Broyden's method for nonlinear eigenproblems

TL;DR

This paper extends Broyden's method to nonlinear eigenvalue problems, focusing on computational efficiency, convergence properties, and the ability to compute multiple eigenvalues, demonstrated through a machine tool milling PDE example.

Contribution

It introduces a Broyden-based algorithm tailored for nonlinear eigenproblems with expensive matrix-vector products, incorporating Jacobian structure and deflation techniques.

Findings

The method achieves local superlinear convergence for simple eigenvalues.

Incorporating Jacobian structure improves convergence.

The approach successfully computes multiple eigenvalues in a PDE-based problem.

Abstract

Broyden's method is a general method commonly used for nonlinear systems of equations, when very little information is available about the problem. We develop an approach based on Broyden's method for nonlinear eigenvalue problems. Our approach is designed for problems where the evaluation of a matrix vector product is computationally expensive, essentially as expensive as solving the corresponding linear system of equations. We show how the structure of the Jacobian matrix can be incorporated into the algorithm to improve convergence. The algorithm exhibits local superlinear convergence for simple eigenvalues, and we characterize the convergence. We show how deflation can be integrated and combined such that the method can be used to compute several eigenvalues. A specific problem in machine tool milling, coupled with a PDE is used to illustrate the approach. The simulations are done in the julia programming language, and are provided as publicly available module for reproducability.