result is similar to that of Coley et al. who proved their result for general spacetimes under the assumption that all scalar invariants constructed from the curvature tensor and all its derivatives were constant. Some preliminary results are also presented f...
Numerical Optimization of Eigenvalues of Hermitian Matrix Functions
Mengi, Emre; Yıldırım, Emre Alper; Kılıç, Mustafa
2011-01-01
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. SIAM J. MATRIX ANAL. APPL. c 2014 Society for Industrial and Applied Mathematics Vol. 35, No. 2, pp. 699–724 NUMERICAL OPTIMIZATION OF EIGENVALUES OF HERMITIAN MATRIX FUNCTIONS∗ EMRE MENGI†, E. ALPER YILDIRIM ‡ , AND MUSTAFA KILIC¸ † Abstract. This work concerns the global minimization of a prescribed eigenvalue or a weighted sum of prescribed eigenvalues of a Hermitian matrix-valued func...
Strong Linear Correlation Between Eigenvalues and Diagonal Matrix Elements
Shen, J J; Zhao, Y M; Yoshinaga, N
2008-01-01
We investigate eigenvalues of many-body systems interacting by two-body forces as well as those of random matrices. We find a strong linear correlation between eigenvalues and diagonal matrix elements if both of them are sorted from the smaller values to larger ones. By using this linear correlation we are able to predict reasonably all eigenvalues of given shell model Hamiltonian without complicated iterations.
Derivatives of eigenvalues and eigenvectors for a general matrix
Rudisill, C. S.
1974-01-01
Expressions are obtained for the derivatives of the eigenvalues and eigenvectors which are expressions of only one left-hand and one right-hand eigenvector. The approach described makes use of a Choleski decomposition or some other decomposition method. The method may be extended to find any order of derivative of the eigenvalue and eigenvector. The expressions obtained for finding the derivatives of eigenvalues and eigenvectors for nonself-adjoint systems may be applied to self-adjoint systems.
The Moment Convergence Rates for Largest Eigenvalues of β Ensembles
Institute of Scientific and Technical Information of China (English)
Jun Shan XIE
2013-01-01
The paper focuses on the largest eigenvalues of the β-Hermite ensemble and theβ-Laguerre ensemble.In particular,we obtain the precise moment convergence rates of their largest eigenvalues.The results are motivated by the complete convergence for partial sums of i.i.d.random variables,and the proofs depend on the small deviations for largest eigenvalues of the β ensembles and tail inequalities of the general β Tracy-Widom law.
Arbogast, Todd
2010-05-01
Tracer transport is governed by a convection-diffusion problem modeling mass conservation of both tracer and ambient fluids. Numerical methods should be fully conservative, enforcing both conservation principles on the discrete level. Locally conservative characteristics methods conserve the mass of tracer, but may not conserve the mass of the ambient fluid. In a recent paper by the authors [T. Arbogast, C. Huang, A fully mass and volume conserving implementation of a characteristic method for transport problems, SIAM J. Sci. Comput. 28 (2006) 2001-2022], a fully conservative characteristic method, the Volume Corrected Characteristics Mixed Method (VCCMM), was introduced for potential flows. Here we extend and apply the method to problems with a solenoidal (i.e., divergence-free) flow field. The modification is a computationally inexpensive simplification of the original VCCMM, requiring a simple adjustment of trace-back regions in an element-by-element traversal of the domain. Our numerical results show that the method works well in practice, is less numerically diffuse than uncorrected characteristic methods, and can use up to at least about eight times the CFL limited time step. © 2010 Elsevier Inc.
Sums of Laplace eigenvalues - rotationally symmetric maximizers in the plane
Laugesen, R S
2010-01-01
The sum of the first $n \\geq 1$ eigenvalues of the Laplacian is shown to be maximal among triangles for the equilateral triangle, maximal among parallelograms for the square, and maximal among ellipses for the disk, provided the ratio $\\text{(area)}^3/\\text{(moment of inertia)}$ for the domain is fixed. This result holds for both Dirichlet and Neumann eigenvalues, and similar conclusions are derived for Robin boundary conditions and Schr\\"odinger eigenvalues of potentials that grow at infinity. A key ingredient in the method is the tight frame property of the roots of unity. For general convex plane domains, the disk is conjectured to maximize sums of Neumann eigenvalues.
