A Fast Block Low-Rank Dense Solver with Applications to Finite-Element Matrices

1. Abstract. This article presents a fast dense solver for hierarchically off-diagonal low-rank (HODLR) matrices. This solver uses algebraic techniques such as the adaptive cross approximation (ACA) algorithm to construct the low-rank approximation of the off-diagonal matrix blocks. This allows us to apply the solver to any dense matrix that has an off-diagonal low-rank structure without any prior knowledge of the problem. Using this solver, we propose an algorithm to lower the computational cost of the multifrontal sparse solve process for finite-element matrices arising out of 2D elliptic PDEs. This algorithm relies on the fact that dense “frontal” matrices that arise from the sparse elimination process can be efficiently represented as a hierarchically off-diagonal low-rank (HODLR) matrix. We also present an extended review of the literature on fast direct solvers for dense and sparse matrices.