On the Complexity of the Block Low-rank Multifrontal

Matrices coming from elliptic Partial Differential Equations have been shown to have a low4 rank property: well defined off-diagonal blocks of their Schur complements can be approximated by low-rank 5 products and this property can be efficiently exploited in multifrontal solvers to provide a substantial reduction 6 of their complexity. Among the possible low-rank formats, the Block Low-Rank format (BLR) is easy to use 7 in a general purpose multifrontal solver and has been shown to provide significant gains compared to full-rank 8 on practical applications. However, unlike hierarchical formats, such as H and HSS, its theoretical complexity 9 was unknown. In this paper, we extend the theoretical work done on hierarchical matrices in order to compute 10 the theoretical complexity of the BLR multifrontal factorization. We then present several variants of the BLR 11 multifrontal factorization, depending on the strategies used to perform the updates in the frontal matrices and 12 on the constraints on how numerical pivoting is handled. We show how these variants can further reduce the 13 complexity of the factorization. In the best case (3D, constant ranks), we obtain a complexity of the order of 14 O(n4/3). We provide an experimental study with numerical results to support our complexity bounds. 15