Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem: global convergence and computational efficiency
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- December 2018
- DOI:
- 10.48550/arXiv.1812.00835
- arXiv:
- arXiv:1812.00835
- Bibcode:
- 2018arXiv181200835H
- Keywords:
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- Mathematics - Numerical Analysis;
- 35Q55;
- 65N12;
- 65N25;
- 65N30;
- 81Q05