Journal Article10.1002/NME.5441
Solving large‐scale nonlinear eigenvalue problems by rational interpolation and resolvent sampling based Rayleigh–Ritz method
23
TL;DR: A rational interpolation approach (RIA) is proposed based on the Keldysh theorem for holomorphic matrix functions based on which a robust eigen-solver, called RSRR, is developed for solving general NEPs.
read more
Abstract: Summary
Numerical solution of nonlinear eigenvalue problems (NEPs) is frequently encountered in computational science and engineering. The applicability of most existing methods is limited by the matrix structures, properties of the eigen-solutions, sizes of the problems, etc. This paper aims to remove those limitations and develop robust and universal NEP solvers for large-scale engineering applications. The novelty lies in two aspects. First, a rational interpolation approach (RIA) is proposed based on the Keldysh theorem for holomorphic matrix functions. Comparing with the existing contour integral approach, the RIA provides the possibility to select sampling points in more general regions and has advantages in improving the accuracy and reducing the computational cost. Second, a resolvent sampling scheme using the RIA is proposed to construct reliable search spaces for the Rayleigh-Ritz procedure, based on which a robust eigen-solver, called RSRR, is developed for solving general NEPs. The RSRR can be easily implemented and parallelized. The advantages of the RIA and the performance of the RSRR are demonstrated by a variety of benchmark and application examples. This article is protected by copyright. All rights reserved.
read more
Chat with Paper
AI Agents for this Paper
Find similar papers on Google Scholar, PubMed and Arxiv
Write a critical review of this paper
Analyze citations of this paper to find unaddressed research gaps
Citations
The nonlinear eigenvalue problem
Stefan Güttel,Françoise Tisseur +1 more
TL;DR: This article surveys nonlinear eigenvalue problems associated with matrix-valued functions which depend nonlinearly on a single scalar parameter, with a particular emphasis on their mathematical properties and available numerical solution techniques.
The Boundary Element Method in Acoustics: A Survey
TL;DR: The boundary element method (BEM) was initially developed for Laplace's equation and has been applied in the context of acoustics or Helmholtz problems as mentioned in this paper, including half-space problems and modeling the acoustic field surrounding thin screens.
147
A robust Bézier based solution for nonlinear vibration and post-buckling of random checkerboard graphene nano-platelets reinforced composite beams
TL;DR: In this article, an accurate Bezier-based multi-step method is developed and implemented to find the nonlinear vibration and postbuckling configurations of Euler-Bernoulli composite beams reinforced with graphene nano-platelets.
127
Controlling inner iterations in the Jacobi-Davidson method
Michiel E. Hochstenbach,Yvan Notay +1 more
- 01 Jan 2007
TL;DR: A relation between the residualnorm of the inner linear system and the residual norm of the eigenvalue problem is proved and it is shown that the latter may be estimated inexpensively during the inner iterations.
31
Designing rational filter functions for solving eigenvalue problems by contour integration
Marc Van Barel
- 25 Mar 2015
TL;DR: In this article, 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.
26
References
The Quadratic Eigenvalue Problem
TL;DR: This work surveys the quadratic eigenvalue problem, treating its many applications, its mathematical properties, and a variety of numerical solution techniques.
Barycentric Lagrange Interpolation
TL;DR: Barycentric interpolation is a variant of Lagrange polynomial interpolation that is fast and stable and deserves to be known as the standard method of polynometric interpolation.
Density-Matrix-Based Algorithm for Solving Eigenvalue Problems
TL;DR: A new numerical algorithm for solving the symmetric eigenvalue problem is presented, which takes its inspiration from the contour integration and density matrix representation in quantum mechanics.
468
A projection method for generalized eigenvalue problems using numerical integration
Tetsuya Sakurai,Hiroshi Sugiura +1 more
TL;DR: In this article, a method for finding certain eigenvalues of a generalized eigenvalue problem that lie in a given domain of the complex plane is proposed, which projects the matrix pencil onto a subspace associated with the eigen values that are located in the domain via numerical integration.
402
NLEVP: A Collection of Nonlinear Eigenvalue Problems
TL;DR: 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.