The waveguide eigenvalue problem and the tensor infinite Arnoldi method

TL;DR

This paper introduces the tensor infinite Arnoldi method (TIAR), a new iterative algorithm for large-scale nonlinear eigenvalue problems, specifically applied to waveguide problems, with proven efficiency and complexity analysis.

Contribution

The paper develops the TIAR algorithm for general NEPs and specializes it to waveguide eigenvalue problems, demonstrating its effectiveness and analyzing its computational complexity.

Findings

Successfully applied to benchmark and complex waveguide problems

Established the algorithm's complexity as O(nm^2 + sqrt(n) m^3)

TIAR's asymptotic complexity matches that of Arnoldi's method for standard eigenproblems

Abstract

We present a new computational approach for a class of large-scale nonlinear eigenvalue problems (NEPs) that are nonlinear in the eigenvalue. The contribution of this paper is two-fold. We derive a new iterative algorithm for NEPs, the tensor infinite Arnoldi method (TIAR), which is applicable to a general class of NEPs, and we show how to specialize the algorithm to a specific NEP: the waveguide eigenvalue problem. The waveguide eigenvalue problem arises from a finite-element discretization of a partial differential equation (PDE) used in the study waves propagating in a periodic medium. The algorithm is successfully applied to accurately solve benchmark problems as well as complicated waveguides. We study the complexity of the specialized algorithm with respect to the number of iterations m and the size of the problem n, both from a theoretical perspective and in practice. For the waveguide eigenvalue problem, we establish that the computationally dominating part of the algorithm has complexity O(nm^2 + sqrt(n) m^3). Hence, the asymptotic complexity of TIAR applied to the waveguide eigenvalue problem, for n that goes to infinity, is the same as for Arnoldi's method for standard eigenvalue problems.