A Multiscale Reduced Basis Method for the Schrödinger Equation With Multiscale and Random Potentials

Abstract

The semiclassical Schrödinger equation with multiscale and random potentials often appears when studying electron dynamics in heterogeneous quantum systems. As time evolves, the wave function develops high-frequency oscillations in both the physical space and the random space, which poses severe challenges for numerical methods. To address this problem, in this paper we propose a multiscale reduced basis method, where we construct multiscale reduced basis functions using an optimization method and the proper orthogonal decomposition method in the physical space and employ the quasi-Monte Carlo method in the random space. Our method is verified to be efficient: the spatial grid size is only proportional to the semiclassical parameter and (under suitable conditions) an almost first-order convergence rate is achieved in the random space with respect to the sample number. Several theoretical aspects of the proposed method, including how to determine the number of samples in the construction of multiscale reduced basis and convergence analysis, are studied with numerical justification. In addition, we investigate the Anderson localization phenomena for the Schrödinger equation with correlated random potentials in both 1-dimensional and 2-dimensional space.