Gradient flows and nonlinear power methods for the computation of nonlinear eigenfunctions

Bungert L, Burger M (2022)

Publication Type: Book chapter / Article in edited volumes

Publication year: 2022

Journal

Handbook of Numerical Analysis Elsevier BV

Publisher: Elsevier B.V.

Series: Handbook of Numerical Analysis

Book Volume: 23

Pages Range: 427-465

DOI: 10.1016/bs.hna.2021.12.013

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 these can be approximated using Γ-convergence. 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 we demonstrate their convergence to nonlinear eigenfunctions. Finally, we prove that Γ-convergence of functionals implies convergence of their ground states.

Authors with CRIS profile

Martin Burger Lehrstuhl für Angewandte Mathematik (Modellierung und Numerik)

Involved external institutions

Rheinische Friedrich-Wilhelms-Universität Bonn DE Germany (DE)

How to cite

APA:

Bungert, L., & Burger, M. (2022). Gradient flows and nonlinear power methods for the computation of nonlinear eigenfunctions. In Emmanuel Trélat, Enrique Zuazua, Enrique Zuazua, Enrique Zuazua (Eds.), (pp. 427-465). Elsevier B.V..

MLA:

Bungert, Leon, and Martin Burger. "Gradient flows and nonlinear power methods for the computation of nonlinear eigenfunctions." Ed. Emmanuel Trélat, Enrique Zuazua, Enrique Zuazua, Enrique Zuazua, Elsevier B.V., 2022. 427-465.

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