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

We propose to use the Łojasiewicz 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 using several examples, in particular a nonlinear Schrödinger eigenproblem with an extra high-order interaction term. Numerical experiments are presented for these problems.

Keywords

Sobolev gradient descent, Gross–Pitaevskii eigenproblem, Łojasiewicz inequality, Schrödinger equation, nonlinear eigenproblems

2010 Mathematics Subject Classification

35P30, 47J10, 65K10, 65N25, 81Q05

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Received 23 September 2020

Received revised 19 May 2021

Accepted 10 July 2021

Published 28 January 2022