On the convergence of Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem

TL;DR

This paper analyzes the convergence properties of three Sobolev gradient flow methods for finding the ground state of the Gross-Pitaevskii eigenvalue problem, establishing energy dissipation and exponential convergence results.

Contribution

It introduces and compares three projected Sobolev gradient flows with different metrics, proving their convergence and energy dissipation properties.

Findings

Proved energy dissipation for discrete-time flows.

Established global convergence to critical points.

Proved local exponential convergence to the ground state.

Abstract

We study the convergences of three projected Sobolev gradient flows to the ground state of the Gross-Pitaevskii eigenvalue problem. They are constructed as the gradient flows of the Gross-Pitaevskii energy functional with respect to the $H^1_0$-metric and two other equivalent metrics on $H_0^1$, including the iterate-independent $a_0$-metric and the iterate-dependent $a_u$-metric. We first prove the energy dissipation property and the global convergence to a critical point of the Gross-Pitaevskii energy for the discrete-time $H^1$ and $a_0$-gradient flow. We also prove local exponential convergence of all three schemes to the ground state.