论文标题
具有任意来源的不均匀媒体中的高频Helmholtz方程的快速蝴蝶压缩的Hadamard-Babich积分器
A Fast Butterfly-compressed Hadamard-Babich Integrator for High-Frequency Helmholtz Equations in Inhomogeneous Media with Arbitrary Sources
论文作者
论文摘要
我们提出了Hadamard-Babich（HB）Ansatz的蝴蝶压缩表示形式，以平滑不均匀介质中高频Helmholtz方程的绿色功能。 For a computational domain discretized with $N_v$ discretization cells, the proposed algorithm first solves and tabulates the phase and HB coefficients via eikonal and transport equations with observation points and point sources located at the Chebyshev nodes using a set of much coarser computation grids, and then butterfly compresses the resulting HB interactions from all $N_v$ cell centers to each other.对于具有任意激发源的任何有限的2D域而言，总体CPU时间和内存要求量表为$ O（N_V \ Log^2n_v）$。该方案向有限的3D域的直接扩展产生了$ O（N_V^{4/3}）$ CPU复杂性，可以将其进一步降低为具有建议的补救措施的准线性复杂性。该方案还可以有效处理涉及不均匀媒体包含物的散射问题。尽管当前的HB集成剂的构造不适应苛性碱，但所得的HB积分器本身可以应用于某些来源，例如凹形来源，以产生苛性造成的效果。与有限差异频域（FDFD）方法相比，提出的HB积分器没有数值分散体，并且每个波长的离散点更少。结果，它可以解决远远超出现有求解器的能力的波传播问题。值得注意的是，所提出的方案可以准确地模拟2D域中的波浪传播，每个方向具有640个波长，在3D域中，在劳伦斯·伯克利国家实验室的一项状态超级计算机上，每个方向的波长为54个波长。
We present a butterfly-compressed representation of the Hadamard-Babich (HB) ansatz for the Green's function of the high-frequency Helmholtz equation in smooth inhomogeneous media. For a computational domain discretized with $N_v$ discretization cells, the proposed algorithm first solves and tabulates the phase and HB coefficients via eikonal and transport equations with observation points and point sources located at the Chebyshev nodes using a set of much coarser computation grids, and then butterfly compresses the resulting HB interactions from all $N_v$ cell centers to each other. The overall CPU time and memory requirement scale as $O(N_v\log^2N_v)$ for any bounded 2D domains with arbitrary excitation sources. A direct extension of this scheme to bounded 3D domains yields an $O(N_v^{4/3})$ CPU complexity, which can be further reduced to quasi-linear complexities with proposed remedies. The scheme can also efficiently handle scattering problems involving inclusions in inhomogeneous media. Although the current construction of our HB integrator does not accommodate caustics, the resulting HB integrator itself can be applied to certain sources, such as concave-shaped sources, to produce caustic effects. Compared to finite-difference frequency-domain (FDFD) methods, the proposed HB integrator is free of numerical dispersion and requires fewer discretization points per wavelength. As a result, it can solve wave-propagation problems well beyond the capability of existing solvers. Remarkably, the proposed scheme can accurately model wave propagation in 2D domains with 640 wavelengths per direction and in 3D domains with 54 wavelengths per direction on a state-the-art supercomputer at Lawrence Berkeley National Laboratory.
