The Experts below are selected from a list of 70383 Experts worldwide ranked by ideXlab platform
Volker Mehrmann - One of the best experts on this subject based on the ideXlab platform.
-
nlevp a collection of nonlinear Eigenvalue Problems
ACM Transactions on Mathematical Software, 2013Co-Authors: Timo Betcke, Volker Mehrmann, Nicholas J Higham, Christian Schroder, Francoise TisseurAbstract:We present a collection of 52 nonlinear Eigenvalue Problems in the form of a MATLAB toolbox. The collection contains Problems from models of real-life applications as well as ones constructed specifically to have particular properties. A classification is given of polynomial Eigenvalue Problems according to their structural properties. Identifiers based on these and other properties can be used to extract particular types of Problems from the collection. A brief description of each problem is given. NLEVP serves both to illustrate the tremendous variety of applications of nonlinear Eigenvalue Problems and to provide representative Problems for testing, tuning, and benchmarking of algorithms and codes.
-
NLEVP: A Collection of Nonlinear Eigenvalue Problems
ACM Transactions on Mathematical Software, 2013Co-Authors: Timo Betcke, Volker Mehrmann, Nicholas J Higham, Christian Schroder, Francoise TisseurAbstract:We present a collection of 52 nonlinear Eigenvalue Problems in the form of a MATLAB toolbox. The collection contains Problems from models of real-life applications as well as ones constructed specifically to have particular properties. A classification is given of polynomial Eigenvalue Problems according to their structural properties. Identifiers based on these and other properties can be used to extract particular types of Problems from the collection. A brief description of each problem is given. NLEVP serves both to illustrate the tremendous variety of applications of nonlinear Eigenvalue Problems and to provide representative Problems for testing, tuning, and benchmarking of algorithms and codes.
-
Adaptive solution of elliptic PDE-Eigenvalue Problems.
Pamm, 2009Co-Authors: Volker Mehrmann, Agnieszka MiedlarAbstract:In this paper we introduce a new adaptive algorithm (AFEMLA) for elliptic PDE-Eigenvalue Problems. In contrast to other approaches the algebraic Eigenvalue problem does not have to be solved to full accuracy. We incorporate the iterative solution of the resulting finite dimensional algebraic Eigenvalue Problems in the adaptation process in order to balance the cost with the costs for the iterative Eigenvalue method. We present error estimates that incorporate the discretization errors, approximation errors in the Eigenvalue solver and roundoff errors. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)
-
nonlinear Eigenvalue Problems a challenge for modern Eigenvalue methods
Gamm-mitteilungen, 2004Co-Authors: Volker Mehrmann, Heinrich VossAbstract:We discuss the state of the art in numerical solution methods for large scale polynomial or rational Eigenvalue Problems. We present the currently available solution methods such as the Jacobi-Davidson, Arnoldi or the rational Krylov method and analyze their properties. We briefly introduce a new linearization technique and demonstrate how it can be used to improve structure preservation and with this the accuracy and efficiency of linearization based methods. We present several recent applications where structured and unstructured nonlinear Eigenvalue Problems arise and some numerical results. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
-
ON THE SOLUTION OF PALINDROMIC Eigenvalue Problems
2004Co-Authors: A. Hilliges, Christian Mehl, Volker MehrmannAbstract:A rational Eigenvalue problem of the form 1 (A T + •A0 + • 2 A1)x = 0 arising in the vibration analysis of rail tracks under periodic excitation is investigated. This Eigenvalue problem is a special case of a more general class of Problems referred to as palindromic Eigenvalue Problems. The paper provides the theoretical background of these Eigenvalue Problems and proposes a Jacobi-like algorithm for their numerical solution.
Ren-cang Li - One of the best experts on this subject based on the ideXlab platform.
-
Recent Progress in Linear Response Eigenvalue Problems
Lecture Notes in Computational Science and Engineering, 2017Co-Authors: Ren-cang LiAbstract:Linear response Eigenvalue Problems arise from the calculation of excitation states of many-particle systems in computational materials science. In this paper, from the point of view of numerical linear algebra and matrix computations, we review the progress of linear response Eigenvalue Problems in theory and algorithms since 2012.
