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.

翻译：本文通过发展一种基于Curvelet的方法，提出了振荡核的近最优显式稀疏表示。首先构造了多层级类Curvelet函数作为原始节点基的变换，随后基于Curvelet基导出了新标准形式的系统矩阵。由于振荡核的方向性低秩特性，该矩阵具有近最优稀疏性，并通过后验压缩进一步增强了其稀疏性。最后，数值结果验证了其在可控精度下的近最优对数线性计算复杂度。这种显式稀疏表示有望为高频问题的快速直接求解器和高效预处理器奠定基础，同时可视为基于小波方法向高频情形的推广，并作为波动问题的新型宽带快速算法。