Complex moment-based methods for differential eigenvalue problems

TL;DR

This paper introduces higher-order complex moment-based methods for differential eigenvalue problems, significantly reducing computational costs while maintaining high accuracy, thereby advancing the 'solve-then-discretize' paradigm.

Contribution

It develops operation analogues of Sakurai-Sugiura-type eigensolvers using higher-order moments, reducing the number of ODEs solved without sacrificing accuracy.

Findings

Methods are over five times faster than existing operator FEAST.

Achieved similar high accuracy with fewer ODE solves.

Numerical results confirm efficiency and precision improvements.

Abstract

This paper considers computing partial eigenpairs of differential eigenvalue problems (DEPs) such that eigenvalues are in a certain region on the complex plane. Recently, based on a "solve-then-discretize" paradigm, an operator analogue of the FEAST method has been proposed for DEPs without discretization of the coefficient operators. Compared to conventional "discretize-then-solve" approaches that discretize the operators and solve the resulting matrix problem, the operator analogue of FEAST exhibits much higher accuracy; however, it involves solving a large number of ordinary differential equations (ODEs). In this paper, to reduce the computational costs, we propose operation analogues of Sakurai-Sugiura-type complex moment-based eigensolvers for DEPs using higher-order complex moments and analyze the error bound of the proposed methods. We show that the number of ODEs to be solved can be reduced by a factor of the degree of complex moments without degrading accuracy, which is verified by numerical results. Numerical results demonstrate that the proposed methods are over five times faster compared with the operator analogue of FEAST for several DEPs while maintaining almost the same high accuracy. This study is expected to promote the "solve-then-discretize" paradigm for solving DEPs and contribute to faster and more accurate solutions in real-world applications.