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.

翻译：本文研究含随机参数的半经典薛定谔方程，并发展了几种鲁棒的多保真度方法。我们采用时间分裂傅里叶伪谱（TSFP）方法作为高保真求解器，同时考虑了多种低保真求解器，包括用于薛定谔方程半经典极限的无网格方法（如冻结高斯近似FGA）以及水平集（LS）方法。通过对低保真模型的审慎选择，我们获得了双保真度方法的误差估计。通过比较一系列双保真度与三保真度近似方法的性能，我们开展了大量数值实验，验证了所提多保真度方法的精度与效率。