Exponential convergence of Sobolev gradient descent for a class of nonlinear eigenproblems

TL;DR

This paper introduces a Sobolev gradient descent method utilizing the {\

Contribution

It presents a novel convergence analysis using the {\

Findings

Exponential convergence to the ground state for the Gross-Pitaevskii eigenproblem.

Extension of the method to high-degree nonlinear eigenproblems.

Numerical validation on various nonlinear Schrödinger problems.

Abstract

We propose to use the {\L}ojasiewicz inequality as a general tool for analyzing the convergence rate of gradient descent on a Hilbert manifold, without resorting to the continuous gradient flow. Using this tool, we show that a Sobolev gradient descent method with adaptive inner product converges exponentially fast to the ground state for the Gross-Pitaevskii eigenproblem. This method can be extended to a class of general high-degree optimizations or nonlinear eigenproblems under certain conditions. We demonstrate this generalization by several examples, in particular a nonlinear Schr\"odinger eigenproblem with an extra high-order interaction term. Numerical experiments are presented for these problems.