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

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 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 resolvent sampling based Rayleigh-Ritz method (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.Generally speaking, although there exist a number of numerical methods for NEPs in literature, most of them are restricted by the matrix structures, properties of eigenvalues, computational costs, etc. Accurate and efficient methods that can robustly and reliably calculate all the eigenpairs within a given region, and at the same time, can be applied to large-scale computations are still lacking [3,4,[7][8][9][10][11].In recent years, the contour integral approach (CIA) based on contour integrals of the probed resolvent T .´/ 1 U has attracted great attention [12][13][14][15][16][17][18][19], where the constant matrix U 2 C n L consists of L random column vectors used to probe T .´/ 1 . The CIA is first developed for solving generalized eigenvalue problems [12,13], and later, extended to NEPs in [20] and [15] based on the Smith form and Keldysh's theorem for analytic matrix-valued functions, respectively. Because [20] and [15] result in similar algorithms, we consider the block Sakurai-Sugiura (SS) algorithm in [20]. This algorithm transforms the original NEP into a small-sized generalized eigenvalue problem involving block Hankel matrices by using the monomial moments of the probed resolvent T .´/ 1 U on a given contour. It becomes unstable and inaccurate when higher order contour moments of T .´/ 1 U are used.To circumvent this problem, the SS method with the Rayleigh-Ritz projection (SSRR) has been recently proposed [16]. The eigenspace of the SSRR is constructed from a matrix M 2 C n K L collecting all the monomial moments of T .´/ 1 U up to a given order K. We refer to this as the resolvent moment scheme (or simply moment scheme) for generating eigenspaces. Numerical results in [16] show that the SSRR has a much better stability and accuracy than the SS algorithm when a small value of L is used. Besides, the...

The nonconforming Trefftz virtual element method: general setting, applications, and dispersion analysis for the Helmholtz equation

“…, u Ξ ) T ∈ C Ξ . We highlight that T is a holomorphic map and ( 49) is a small nonlinear eigenvalue problem, which can be solved using, e.g., an iterative method as done in [24], or a direct method based on a rational interpolation procedure [47] or on a contour integral approach [12,5]. For the numerical experiments presented in Section 6.3, we will make use of the latter, which we will denote by contour integral method (CIM) in the sequel.…”

The nonlinear eigenvalue problem

“…Similar ideas can be used to compute roots and poles of a meromorphic function inside (Kravanja, Van Barel and Haegemans 1999 b ); see also Austin, Kravanja and Trefethen (2014), who discuss connections of contour integration with (rational) interpolation and provide several MATLAB code snippets. An NEP solver based on rational interpolation and resolvent sampling is presented and applied in Xiao, Zhang, Huang and Sakurai (2016 a ) and Xiao, Zhou, Zhang and Xu (2016 b ).…”

A numerical study of the dispersion and dissipation properties of virtual element methods for the Helmholtz problem

“…We highlight that T is a holomorphic map, and that ( 10) is a small nonlinear eigenvalue problem, which can be solved using e.g. an iterative method as done in [15], or a direct method based on a rational interpolation procedure [29] or on a contour integral approach [4,10]. For the numerical experiments presented in Section 4, we will make use of the latter, which we will denote by contour integral method (CIM) in the sequel.…”