论文标题
标量波方程的冷冻高斯抽样
Frozen Gaussian Sampling for Scalar Wave Equations
论文作者
论文摘要
在本文中，我们介绍了冷冻的高斯采样（FGS）算法，以求解高频制度中的标量波方程。 FGS算法是基于冷冻高斯近似的蒙特卡洛采样策略，可大大减少波传播和重建中的计算工作量。在这项工作中，我们提出了可行且详细的程序，以分别使用高斯初始条件和WKB初始条件来实施FGS算法以近似标量波方程。对于两个初始数据案例，我们严格分析将此算法应用于维度方程$ d \ geq 3 $的错误。在高斯初始数据案例中，我们证明，由于蒙特卡洛法引起的采样误差与典型的波数无关。我们还得出了WKB初始数据案例中采样误差的定量结合。最后，我们通过各种数值示例验证了FGS的性能以及有关采样误差的理论估计，其中包括使用FGS求解具有高速度和WKB尺寸的初始数据$ d = 1、2 $和$ 3 $的波浪方程。
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 $3$.
