Gradient Flows and Nonlinear Power Methods for the Computation of Nonlinear Eigenfunctions

TL;DR

This paper explores gradient flows and nonlinear power methods in Banach spaces to compute nonlinear eigenfunctions, establishing convergence results and linking various flows to normalized gradient flows.

Contribution

It introduces a unified framework connecting gradient flows and nonlinear power methods, and proves convergence of discretized algorithms and ground states via $ extGamma$-convergence.

Findings

Implicit Euler discretization leads to a nonlinear power method with proven convergence.

Normalized gradient flows relate to several existing methods in literature.

Convergence of discrete approximations is guaranteed through $ extGamma$-convergence.

Abstract

This chapter describes how gradient flows and nonlinear power methods in Banach spaces can be used to solve nonlinear eigenvector-dependent eigenvalue problems, and how convergence of (discretized) approximations can be verified. We review several flows from literature, which were proposed to compute nonlinear eigenfunctions, and show that they all relate to normalized gradient flows. Furthermore, we show that the implicit Euler discretization of gradient flows gives rise to a nonlinear power method of the proximal operator and prove their convergence to nonlinear eigenfunctions. Finally, we prove that $\Gamma$-convergence of functionals implies convergence of their ground states, which is important for discrete approximations.