Frozen Gaussian sampling: a mesh-free Monte Carlo method for approximating semiclassical Schrödinger equations

In this paper, we develop a Monte Carlo algorithm named the Frozen Gaussian Sampling (FGS) to solve the semiclassical Schrödinger equation based on the frozen Gaussian approximation. Due to the highly oscillatory structure of the wave function, traditional mesh-based algorithms suffer from “the curse of dimensionality,” which gives rise to more severe computational burden when the semiclassical parameter $\varepsilon$ is small. The Frozen Gaussian sampling outperforms the existing algorithms in that it is mesh-free in computing the physical observables and is suitable for high dimensional problems. In this work, we provide detailed procedures to implement the FGS for both Gaussian and WKB initial data cases, where the sampling strategies on the phase space balance the need for variance reduction and sampling convenience. Moreover, we rigorously prove that the number of samples needed for the FGS is independent of the scaling parameter $\varepsilon$ to reach a certain accuracy. Furthermore, the complexity of the FGS algorithm is of a sublinear scaling concerning the microscopic degrees of freedom and, in particular, is insensitive to the dimension number. The performance of the FGS is validated through several typical numerical experiments, including simulating scattering by the barrier potential, formation of the caustics, and computing the high-dimensional physical observables without mesh.

Keywords

semiclassical Schrödinger equation, frozen Gaussian approximation, frozen Gaussian sampling, Monte Carlo method, variance reduction

2010 Mathematics Subject Classification

65C05, 65M75, 81Q05, 81Q20

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Received 13 December 2021

Accepted 28 October 2023

Published 12 July 2024