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 $3$.

翻译：摘要：本文提出冻结高斯采样（FGS）算法以求解高频区域中的标量波动方程。FGS算法是一种基于冻结高斯近似的蒙特卡洛采样策略，该算法大幅降低了波传播与重构过程中的计算量。本研究针对高斯初始条件与WKB初始条件，分别提出了实现FGS算法近似标量波动方程的可行且详细步骤。对于两种初始数据情形，我们严格分析了该方法在维度$d \geq 3$的波动方程中的误差。在高斯初始数据情形下，我们证明蒙特卡洛方法产生的采样误差与典型波数无关；在WKB初始数据情形下，我们推导出采样误差的定量界。最后，通过多个数值算例（包括使用FGS求解维度$d=1,2,3$的高斯与WKB初始数据波动方程），验证了FGS算法的性能及关于采样误差的理论估计。