A nearly optimal explicitly-sparse representation for oscillatory kernels with curvelet-like functions

TL;DR

The paper introduces a curvelet-based method (CBM) to obtain a nearly optimal explicitly-sparse representation for oscillatory kernels in $N$-body problems. It combines a multilevel wavelet framework for low frequencies with a directional curvelet extension for high frequencies, producing a non-standard sparse system that scales as $O(N \log N)$ in both storage and matrix-vector products. Core contributions include transforming the nodal basis into wavelets, performing parabolic-separation–driven curvelet constructions per directional cone, and applying a-posteriori compression to sharpen sparsity while controlling accuracy. The approach unifies and extends wavelet- and FMM-like strategies, enabling fast direct solvers and efficient preconditioners for high-frequency wave problems across varied geometries and kernel layers.

Abstract

A nearly optimal explicitly-sparse representation for oscillatory kernels is presented in this work by developing a curvelet based method. Multilevel curvelet-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 curvelet basis, which would be nearly optimally sparse due to the directional low rank property of the oscillatory kernel. Its sparsity is further enhanced via a-posteriori compression. Finally its nearly optimial 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. It 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.

Figures (7)

• Figure 1: Parabolic separation condition.
• Figure 2: Construction of multilevel curvelets, where subscripts $(l, \gamma)$ denotes the $\gamma$-th directional cone in the $l$-th level.
• Figure 3: A sphere with $\kappa D = 201.1$.
• Figure 4: Computational cost of the summation on a unit sphere with the single layer kernel.
• Figure 5: Increasement of the number of nonempty directional cones with frequency for the same $w/\lambda$. Illustrated with a two-dimensional case. Three-dimensional cases would be similar.