标题： 冻结高斯网格点修正用于半经典薛定谔方程 显示英文标题

标题： Frozen Gaussian Grid-point Correction For Semi-classical Schrödinger Equation

摘要： 我们提出了一种名为冻结高斯网格点校正（FGGC）的高效重构算法，用于使用冻结高斯近似（FGA）计算半经典领域的薛定谔方程。FGA已经展示了其在处理半经典问题和高频波传播方面的卓越效率。然而，从大量的高斯波包重构波函数通常计算量很大。这种困难源于这些波包沿着FGA轨迹传播到非网格位置，使得快速傅里叶变换的应用变得不可行。在这项工作中，我们引入了“网格点校正”的概念，并推导出高斯波包最小二乘逼近的公式，同时提供了FGGC算法的详细过程。此外，我们严格证明了最小二乘逼近在每个高斯波包上引入的误差与半经典参数$\varepsilon$无关。数值实验表明，FGGC算法可以显著提高重构效率，同时仅引入可忽略的误差，使其成为求解半经典薛定谔方程的强大工具，特别是在需要准确性和效率的应用中。 显示英文摘要

摘要： We propose an efficient reconstruction algorithm named the frozen Gaussian grid-point correction (FGGC) for computing the Schr\"odinger equation in the semi-classical regime using the frozen Gaussian approximation (FGA). The FGA has demonstrated its superior efficiency in dealing with semi-classical problems and high-frequency wave propagations. However, reconstructing the wave function from a large number of Gaussian wave-packets is typically computationally intensive. This difficulty arises because these wave-packets propagate along the FGA trajectories to non-grid positions, making the application of the fast Fourier transform infeasible. In this work, we introduce the concept of ``on-grid correction'' and derive the formulas for the least squares approximation of Gaussian wave-packets, and also provide a detailed process of the FGGC algorithm. Furthermore, we rigorously prove that the error introduced by the least squares approximation on each Gaussian wave-packet is independent of the semi-classical parameter $\varepsilon$. Numerical experiments show that the FGGC algorithm can significantly improve reconstruction efficiency while introducing only negligible error, making it a powerful tool for solving the semi-classical Schr\"odinger equation, especially in applications requiring both accuracy and efficiency.