EIGENVALUE FUNCTIONS IN EXCITATORY-INHIBITORY NEURONAL NETWORKS
Institute of Scientific and Technical Information of China (English)
Zhang Linghai
2004-01-01
We study the exponential stability of traveling wave solutions of nonlinear systems of integral differential equations arising from nonlinear, nonlocal, synaptically coupled, excitatory-inhibitory neuronal networks. We have proved that exponential stability of traveling waves is equivalent to linear stability. Moreover, if the real parts of nonzero spectrum of an associated linear differential operator have a uniform negative upper bound, namely, max{Reλ: λ∈σ(L), λ≠ 0} ≤ -D, for some positive constant D, and λ = 0 is an algebraically simple eigenvalue of , then the linear stability follows, where is the linear differential operator obtained by linearizing the nonlinear system about its traveling wave and σ(L) denotes the spectrum of . The main aim of this paper is to construct complex analytic functions (also called eigenvalue or Evans functions) for exploring eigenvalues of linear differential operators to study the exponential stability of traveling waves. The zeros of the eigenvalue functions coincide with the eigenvalues of(L) .When studying multipulse solutions, some components of the traveling waves cross their thresholds for many times. These crossings cause great difficulty in the construction of the eigenvalue functions. In particular, we have to solve an over-determined system to construct the eigenvalue functions. By investigating asymptotic behaviors as z → -co of candidates for eigenfunctions, we find a way to construct the eigenvalue functions.By analyzing the zeros of the eigenvalue functions, we can establish the exponential stability of traveling waves arising from neuronal networks.
Bai, Lihua
2012-01-01
In Bai and Paulsen (SIAM J. Control optim. 48, 2010) the optimal dividend problem under transaction costs was analyzed for a rather general class of diffusion processes. It was divided into several subclasses, and for the majority of subclasses the optimal policy is a simple barrier policy; whenever the process hits an upper barrier $\\bar{u}^*$, reduce it to $\\bar{u}^*-\\xi$ through a dividend payment. After transaction costs, the shareholder receives $k\\xi-K$. It was proved that a simple barrier strategy is not always optimal, and here these more difficult cases are solved. The optimal solutions are rather complicated, but interesting.
Laser Induced Heat Diffusion Limited Tissue Coagulation Problem and General Properties
Lubashevsky, I A; Priezzhev, A V
2001-01-01
Previously we have developed a free boundary model for local thermal coagulation induced by laser light absorption when the tissue region affected directly by laser light is sufficiently small and heat diffusion into the surrounding tissue governs the necrosis growth. In the present paper surveying the obtained results we state the point of view on the necrosis formation under these conditions as the basis of an individual laser therapy mode exhibiting specific properties. In particular, roughly speaking, the size of the resulting necrosis domain is determined by the physical characteristics of the tissue and its response to local heating, and by the applicator form rather than the treatment duration and the irradiation power.
ON SOME MODEL DIFFUSION PROBLEMS WITH A NONLOCAL LOWER ORDER TERM
Institute of Scientific and Technical Information of China (English)
无
2003-01-01
The authors consider a class of nonlinear parabolic problems where the lower order term isdepending on a weighted integral of the solution, and address the issues of existence, uniqueness,stationary solutions and in some cases asymptotic behaviour.
Bloemen, S; Aerts, C; Dupret, M A; Østensen, R H; Degroote, P; Müller-Ringat, E; Rauch, T
2014-01-01
We have computed a new grid of evolutionary subdwarf B star (sdB) models from the start of central He burning, taking into account atomic diffusion due to radiative levitation, gravitational settling, concentration diffusion, and thermal diffusion. We have computed the non-adiabatic pulsation properties of the models and present the predicted p-mode and g-mode instability strips. In previous studies of the sdB instability strips, artificial abundance enhancements of Fe and Ni were introduced in the pulsation driving layers. In our models, the abundance enhancements of Fe and Ni occur naturally, eradicating the need to use artificial enhancements. We find that the abundance increases of Fe and Ni were previously underestimated and show that the instability strip predicted by our simulations solves the so-called blue edge problem of the subdwarf B star g-mode instability strip. The hottest known g-mode pulsator, KIC 10139564, now resides well within the instability strip {even when only modes with low spherical...
Methods for eigenvalue problems with applications in model order reduction
Rommes, J.
