A Directional Fast Algorithm for Oscillatory Kernels with Curvelet-Like Functions

Interactions of multiple points with oscillatory kernels are widely encountered in wave analysis. For large scale problems, its direct evaluation is prohibitive since the computational cost increases quadratically number points. Various fast algorithms have been constructed by exploiting specific properties kernel function. Early algorithms, such as multipole method (FMM) and variants, H2-matrix, adaptive cross approximation (ACA), wavelet-based method, etc., generally developed for that asymptotically smooth when source target well separated. kernels, however, asymptotic smoothness criteria only satisfied oscillation insignificant, thus these suitable low frequency cases. Fortunately, later directional rank property highly found, based on which various wideband problems proposed, include FMM, dirH2-ACA, algebraic etc. They may be viewed generalization early low-frequency to high cases. In this work, a curvelet-based generalizing Multilevel curvelet-like functions transform original nodal basis. Then system matrix new non-standard form derived under curvelet basis, would nearly optimally sparse due kernel. Its sparsity further enhanced via a-posteriori compression. Finally, log-linear complexity controllable accuracy demonstrated numerical results. This work provides another efficient algorithm problems. Moreover, it gives an explicitly-sparse representation matrix, expected beneficial development solvers preconditioners.