Multigrid Optimization Schemes for Solving Bose–Einstein Condensate Control Problems

Abstract

The control of the transport of Bose–Einstein condensates in magnetic microtraps is formulated within the framework of optimal control theory and solved by multigrid optimization schemes. The time evolution of the wave function of Bose–Einstein condensates is governed by the Gross–Pitaevskii equation and can be manipulated through variation of a controllable magnetic confinement potential. In order to define an optimal control strategy, an appropriate cost functional is introduced that must be minimized under the constraint given by the dynamic equation. The resulting optimality system consists of two nonlinear Schrödinger-type equations with opposite time orientation coupled with an elliptic equation for the control function. These equations are approximated by using a time-splitting pseudospectral method and finite differences. To solve the resulting problem a cascadic nonlinear conjugate gradient scheme and a multigrid optimization scheme are considered. The convergence properties of these two schemes are investigated theoretically, and their computational performance is discussed based on results of numerical experiments. It appears that the multigrid optimization scheme provides a robust optimization strategy.