Sobolev Gradient Flow for the Gross--Pitaevskii Eigenvalue Problem: Global Convergence and Computational Efficiency
Year: 2020
Type: article
Abstract: We propose a new normalized Sobolev gradient flow for the Gross--Pitaevskii eigenvalue problem based on an energy inner product that depends on time through the density of the flow itself. The gradient flow is well-defined and converges to an eigenfunction. For ground states we can quantify the convergence speed as exponentially fast where the rate depends on spectral gaps of a linearized operator. The forward Euler time discretization of the flow yields a numerical method which generalizes the inverse iteration for the nonlinear eigenvalue problem. For sufficiently small time steps, the method reduces the energy in every step and converges globally in $H^1$ to an eigenfunction. In particular, for any nonnegative starting value, the ground state is obtained. A series of numerical experiments demonstrates the computational efficiency of the method and its competitiveness with established discretizations arising from other gradient flows for this problem.
DOI: 10.1137/18m1230463
Authors: Patrick Henning, Daniel Peterseim
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2903259461