2 research outputs found
Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem: global convergence and computational efficiency
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 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