Gradient Flows, Nonlinear Power Methods, and Computation of Nonlinear Eigenfunctions

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 $\Gamma$-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 prove their convergence to nonlinear eigenfunctions. Finally, we prove that $\Gamma$-convergence of functionals implies convergence of their ground states.