2007-01-01
Physical structures and processes are modeled by dynamical systems in a wide range of application areas. The increasing demand for complex components and large structures, together with an increasing demand for detail and accuracy, makes the models larger and more complicated. To be able to simulate
On the nonnegative inverse eigenvalue problem of traditional matrices
Directory of Open Access Journals (Sweden)
Alimohammad Nazari
2014-07-01
Full Text Available In this paper, at first for a given set of real or complex numbers $\\sigma$ with nonnegativesummation, we introduce some special conditions that with them there is no nonnegativetridiagonal matrix in which $\\sigma$ is its spectrum. In continue we present some conditions forexistence such nonnegative tridiagonal matrices.
From homogeneous eigenvalue problems to two-sex population dynamics.
Thieme, Horst R
2017-03-08
Enclosure theorems are derived for homogeneous bounded order-preserving operators and illustrated for operators involving pair-formation functions introduced by Karl-Peter Hadeler in the late 1980s. They are applied to a basic discrete-time two-sex population model and to the relation between the basic turnover number and the basic reproduction number.
Sensitivity of Eigenvalues to Nonsymmetrical, Dissipative Control Matrices
Directory of Open Access Journals (Sweden)
Vernon H. Neubert
1993-01-01
Full Text Available Dissipation of energy in vibrating structures can be accomplished with a combination of passive damping and active, constant gain, closed loop control forces. The matrix equations are Mz¨+Cz˙+Kz=−Gz˙. With conventional viscous damping, the damping force is proportional to relative velocity, with Fi=Ciiz˙i−Cijz˙j, where Cii=Cij but the subscripts show the position of the number in the C matrix. For a dashpot connected directly to ground, Fi=Ciiz˙i. Thus there is a definite pattern to the positions of numbers in the ith and jth rows of the C matrix, that is, a positive number on the diagonal is paired with an equal negative number, or zero, off the diagonal. With the control matrix G, it is here assumed that the positioning of individual controllers and sensors is flexible, with Fi=Ciiz˙i or Fi=Cijz˙j, the latter meaning that the control force at i is proportional to the velocity sensed at j. Thus the problem addressed herein is how the individual elements in the G matrix affect the modal eigenvalues. Two methods are discussed for finding the sensitivities, the classical method based on the products of eigenvectors and a new method, derived during the present study, involving the derivatives of the invariants in the similarity transformation. Examples are presented for the sensitivities of the complex eigenvalues of the form λr=−ζrwr+iwDr to individual elements in the G matrix, to combinations of elements, and to a combination of passive damping and active control. Systems with two, three, and eight degrees of freedom are investigated.
Eigenvalues properties of terms correspondences matrix
Bondarchuk, Dmitry; Timofeeva, Galina
2016-12-01
Vector model representations of text documents are widely used in the intelligent search. In this approach a collection of documents is represented in the form of the term-document matrix, reflecting the frequency of terms. In the latent semantic analysis the dimension of the vector space is reduced by the singular value decomposition of the term-document matrix. Authors use a matrix of terms correspondences, reflecting the relationship between the terms, to allocate a semantic core and to obtain more simple presentation of the documents. With this approach, reducing the number of terms is based on the orthogonal decomposition of the matrix of terms correspondences. Properties of singular values of the term-document matrix and eigenvalues of the matrix of terms correspondences are studied in the case when documents differ substantially in length.
Codazzi Tensors with Two Eigenvalue Functions
Merton, Gabe
2011-01-01
This paper addresses a gap in the classifcation of Codazzi tensors with exactly two eigenfunctions on a Riemannian manifold of dimension three or higher. Derdzinski proved that if the trace of such a tensor is constant and the dimension of one of the the eigenspaces is $n-1$, then the metric is a warped product where the base is an open interval- a conclusion we will show to be true under a milder trace condition. Furthermore, we construct examples of Codazzi tensors having two eigenvalue functions, one of which has eigenspace dimension $n-1$, where the metric is not a warped product with interval base, refuting a remark in \\cite{Besse} that the warped product conclusion holds without any restriction on the trace.
Directory of Open Access Journals (Sweden)
De-Lei Sheng
2014-01-01
Full Text Available This paper investigates the excess-of-loss reinsurance and investment problem for a compound Poisson jump-diffusion risk process, with the risk asset price modeled by a constant elasticity of variance (CEV model. It aims at obtaining the explicit optimal control strategy and the optimal value function. Applying stochastic control technique of jump diffusion, a Hamilton-Jacobi-Bellman (HJB equation is established. Moreover, we show that a closed-form solution for the HJB equation can be found by maximizing the insurer’s exponential utility of terminal wealth with the independence of two Brownian motions W(t and W1(t. A verification theorem is also proved to verify that the solution of HJB equation is indeed a solution of this optimal control problem. Then, we quantitatively analyze the effect of different parameter impacts on optimal control strategy and the optimal value function, which show that optimal control strategy is decreasing with the initial wealth x and decreasing with the volatility rate of risk asset price. However, the optimal value function V(t;x;s is increasing with the appreciation rate μ of risk asset.
