An explicitly-sparse representation for oscillatory kernels with wave atom-like functions

Abstract

An explicitly-sparse representation for oscillatory kernels is presented in this work by developing a wave atom based method. Multilevel wave atom-like functions are constructed as the transform of the original nodal basis. Then the system matrix in a new non-standard form is derived with respect to the wave atom basis. The wave atom representation is explicitly-sparse in the sense that the system matrix is sparse and computed explicitly. Its sparsity is further enhanced via a-posteriori compression. Finally its log-linear computational complexity with controllable accuracy is demonstrated with numerical results. This explicitly-sparse representation is expected to lay ground to future work related to fast direct solvers and effective preconditioners for high frequency problems. The algorithm may also be viewed as the generalization of wavelet based methods to high frequency cases, and used as a new wideband fast algorithm for wave problems.