-
Structured backward error for palindromic polynomial Eigenvalue Problems
Numerische Mathematik, 2010Co-Authors: Ren-cang Li, Wen-wei Lin, Chern-shuh WangAbstract:A detailed structured backward error analysis for four kinds of palindromic polynomial Eigenvalue Problems (PPEP) $$ \left(\sum_{\ell=0}^d A_{\ell} \lambda^{\ell} \right)x=0, \quad A_{d-\ell}=\varepsilon A_{\ell}^{\star} \quad{\rm for}\,\ell=0,1,\ldots,\lfloor d/2\rfloor, $$ where $${\star}$$ is one of the two actions: transpose and conjugate transpose, and $${\varepsilon\in\{\pm 1\}}$$ . Each of them has its application background with the case $${\star}$$ taking transpose and ε = 1 attracting a great deal of attention lately because of its application in the fast train modeling. Computable formulas and bounds for the structured backward errors are obtained. The analysis reveals distinctive features of PPEP from general polynomial Eigenvalue Problems (PEP) investigated by Tisseur (Linear Algebra Appl 309:339–361, 2000) and by Liu and Wang (Appl Math Comput 165:405–417, 2005).
-
Structured backward error for palindromic polynomial Eigenvalue Problems
Numerische Mathematik, 2010Co-Authors: Ren-cang Li, Chern-shuh WangAbstract:A detailed structured backward error analysis for four kinds of palindromic polynomial Eigenvalue Problems (PPEP) $$ \left(\sum_{\ell=0}^d A_{\ell} \lambda^{\ell} \right)x=0, \quad A_{d-\ell}=\varepsilon A_{\ell}^{\star} \quad{\rm for}\,\ell=0,1,\ldots,\lfloor d/2\rfloor, $$ is one of the two actions: transpose and conjugate transpose, and $${\varepsilon\in\{\pm 1\}}$$. Each of them has its application background with the case $${\star}$$taking transpose and e = 1 attracting a great deal of attention lately because of its application in the fast train modeling. Computable formulas and bounds for the structured backward errors are obtained. The analysis reveals distinctive features of PPEP from general polynomial Eigenvalue Problems (PEP) investigated by Tisseur (Linear Algebra Appl 309:339–361, 2000) and by Liu and Wang (Appl Math Comput 165:405–417, 2005).
Chern-shuh Wang - One of the best experts on this subject based on the ideXlab platform.
-
Structured backward error for palindromic polynomial Eigenvalue Problems
Numerische Mathematik, 2010Co-Authors: Ren-cang Li, Wen-wei Lin, Chern-shuh WangAbstract:A detailed structured backward error analysis for four kinds of palindromic polynomial Eigenvalue Problems (PPEP) $$ \left(\sum_{\ell=0}^d A_{\ell} \lambda^{\ell} \right)x=0, \quad A_{d-\ell}=\varepsilon A_{\ell}^{\star} \quad{\rm for}\,\ell=0,1,\ldots,\lfloor d/2\rfloor, $$ where $${\star}$$ is one of the two actions: transpose and conjugate transpose, and $${\varepsilon\in\{\pm 1\}}$$ . Each of them has its application background with the case $${\star}$$ taking transpose and ε = 1 attracting a great deal of attention lately because of its application in the fast train modeling. Computable formulas and bounds for the structured backward errors are obtained. The analysis reveals distinctive features of PPEP from general polynomial Eigenvalue Problems (PEP) investigated by Tisseur (Linear Algebra Appl 309:339–361, 2000) and by Liu and Wang (Appl Math Comput 165:405–417, 2005).
-
Structured backward error for palindromic polynomial Eigenvalue Problems
Numerische Mathematik, 2010Co-Authors: Ren-cang Li, Chern-shuh WangAbstract:A detailed structured backward error analysis for four kinds of palindromic polynomial Eigenvalue Problems (PPEP) $$ \left(\sum_{\ell=0}^d A_{\ell} \lambda^{\ell} \right)x=0, \quad A_{d-\ell}=\varepsilon A_{\ell}^{\star} \quad{\rm for}\,\ell=0,1,\ldots,\lfloor d/2\rfloor, $$ is one of the two actions: transpose and conjugate transpose, and $${\varepsilon\in\{\pm 1\}}$$. Each of them has its application background with the case $${\star}$$taking transpose and e = 1 attracting a great deal of attention lately because of its application in the fast train modeling. Computable formulas and bounds for the structured backward errors are obtained. The analysis reveals distinctive features of PPEP from general polynomial Eigenvalue Problems (PEP) investigated by Tisseur (Linear Algebra Appl 309:339–361, 2000) and by Liu and Wang (Appl Math Comput 165:405–417, 2005).