A RBF Based Local Gridfree Scheme for Unsteady Convection-Diffusion Problems
Directory of Open Access Journals (Sweden)
Sanyasiraju VSS Yedida
2009-12-01
Full Text Available In this work a Radial Basis Function (RBF based local gridfree scheme has been presented for unsteady convection diffusion equations. Numerical studies have been made using multiquadric (MQ radial function. Euler and a three stage Runge-Kutta schemes have been used for temporal discretization. The developed scheme is compared with the corresponding finite difference (FD counterpart and found that the solutions obtained using the former are more superior. As expected, for a fixed time step and for large nodal densities, thought the Runge-Kutta scheme is able to maintain higher order of accuracy over the Euler method, the temporal discretization is independent of the improvement in the solution which in the developed scheme has been achived by optimizing the shape parameter of the RBF.
Energy Technology Data Exchange (ETDEWEB)
Carrozini, B; Cascarano, G; De Caro, L; Giacovazzo, C; Marchesini, S; Chapman, H N; Howells, M R; He, H; Wu, J S; Weiestrall, U; Spence, J H
2004-03-18
A new phasing algorithm has been used to determine the phases of diffuse elastic X-ray scattering from a non-periodic array of gold balls of 50 nm diameter. Two-dimensional real-space images , showing the charge-density distribution of the balls, have been reconstructed at 50 nm resolution from transmission diffraction patterns recorded at 550 eV energy. The reconstructed image fits well with scanning electron microscope (SEM) image of the same sample. The algorithm, which uses only the density modification portion of the SIR2002 program, is compared with the results obtained via the Gerchberg-Saxton-Fienup HiO algorithm. The new algorithm requires no knowledge of the object's boundary, and proceeds from low to high resolution. In this way the relationship between density modification in crystallography and the HiO algorithm used in signal and image processing is elucidated.
Partition function for the eigenvalues of the Wilson line
Gocksch, A
1993-01-01
In a gauge theory at nonzero temperature the eigenvalues of the Wilson line form a set of gauge invariant observables. By constructing the corresponding partition function for the phases of these eigenvalues, we prove that the trivial vacuum, where the phases vanish, is a minimum of the free energy.
Matrix eigenvalue model: Feynman graph technique for all genera
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Chekhov, Leonid [Steklov Mathematical Institute, ITEP and Laboratoire Poncelet, Moscow (Russian Federation); Eynard, Bertrand [SPhT, CEA, Saclay (France)
2006-12-15
We present the diagrammatic technique for calculating the free energy of the matrix eigenvalue model (the model with arbitrary power {beta} by the Vandermonde determinant) to all orders of 1/N expansion in the case where the limiting eigenvalue distribution spans arbitrary (but fixed) number of disjoint intervals (curves)
STABILIZED FEM FOR CONVECTION-DIFFUSION PROBLEMS ON LAYER-ADAPTED MESHES
Institute of Scientific and Technical Information of China (English)
Hans-G(o)rg Roos
2009-01-01
The application of a standard Galerkin finite element method for convection-diflusion problems leads to oscillations in the discrete solution,therefore stabilization seems to be necessary.We discuss several recent stabilization methods,especially its combination with a Galerkin method on layer-adapted meshes.Supercloseness results obtained allow an improvement of the discrete solution using recovery techniques.
Lanser, D.; Verwer, J.G.
1998-01-01
Operator or time splitting is often used in the numerical solution of initial boundary value problems for differential equations. It is, for example, standard practice in computational air pollution modelling where we encounter systems of three-dimensional, time-dependent partial differential equati
Shape optimization for low Neumann and Steklov eigenvalues
Girouard, Alexandre
2008-01-01
We give an overview of results on shape optimization for low eigenvalues of the Laplacian on bounded planar domains with Neumann and Steklov boundary conditions. These results share a common feature: they are proved using methods of complex analysis. In particular, we present modernized proofs of the classical inequalities due to Szego and Weinstock for the first nonzero Neumann and Steklov eigenvalues. We also extend the inequality for the second nonzero Neumann eigenvalue, obtained recently by Nadirashvili and the authors, to non-homogeneous membranes with log-subharmonic densities. In the homogeneous case, we show that this inequality is strict, which implies that the maximum of the second nonzero Neumann eigenvalue is not attained in the class of simply-connected membranes of a given mass. The same is true for the second nonzero Steklov eigenvalue, as follows from our results on the Hersch-Payne-Schiffer inequalities.
