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

Sparse Approximate Multifrontal Factorization with Butterfly Compression for High Frequency Wave Equations