Nonlinear eigenvalue problems and contour integrals

Barel, Marc Van; Kravanja, Peter

doi:10.1016/j.cam.2015.07.012

Cited by 39 publications

(54 citation statements)

References 24 publications

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“…In [23], we designed a variant of Beyn's algorithm [5] showing that this eigenvalue problem can be solved via contour integration of the resolvent function T (z) 1 . Consider the following contour integrals, called moments, based on the resolvent function applied to a rectangular matrixV :…”

Section: Algorithm Based On Approximate Contour Integrationmentioning

confidence: 99%

“…For a detailed overview of the history and current research on solving eigenvalue problems by contour integration, we refer the interested reader to the introduction section of [23]. Here, only some key references are mentioned without having the intention of being complete.…”

Section: Introductionmentioning

confidence: 99%

“…In [23], we presented an algorithm based on contour integration to solve the corresponding eigenvalue problem. We showed that the so-called filter function plays an important role in the e↵ectiveness of the contour integration approach.…”

Section: Introductionmentioning

confidence: 99%

“…We showed that the so-called filter function plays an important role in the e↵ectiveness of the contour integration approach. To compute the eigenvalues in the neighborhood of a branch cut, we used in [23] a filter function developed in [8] by Hale, Higham and Trefethen using detailed knowledge of complex analysis. Because this knowledge is not always readily available for someone who wants to solve a specific eigenvalue problem, we will design in this paper e↵ective filter functions using an optimization algorithm.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 2 we start by summarizing the algorithm as developed in [23]. Section 3 describes the properties an e↵ective filter function should have.…”

Section: Introductionmentioning

confidence: 99%

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Designing rational filter functions for solving eigenvalue problems by contour integration

Barel

2016

Linear Algebra and its Applications

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Solving (nonlinear) eigenvalue problems by contour integration, requires an e↵ective discretization for the corresponding contour integrals. In this paper it is shown that good rational filter functions can be computed using (nonlinear least squares) optimization techniques as opposed to designing those functions based on a thorough understanding of complex analysis. The conditions that such an effective filter function should satisfy, are derived and translated in a nonlinear least squares optimization problem solved by optimization algorithms from Tensorlab. Numerical experiments illustrate the validity of this approach.Keywords : linear, polynomial, nonlinear eigenvalue problems; contour integration; filter function; rational approximation; nonlinear least squares; resolvent. MSC : Primary : 65-F15, Secondary : 47-J10, 30-E05. Designing rational filter functions for solving eigenvalue problems by contour integrationMarc Van Barel ⇤ January 20, 2015Abstract Solving (nonlinear) eigenvalue problems by contour integration, requires an e↵ective discretization for the corresponding contour integrals. In this paper it is shown that good rational filter functions can be computed using (nonlinear least squares) optimization techniques as opposed to designing those functions based on a thorough understanding of complex analysis. The conditions that such an e↵ective filter function should satisfy, are derived and translated in a nonlinear least squares optimization problem solved by optimization algorithms from Tensorlab. Numerical experiments illustrate the validity of this approach.

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Section: Algorithm Based On Approximate Contour Integrationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…In Section 2 we start by summarizing the algorithm as developed in [23]. Section 3 describes the properties an e↵ective filter function should have.…”

Section: Introductionmentioning

confidence: 99%

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Designing rational filter functions for solving eigenvalue problems by contour integration

Barel

2016

Linear Algebra and its Applications

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Solving Maxwell's eigenvalue problem via isogeometric boundary elements and a contour integral method

Kurz

Schöps

Unger

et al. 2021

Math Methods in App Sciences

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Domain decomposition approaches for accelerating contour integration eigenvalue solvers for symmetric eigenvalue problems

Kalantzis

Kestyn

Polizzi

et al. 2018

Numerical Linear Algebra App

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Summary This paper discusses techniques for computing a few selected eigenvalue–eigenvector pairs of large and sparse symmetric matrices. A recently developed class of techniques to solve this type of problems is based on integrating the matrix resolvent operator along a complex contour that encloses the interval containing the eigenvalues of interest. This paper considers such contour integration techniques from a domain decomposition viewpoint and proposes two schemes. The first scheme can be seen as an extension of domain decomposition linear system solvers in the framework of contour integration methods for eigenvalue problems, such as FEAST. The second scheme focuses on integrating the resolvent operator primarily along the interface region defined by adjacent subdomains. A parallel implementation of the proposed schemes is described, and results on distributed computing environments are reported. These results show that domain decomposition approaches can lead to reduced run times and improved scalability.

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Nonlinear eigenvalue problems and contour integrals

Cited by 39 publications

References 24 publications

Designing rational filter functions for solving eigenvalue problems by contour integration

Designing rational filter functions for solving eigenvalue problems by contour integration

Solving Maxwell's eigenvalue problem via isogeometric boundary elements and a contour integral method

Domain decomposition approaches for accelerating contour integration eigenvalue solvers for symmetric eigenvalue problems

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