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In this paper, we develop a Monte Carlo algorithm named the Frozen GaussianSampling (FGS) to solve the semiclassical Schr\"odinger equation based on thefrozen Gaussian approximation. Due to the highly oscillatory structure of thewave function, traditional mesh-based algorithms suffer from "the curse ofdimensionality", which gives rise to more severe computational burden when thesemiclassical parameter \(\ep\) is small. The Frozen Gaussian samplingoutperforms the existing algorithms in that it is mesh-free in computing thephysical observables and is suitable for high dimensional problems. In thiswork, we provide detailed procedures to implement the FGS for both Gaussian andWKB initial data cases, where the sampling strategies on the phase spacebalance the need of variance reduction and sampling convenience. Moreover, werigorously prove that, to reach a certain accuracy, the number of samplesneeded for the FGS is independent of the scaling parameter \(\ep\).Furthermore, the complexity of the FGS algorithm is of a sublinear scaling withrespect to the microscopic degrees of freedom and, in particular, isinsensitive to the dimension number. The performance of the FGS is validatedthrough several typical numerical experiments, including simulating scatteringby the barrier potential, formation of the caustics and computing thehigh-dimensional physical observables without mesh.

Frozen Gaussian Sampling: A Mesh-free Monte Carlo Method For Approximating Semiclassical Schr\"odinger Equations