Emil Ringh: Sylvester-based Preconditioner for the Waveguide Eigenvalue Problem

Abstract:

We study 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 NEP based on residual inverse iteration with preconditioned iterative solves. The construction of an accurate and efficient preconditioner is crucial when applying this to large-scale problems. Our preconditioner is based on a generalization of the Sylvester equation \(AX + XB + K*X = C\), where \(*\)denotes the Hadamard (direct) product. This matrix equation arises naturally in relation to the discretization of the PDE.

We use the matrix equation version of the Sherman-Morrison-Woodbury formula, \(AX + XB + E_1 W_1(X) + ... + E_m W_m(X) = C\), where the \(E_k\) are matrices and \(W_k\) are linear functionals, and approximate the matrix equation in this form to construct the preconditioner. This has a natural relation to approximations in the original PDE. We also show how to integrate the preconditioner in the setting of the algorithm for NEP to minimize computational effort. The results are illustrated by applying the method to large-scale benchmark problems as well as more complicated waveguides.