On a class of multi-fidelity methods for the semiclassical Schr\"odinger equation with uncertainties

TL;DR

This paper develops robust multi-fidelity methods for solving the semiclassical Schrödinger equation with uncertainties, combining high- and low-fidelity models to improve efficiency and accuracy.

Contribution

The paper introduces a class of multi-fidelity methods using various low-fidelity models alongside a high-fidelity solver for the semiclassical Schrödinger equation.

Findings

The methods achieve high accuracy with reduced computational cost.

Numerical experiments validate the effectiveness of bi- and tri-fidelity approximations.

Error estimates are provided for the bi-fidelity method.

Abstract

In this paper, we study the semiclassical Schr\"odinger equation with random parameters and develop several robust multi-fidelity methods. We employ the time-splitting Fourier pseudospectral (TSFP) method for the high-fidelity solver, and consider different low-fidelity solvers including the meshless method like frozen Gaussian approximation (FGA) and the level set (LS) method for the semiclassical limit of the Schr\"odinger equation. With a careful choice of the low-fidelity model, we obtain an error estimate for the bi-fidelity method. We conduct numerous numerical experiments and validate the accuracy and efficiency of our proposed multi-fidelity methods, by comparing the performance of a class of bi-fidelity and tri-fidelity approximations.