Superfast Algorithm for Computing Arbitrary Entries of the Inverse of a Sparse Matrix with Application to the Hessian in Time-harmonic Fwi

Abstract. We extend a structured selected inversion method to the extraction of any arbitrary entry of the inverse of a large sparse matrix A. For various discretized PDEs such as Helmholtz equations, a structured factorization yields a sequence of data-sparse factors of about O(n) nonzero entries, where n is the matrix size. We are then able to extract any arbitrary entry of A 1 in about O(log2 n) flops for both two and three dimensions. On the contrary, even the latest developments are either too expensive or have severe di culties in this. Our method also uses ULV factorizations instead of explicit hierarchically semiseparable inversions to enhance the stability and accuracy. We fully take advantage of the structures (e.g., common nested bases in the blocks) to achieve the high e ciency. The method can have a substantial impact on Gauss-Newton iterations, where preliminary studies of the Hessian matrices are made. Numerical tests indicate significant advantages over exact inversions.