Anderson Acceleration of Nonlinear Solvers for the Stationary Gross-Pitaevskii Equation

Abstract

We consider Anderson acceleration (AA) applied to two nonlinear solvers for the stationary Gross-Pitaevskii equation: a Picard type nonlinear iterative solver and a normalized gradient flow method. We formulate the solvers as fixed point problems and show that they both fit into the recently developed AA analysis framework. This allows us to prove that both methods' linear convergence rates are improved by a factor (less than one) from the gain of the AA optimization problem at each step. Numerical tests for finding ground state solutions in 1D and 2D show that AA significantly improves convergence behavior in both solvers, and additionally some comparisons between the solvers are drawn. A local convergence analysis for both methods are also provided.