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Cordier, G [Ecole Catholique d' Arts et Metiers, Lyon (France); Choi, J; Raguin, L G [Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824-1226 (United States)], E-mail: guy.raguin@mines-nancy.org
2008-11-01
Skin microcirculation plays an important role in diseases such as chronic venous insufficiency and diabetes. Magnetic resonance imaging (MRI) can provide quantitative information with a better penetration depth than other noninvasive methods, such as laser Doppler flowmetry or optical coherence tomography. Moreover, successful MRI skin studies have recently been reported. In this article, we investigate three potential inverse models to quantify skin microcirculation using diffusion-weighted MRI (DWI), also known as q-space MRI. The model parameters are estimated based on nonlinear least-squares (NLS). For each of the three models, an optimal DWI sampling scheme is proposed based on D-optimality in order to minimize the size of the confidence region of the NLS estimates and thus the effect of the experimental noise inherent to DWI. The resulting covariance matrices of the NLS estimates are predicted by asymptotic normality and compared to the ones computed by Monte-Carlo simulations. Our numerical results demonstrate the effectiveness of the proposed models and corresponding DWI sampling schemes as compared to conventional approaches.
Mena, Andres; Ferrero, Jose M.; Rodriguez Matas, Jose F.
2015-11-01
Solving the electric activity of the heart possess a big challenge, not only because of the structural complexities inherent to the heart tissue, but also because of the complex electric behaviour of the cardiac cells. The multi-scale nature of the electrophysiology problem makes difficult its numerical solution, requiring temporal and spatial resolutions of 0.1 ms and 0.2 mm respectively for accurate simulations, leading to models with millions degrees of freedom that need to be solved for thousand time steps. Solution of this problem requires the use of algorithms with higher level of parallelism in multi-core platforms. In this regard the newer programmable graphic processing units (GPU) has become a valid alternative due to their tremendous computational horsepower. This paper presents results obtained with a novel electrophysiology simulation software entirely developed in Compute Unified Device Architecture (CUDA). The software implements fully explicit and semi-implicit solvers for the monodomain model, using operator splitting. Performance is compared with classical multi-core MPI based solvers operating on dedicated high-performance computer clusters. Results obtained with the GPU based solver show enormous potential for this technology with accelerations over 50 × for three-dimensional problems.
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Abbas, Ibrahim A., E-mail: aabbas5@kau.edu.sa [Department of Mathematics, Faculty of Science and Arts-Khulais, King Abdulaziz University, Jeddah (Saudi Arabia); Department of mathematics, Faculty of Science, Sohag University, Sohag (Egypt)
2015-03-01
In the present work, we consider the problem of fractional order thermoelastic interaction in a material placed in a magnetic field and subjected to a moving plane of heat source. The basic equations have been written in the form of a vector–matrix differential equation in the Laplace transform domain, which is then solved by an eigenvalue approach. The inverse Laplace transforms are computed numerically and some comparisons have been shown in figures to estimate the effect of each of the fractional order, heat source velocity, time and the magnetic field and parameters. - Highlights: • The problem of fractional order thermoelastic interaction in a material placed in a magnetic field and subjected to a moving plane of heat source. • The eigenvalue approach gives exact solution in the Laplace domain without any assumed restrictions on the actual physical quantities. • Numerical results for the temperature, displacement and the stress distributions are represented graphically.
Numerical computation of the linear stability of the diffusion model for crystal growth simulation
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Yang, C.; Sorensen, D.C. [Rice Univ., Houston, TX (United States); Meiron, D.I.; Wedeman, B. [California Institute of Technology, Pasadena, CA (United States)
1996-12-31
We consider a computational scheme for determining the linear stability of a diffusion model arising from the simulation of crystal growth. The process of a needle crystal solidifying into some undercooled liquid can be described by the dual diffusion equations with appropriate initial and boundary conditions. Here U{sub t} and U{sub a} denote the temperature of the liquid and solid respectively, and {alpha} represents the thermal diffusivity. At the solid-liquid interface, the motion of the interface denoted by r and the temperature field are related by the conservation relation where n is the unit outward pointing normal to the interface. A basic stationary solution to this free boundary problem can be obtained by writing the equations of motion in a moving frame and transforming the problem to parabolic coordinates. This is known as the Ivantsov parabola solution. Linear stability theory applied to this stationary solution gives rise to an eigenvalue problem of the form.
