Effective Preconditioners for High Frequency Wave Equations

高频波动方程的有效预调节器

• 批准号: 1521830
• 负责人: Lexing Ying
• 金额: $ 30万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-15 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1521830&HistoricalAwards=false
• 关键词: Effective; Preconditioners; Frequency; Wave; Equations

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.

物理科学和工程中的许多基本现象都是通过高频波来建模的。现代计算方法已成为理解这些现象的重要工具。 在这个项目中，PI将继续他在这一领域的研究，重点是时谐方程。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处理器。研究成果可以提供高效的解决方法并在应用中发挥重要作用。该项目的教育部分涉及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处理器： (1) 波通常沿明确定义的方向传播。这通常允许人们将复杂的波相互作用解耦为简单的组件，每个组件都处于首选方向，并且可以使用低秩和随机技术进行压缩。 (2) 应采用精确的离散化方案来捕获正确的色散关系。 (3) 即使数值离散化的线性系统并不稀疏，也应该利用时谐波算子的稀疏性。利用这些想法，PI 将解决以下技术问题：（1）通过研究更有效的完美匹配层设计、替代分解形式和递归扫描策略来开发更有效的扫描预处理器； (2) 通过有效地将稠密积分系统简化为稀疏形式，开发Lippmann-Schwinger方程的稀疏预条件子； (3) 开发稀疏预处理器，用于计算光子学和电子结构计算中周期结构问题的伪谱近似； (4)通过开发边界积分算子的稀疏表示并引入新的与核无关的方向快速求和方法，构建障碍物散射问题的方向预处理器。