Duality Perspective on Nonlinear Eigenproblems

TL;DR

This paper develops a duality framework for nonlinear eigenproblems in Banach spaces, introducing a dual formulation, analyzing convergence of inverse power methods, and validating results with numerical experiments on the p-Laplacian.

Contribution

It presents a novel dual formulation for nonlinear eigenproblems, establishes convergence results for the inverse power method, and connects flow-based methods to classical algorithms.

Findings

Strong convergence of the inverse power method for p-homogeneous functionals

Duality gap and geometric eigenvector characterization in Banach spaces

Numerical validation on the p-Laplacian problem

Abstract

We investigate nonlinear eigenproblems for a broad class of proper, closed, convex functionals in reflexive Banach spaces. We develop a dual formulation of the nonlinear eigenproblem using the Fenchel conjugate and establish an equivalence to the primal problem. Further, we introduce a duality gap and a geometric characterization of eigenvectors that apply in general Banach spaces. We interpret the dual problem as the eigenproblem for the inverse operator of the primal problem. Concerning numerical methods for solving nonlinear eigenproblems, we analyze the inverse power method, framed as a dual power method, showing strong convergence in the case of absolutely p-homogeneous functionals. Our theoretical results are validated by extensive numerical experiments for the p-Laplacian. We further connect the flow-based proximal power method from the literature to the inverse power method and discuss two numerical approaches to approximate higher-order nonlinear eigenfunctions.