Heinrich Voss - One of the best experts on this subject based on the ideXlab platform.
-
nonlinear Eigenvalue Problems a challenge for modern Eigenvalue methods
Gamm-mitteilungen, 2004Co-Authors: Volker Mehrmann, Heinrich VossAbstract:We discuss the state of the art in numerical solution methods for large scale polynomial or rational Eigenvalue Problems. We present the currently available solution methods such as the Jacobi-Davidson, Arnoldi or the rational Krylov method and analyze their properties. We briefly introduce a new linearization technique and demonstrate how it can be used to improve structure preservation and with this the accuracy and efficiency of linearization based methods. We present several recent applications where structured and unstructured nonlinear Eigenvalue Problems arise and some numerical results. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
Roel Van Beeumen - One of the best experts on this subject based on the ideXlab platform.
-
compact rational krylov methods for nonlinear Eigenvalue Problems
SIAM Journal on Matrix Analysis and Applications, 2015Co-Authors: Roel Van Beeumen, Karl Meerbergen, Wim MichielsAbstract:We propose a new uniform framework of compact rational Krylov (CORK) methods for solving large-scale nonlinear Eigenvalue Problems $A(\lambda) x = 0$. For many years, linearizations were used for solving polynomial and rational Eigenvalue Problems. On the other hand, for the general nonlinear case, $A(\lambda)$ can first be approximated by a (rational) matrix polynomial and then a convenient linearization is used. However, the major disadvantage of linearization-based methods is the growing memory and orthogonalization costs with the iteration count, i.e., in general they are proportional to the degree of the polynomial. Therefore, the CORK family of rational Krylov methods exploits the structure of the linearization pencils by using a generalization of the compact Arnoldi decomposition. In this way, the extra memory and orthogonalization costs due to the linearization of the original Eigenvalue problem are negligible for large-scale Problems. Furthermore, we prove that each CORK step breaks down into an ...
-
Rational Krylov Methods for Nonlinear Eigenvalue Problems
2013Co-Authors: Roel Van BeeumenAbstract:Eigenvalue Problems arise in all fields of science and engineering. The mathematical properties and numerical solution methods for standard, linear Eigenvalue Problems are well understood. However, recent advances in several application areas resulted in a new type of Eigenvalue problem, i.e., the nonlinear Eigenvalue problem which exhibits nonlinearity in the Eigenvalue parameter. The goal of this thesis is to develop new rational Krylov methods for solving both small-scale and large-scale nonlinear Eigenvalue Problems. Firstly, by using polynomial and rational interpolation of the matrix-valued functions, we obtain methods which are globally convergent inside the region of interest. Secondly, linearization of the corresponding polynomial and rational Eigenvalue Problems results in linear pencils. Thirdly, the exploitation of the special structure of the linearization pencils and possibly a low rank structure results in efficient and reliable software which is publicly available. We propose the Compact Rational Krylov (CORK) method as a generic class of numerical methods for solving nonlinear Eigenvalue Problems. CORK is characterized by a uniform and simple representation of structured linearization pencils. The structure of these linearization pencils is fully exploited and the subspace is represented in a compact form. Consequently, we are able to solve Problems of high dimension and high degree in an efficient and reliable way. The family of CORK methods has a lot of flexibility for solving the nonlinear Eigenvalue problem. We discuss three particular types of CORK methods. The first one is the Newton Rational Krylov method which makes use of dynamic polynomial interpolation. The second one is the Fully Rational Krylov method which uses rational interpolation and has three viable variants: a static, dynamic, and hybrid variant. The third one is the Infinite Arnoldi method which uses an operator setting to solve the nonlinear Eigenvalue problem. Finally, the proposed methods are used to solve applications from mechanical engineering, quantum physics, and civil engineering which were not solved earlier with the same efficiency and reliability.