Complex moment-based methods for differential eigenvalue problems

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.

Efficient Analysis of Large-Size Bio-Signals Based on Orthogonal Generalized Laguerre Moments of Fractional Orders and Schwarz–Rutishauser Algorithm

“…It is used basically in solving eigenvalue problems and has been used in image reconstruction applications. In the image reconstruction application, a formulation is shown to obtain an estimated original image from the degraded image moments and the blur parameter [24,25]. Complex Moments (CMs) were initially proposed as a straightforward and uncomplicated approach to reconstruction images.…”