Frozen Gaussian sampling for scalar wave equations

¹ School of Mathematics, Sun Yat-sen University, 135 Xingang Xi Road, Guangzhou 510275, P.R. China
² Department of Mathematics, Duke University, 120 Science Drive, Durham, NC 27708, USA
³ Institute for Theoretical Sciences, Westlake University, 600 Dunyu Road, Hangzhou 310030, P.R. China

^* Corresponding author: chailihui@mail.sysu.edu.cn

Received: 11 November 2022
Accepted: 16 April 2024

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 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 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 ≥ 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.