Butterfly Factorization Via Randomized Matrix-Vector Multiplications

This paper presents an adaptive randomized algorithm for computing the butterfly factorization of $m\times n$ matrix with $m\approx provided that both and its transpose can be rapidly applied to arbitrary vectors. The resulting is composed $\mathcal{O}(\log n)$ sparse factors, each containing $\mathcal{O}(n)$ nonzero entries. attained using $\mathcal{O}(n^{3/2}\log computation $\mathcal{O}(n\log memory resources. proposed implemented in parallel apply matrices strong or weak admissibility conditions arising from surface integral equation solvers as well multi-frontal-based finite-difference, finite-element, finite-volume solvers. A distributed-memory implementation demonstrates excellent scaling behavior.