A Multilevel Stochastic Collocation Method for Schrödinger Equations with a Random Potential

Abstract

We propose and analyze a numerical method for time-dependent linear Schrödinger equations with 5 uncertain parameters in both the potential and the initial data. The random parameters are dis6 cretized by stochastic collocation on a sparse grid, and the sample solutions in the nodes are ap7 proximated with the Strang splitting method. The computational work is reduced by a multi-level 8 strategy, i.e. by combining information obtained from sample solutions computed on different re9 finement levels of the discretization. We prove new error bounds for the time discretization which 10 take the finite regularity in the stochastic variable into account, and which are crucial to obtain 11 convergence of the multi-level approach. The predicted cost savings of the multi-level stochastic 12 collocation method are verified by numerical examples. 13