Quasi-Monte Carlo time-splitting methods for Schr\"odinger equation with Gaussian random potential

TL;DR

This paper introduces a novel quasi-Monte Carlo time-splitting method for efficiently solving the Schrödinger equation with Gaussian random potential, achieving dimension-independent convergence and improved sampling accuracy.

Contribution

The paper develops a new QMC-based numerical scheme for SE-GP that handles unbounded Gaussian variables using weighted Sobolev spaces and proves its near-linear convergence rate.

Findings

Achieves dimension-independent convergence rate

Demonstrates sharp error estimates through numerical experiments

Provides an efficient sampling method for Gaussian random potentials

Abstract

In this paper, we study the Schr\"odinger equation with a Gaussian random potential (SE-GP) and develop an efficient numerical method to approximate the expectation of physical observables. The unboundedness of Gaussian random variables poses significant difficulties in both sampling and error analysis. Under time-splitting discretizations of SE-GP, we establish the regularity of the semi-discrete solution in the random space. Then, we introduce a non-standard weighted Sobolev space with properly chosen weight functions, and obtain a randomly shifted lattice-based quasi-Monte Carlo (QMC) quadrature rule for efficient sampling. This approach leads to a QMC time-splitting (QMC-TS) scheme for solving the SE-GP. We prove that the proposed QMC-TS method achieves a dimension-independent convergence rate that is almost linear with respect to the number of QMC samples. Numerical experiments illustrate the sharpness of the error estimate.