Improving Multifrontal Methods by Means of Block Low-Rank Representations

International audienceMatrices coming from elliptic partial differential equations have been shown to have alow-rank property: well-defined off-diagonal blocks of their Schur complements can be approximatedby low-rank products. Given a suitable ordering of the matrix which gives the blocks a geometricalmeaning, such approximations can be computed using an SVD or a rank-revealing QR factorization.The resulting representation offers a substantial reduction of the memory requirement and givesefficient ways to perform many of the basic dense linear algebra operations. Several strategies,mostly based on hierarchical formats, have been proposed to exploit this property. We study asimple, nonhierarchical, low-rank format called block low-rank (BLR) and explain how it can be usedto reduce the memory footprint and the complexity of sparse direct solvers based on the multifrontalmethod. We present experimental results on matrices coming from elliptic PDEs and from variousother applications. We show that even if BLR-based factorizations are asymptotically less efficientthan hierarchical approaches, they still deliver considerable gains. The BLR format is compatiblewith numerical pivoting, and its simplicity and flexibility make it easy to use in the context of ageneral purpose, algebraic solver