Effective Preconditioners for High Frequency Wave Equations

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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.