A Fast Butterfly-Compressed Hadamard–Babich Integrator for High-Frequency Helmholtz Equations in Inhomogeneous Media with Arbitrary Sources
A Fast Butterfly-Compressed Hadamard–Babich Integrator for High-Frequency Helmholtz Equations in Inhomogeneous Media with Arbitrary Sources
复制标题
任意源非均匀介质中高频亥姆霍兹方程的快速蝶形压缩 Hadamard-Babich 积分器
DOI：
10.1137/21m1450422
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发表时间：
2023
影响因子：
1.6
通讯作者：
Qian, Jianliang
中科院分区：
文献类型：
--
作者：
Liu, Yang;Song, Jian;Burridge, Robert;Qian, Jianliang
我们给出了光滑非均匀介质中高频亥姆霍兹方程的绿色函数的Hadamard-Babich（HB）拟解的蝴蝶压缩表示。该算法首先通过程函方程和输运方程求解相位和HB系数，并将观测点和点源分布在Chebyshev节点上，然后对每个单元中心的HB相互作用进行蝶形压缩。对于具有任意激励源的任何有界二维（2D）域，总体CPU时间和内存需求规模相同。该方案的直接扩展到有界的3D域产生的CPU复杂度，这可以进一步降低到拟线性复杂度与建议的补救措施。该方案也可以有效地处理散射问题，包括在非均匀介质中的夹杂物。虽然我们的HB积分器的当前构造不适应焦散，但由此产生的HB积分器本身可以应用于某些源，例如凹形源，以产生焦散效果。与频域有限差分方法相比，所提出的HB积分器是免费的数值色散，需要更少的离散点每波长。因此，它可以解决波传播问题远远超出现有的解决方案的能力。值得注意的是，所提出的方案可以在劳伦斯伯克利国家实验室最先进的超级计算机上精确地模拟每个方向640个波长的2D域和每个方向54个波长的3D域中的波传播。
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 withdiscretization 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 allcell centers to each other. The overall CPU time and memory requirement scale asfor any bounded two-dimensional (2D) domains with arbitrary excitation sources. A direct extension of this scheme to bounded 3D domains yields anCPU 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 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-of-the-art supercomputer at Lawrence Berkeley National Laboratory.