Open AccessPosted Content
Solving large-scale nonlinear eigenvalue problems by rational interpolation approach and resolvent sampling based Rayleigh-Ritz method
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, denoted by RSRR, is developed for solving general NEPs.
read more
Abstract: Numerical solution of nonlinear eigenvalue problems (NEPs) is frequently encountered in computational science and engineering. The applicability of most existing methods is limited by matrix structures, property of eigen-solutions, size of the problem, etc. This paper aims to break those limitations and to 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 (CIA), the RIA provides the possibility to select sampling points in more general regions and has advantages in improving accuracy and reducing computational cost. Second, a resolvent sampling scheme using the RIA is proposed for constructing reliable search spaces for the Rayleigh-Ritz procedure, based on which a robust eigen-solver, denoted by RSRR, is developed for solving general NEPs. RSRR can be easily implemented and parallelized. The advantages of the RIA and the performance of RSRR are demonstrated by a variety of benchmark and practical problems.
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
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
Algorithms based on analytic function values at roots of unity
Lloyd N. Trefethen,Anthony P. Austin,Peter Kravanja +2 more
- 01 Jan 2013
TL;DR: In this article, the problem of computing the eigenvalues in the disk of a matrix of large dimension is studied. And the power of rational in comparison with polynomial approximations for some of these problems is highlighted.
4
•Journal Article
Non-linear eigensolver-based alternative to traditional SCF methods
Brendan Gavin,Eric Polizzi +1 more
TL;DR: It will be shown that this approach can outperform the traditional SCF mixing-scheme techniques by providing a higher converge rate, convergence to the correct solution regardless of the choice of the initial guess, and a significant reduction of the eigenvalue solve time in simulations.
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.