Optimal control problem for a sixth-order Cahn-Hilliard equation with nonlinear diffusion
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Changchun Liu
2012-08-01
Full Text Available In this article, we study the initial-boundary-value problem for a sixth-order Cahn-Hilliard type equation $$displaylines{ u_t=D^2mu, cr mu=gamma D^4u-a(uD^2u-frac{a'(u}2|D u|^2+f(u+ku_t, }$$ which describes the separation properties of oil-water mixtures, when a substance enforcing the mixing of the phases is added. The optimal control of the sixth order Cahn-Hilliard type equation under boundary condition is given and the existence of optimal solution to the sixth order Cahn-Hilliard type equation is proved.
Minimization of the k-th eigenvalue of the Dirichlet Laplacian
Bucur, Dorin
2012-12-01
For every {k in {N}}, we prove the existence of a quasi-open set minimizing the k-th eigenvalue of the Dirichlet Laplacian among all sets of prescribed Lebesgue measure. Moreover, we prove that every minimizer is bounded and has a finite perimeter. The key point is the observation that such quasi-open sets are shape subsolutions for an energy minimizing free boundary problem.
1986-01-01
1966). 3. Canale, R.P. and S.C. Chapra . Numerical Methods for Engineers with Personnel Computer Applications. New York: McGraw-Hill 509-533, ( 1985...This study looks at numerical % methods from an engineer’s view, a tool to be used in solving problems. This paper has given me much needed experience... numerical method in solving the transient heat conduction equation. The eigenvalue method was compared to five other numerical methods : Runge-Kutta
Eigenvalue solution to the electron-collisional effect on ion-acoustic and entropy waves
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
The linearized electron Fokker-Planck and cold-ion fluid equations are solved as an eigenvalue problem in the quasineutral limit for ionization state,Z=1,8,and 64 for ion-acoustic and entropy waves.The perturbed electron distribution function is written as a moment expansion of eigenvectors,and is used to compute collisionality-dependence macroscopic quantities in the plasma such as the generalized specific heat ratio,and the electron thermal conductivity.
Eigenvalues and eigenvectors: embodied, symbolic and formal thinking
Thomas, Michael O. J.; Stewart, Sepideh
2011-09-01
Many beginning university students struggle with the new approaches to mathematics that they find in their courses due to a shift in presentation of mathematical ideas, from a procedural approach to concept definitions and deductive derivations, and ideas building upon each other in quick succession. This paper highlights this struggle by considering some conceptual processes and difficulties students find in learning about eigenvalues and eigenvectors. We use the theoretical framework of Tall's three worlds of mathematical thinking, along with perspectives from Dubinsky's APOS (action, process, object, schema) theory and Thomas's representational versatility. The results of the study describe thinking about these concepts by several groups of first- and second-year university students. In particular the obstacles they faced, and the emerging links some were constructing between parts of their concept images formed from the embodied, symbolic, and formal worlds are presented. We also identify some fundamental problems with student understanding of the definition of eigenvectors that lead to implementation problems, and some of the concepts underlying such difficulties.
Numerical computations of interior transmission eigenvalues for scattering objects with cavities
Peters, Stefan; Kleefeld, Andreas
2016-04-01
In this article we extend the inside-outside duality for acoustic transmission eigenvalue problems by allowing scattering objects that may contain cavities. In this context we provide the functional analytical framework necessary to transfer the techniques that have been used in Kirsch and Lechleiter (2013 Inverse Problems, 29 104011) to derive the inside-outside duality. Additionally, extensive numerical results are presented to show that we are able to successfully detect interior transmission eigenvalues with the inside-outside duality approach for a variety of obstacles with and without cavities in three dimensions. In this context, we also discuss the advantages and disadvantages of the inside-outside duality approach from a numerical point of view. Furthermore we derive the integral equations necessary to extend the algorithm in Kleefeld (2013 Inverse Problems, 29 104012) to compute highly accurate interior transmission eigenvalues for scattering objects with cavities, which we will then use as reference values to examine the accuracy of the inside-outside duality algorithm.