We present a fast and approximate multifrontal solver for large-scale sparse linear systems arising from finite-difference, finite-volume or finite-element discretization of high-frequency wave equations. The proposed solver leverages the butterfly algorithm and its hierarchical matrix extension for compressing and factorizing large frontal matrices via graph-distance guided entry evaluation or randomized matrix-vector multiplication-based schemes. Complexity analysis and numerical experiments demonstrate $\mathcal{O}(N\log^2 N)$ computation and $\mathcal{O}(N)$ memory complexity when applied to an $N\times N$ sparse system arising from 3D high-frequency Helmholtz and Maxwell problems.

中文翻译：

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用于高频波方程的带有蝴蝶压缩的稀疏近似多面分解

我们为由高频波动方程的有限差分、有限体积或有限元离散化产生的大规模稀疏线性系统提供了一种快速且近似的多前沿求解器。所提出的求解器利用蝴蝶算法及其分层矩阵扩展，通过图距离引导条目评估或基于随机矩阵向量乘法的方案来压缩和分解大型正面矩阵。复杂性分析和数值实验证明了 $\mathcal{O}(N\log^2 N)$ 计算和 $\mathcal{O}(N)$ 内存复杂性，当应用于 $N\times N$ 稀疏系统时，由 3D 产生高频亥姆霍兹和麦克斯韦问题。