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：在这项工作中,通过开发一个基于曲线的方法,可以呈现出一种几乎最优、最优、最优、最优、最优、最优的动脉内核代表。多级曲线式功能作为原始节点基的变换而构建。然后,以新的非标准形式生成的系统矩阵将基于曲线基体,由于螺旋内核的方向性能低,这种基体将几乎最理想地稀疏。通过一个螺旋内核的压缩,其弥漫度将进一步增强。最后,它几乎最优的逻辑-线性计算复杂性及其可控制精确度,将用数字结果来显示。这种明显偏差的表达法预计将为与快速直接求解器和高频率问题的有效先决条件有关的未来工作奠定基础。它也可以被视为波板基方法对高频率案例的普遍化,并用作波子问题的新宽带快速算法。</s>