Frozen Gaussian Sampling for Scalar Wave Equations

TL;DR

This paper introduces the frozen Gaussian sampling (FGS) algorithm for efficiently solving high-frequency scalar wave equations using Monte Carlo methods, with rigorous error analysis and numerical validation.

Contribution

The paper presents a novel FGS algorithm that reduces computational costs for high-frequency wave equations and provides detailed error analysis for different initial conditions.

Findings

Sampling error is independent of wave number for Gaussian initial data.

Quantitative bounds on sampling error for WKB initial data are derived.

Numerical experiments confirm the theoretical error estimates across dimensions 1 to 3.

Abstract

In this article, we introduce the frozen Gaussian sampling (FGS) algorithm to solve the scalar wave equation in the high-frequency regime. The FGS algorithm is a Monte Carlo sampling strategy based on the frozen Gaussian approximation, which greatly reduces the computation workload in the wave propagation and reconstruction. In this work, we propose feasible and detailed procedures to implement the FGS algorithm to approximate scalar wave equations with Gaussian initial conditions and WKB initial conditions respectively. For both initial data cases, we rigorously analyze the error of applying this algorithm to wave equations of dimensionality $d \geq 3$. In Gaussian initial data cases, we prove that the sampling error due to the Monte Carlo method is independent of the typical wave number. We also derive a quantitative bound of the sampling error in WKB initial data cases. Finally, we validate the performance of the FGS and the theoretical estimates about the sampling error through various numerical examples, which include using the FGS to solve wave equations with both Gaussian and WKB initial data of dimensionality $d = 1, 2$, and $3$.