Sylvester-based preconditioning for the waveguide eigenvalue problem

TL;DR

This paper introduces a novel preconditioning method based on Sylvester equations and Sherman-Morrison-Woodbury formula to efficiently solve large-scale nonlinear eigenvalue problems from waveguide PDEs.

Contribution

It develops a Sylvester-based preconditioner integrated with residual inverse iteration for large-scale waveguide eigenvalue problems, improving computational efficiency.

Findings

Effective preconditioner accelerates convergence

Applicable to large-scale waveguide eigenvalue problems

Demonstrated on practical PDE-based models

Abstract

We consider a nonlinear eigenvalue problem (NEP) arising from absorbing boundary conditions in the study of a partial differential equation (PDE) describing a waveguide. We propose a new computational approach for this large scale NEP based on residual inverse iteration (Resinv) with preconditioned iterative solves. Similar to many preconditioned iterative methods for discretized PDEs, this approach requires the construction of an accurate and efficient preconditioner. For the waveguide eigenvalue problem, the associated linear system can be formulated as a generalized Sylvester equation. The equation is approximated by a low-rank correction of a Sylvester equation, which we use as a preconditioner. The action of the preconditioner is efficiently computed using the matrix equation version of the Sherman-Morrison-Woodbury (SMW) formula. We show how the preconditioner can be integrated into Resinv. The results are illustrated by applying the method to large-scale problems.