Effective Preconditioners for High Frequency Wave Equations

物理科学和工程中的许多重要现象都是用高频波来模拟的。现代计算方法已成为理解这些现象的基本工具。在这个项目中，PI将继续他在这个领域的研究，重点是时间谐波方程。高频波的计算问题具有挑战性。PI将为几个关键领域的时谐高频波动方程开发有效的预调节器。研究结果可提供高效的求解方法，具有重要的应用价值。该项目的教育部分包括研究生和本科生的培训以及现代计算数学的课程开发。PI将结合以下思想为时谐高频波动方程开发有效的预调节器：(1)波通常沿明确的方向传播。这通常允许人们将复杂的波相互作用解耦成简单的分量，每个分量都在一个首选的方向上，可以用低秩和随机化技术进行压缩。(2)应采用精确的离散化方案来捕捉正确的色散关系。(3)即使从数值离散得到的线性系统不是稀疏的，也要利用时谐波算子的稀疏性。利用这些思想，PI将解决以下技术问题：(1)通过研究更有效的完全匹配层、替代分解形式和递归扫描策略的设计，开发更有效的扫描预调节器；(2)通过将稠密积分系统有效化简为稀疏形式，开发了Lippmann-Schwinger方程的稀疏化预条件；(3)基于计算光子学和电子结构计算的周期结构赝谱近似问题的稀疏化预处理；(4)通过建立边界积分算子的稀疏表示和引入新的与核无关的方向快速求和方法，构造障碍物散射问题的方向预条件。

Many essential phenomena in physical science and engineering are modeled by high frequency waves. Modern computational methods have become essential tools for understanding these phenomena. In this project, the PI will continue his research in this area, with an emphasis on time-harmonic equations. The computational problems of high frequency waves are challenging. The PI will develop effective preconditioners for time-harmonic high frequency wave equations for several key areas. The research results can allow for highly efficient solution methods and play important roles in applications. The educational component of the project involves graduate and undergraduate student training and curriculum development for modern computational mathematics. The PI will develop effective preconditioners for time-harmonic high frequency wave equations by combining the following ideas: (1) Waves often propagate in well-defined directions. This often allows one to decouple the complicated wave interaction into simple components, each of which is in a preferred direction and can be compressed with low-rank and randomized techniques. (2) Accurate discretization schemes should be utilized to capture the correct dispersion relationship. (3) The sparsity of the time-harmonic wave operator should be exploited even when the linear system from the numerical discretization is not sparse. Using these ideas, the PI will address the following technical problems: (1) Developing more efficient sweeping preconditioners by investigating more efficient designs for perfectly matched layers, alternative factorization forms, and recursive sweeping strategies; (2) Developing sparsifying preconditioners for the Lippmann-Schwinger equation by effectively reducing the dense integral system to a sparse form; (3) Developing sparsifying preconditioners for the pseudospectral approximations of problems on periodic structures from computational photonics and electron structure calculation; and (4) Constructing directional preconditioners for the obstacle scattering problem via developing a sparse representation of the boundary integral operator and introducing a new kernel-independent directional fast summation methods.