Solving large-scale nonlinear eigenvalue problems by rational interpolation approach and resolvent sampling based Rayleigh-Ritz method

TL;DR

This paper introduces a novel rational interpolation approach and a resolvent sampling-based Rayleigh-Ritz method to efficiently solve large-scale nonlinear eigenvalue problems, overcoming limitations of existing techniques.

Contribution

It develops a universal, robust NEP solver combining RIA and RSRR, with improved accuracy and computational efficiency for large-scale applications.

Findings

RIA outperforms contour integral approach in accuracy and cost

RSRR is robust, easy to implement, and parallelizable

Method successfully applied to benchmark and practical problems

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.