IMPROVING MULTIFRONTAL METHODS BY MEANS OF BLOCK LOW-RANK REPRESENTATIONS

Matrices coming from elliptic partial differential equations have been shown to have a low-rank property: well-defined off-diagonal blocks of their Schur complements can be approximated by low-rank products. Given a suitable ordering of the matrix which gives the blocks a geometrical meaning, 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 gives efficient 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 a simple, nonhierarchical, low-rank format called block low-rank (BLR) and explain how it can be used to reduce the memory footprint and the complexity of sparse direct solvers based on the multifrontal method. We present experimental results on matrices coming from elliptic PDEs and from various other applications. We show that even if BLR-based factorizations are asymptotically less efficient than hierarchical approaches, they still deliver considerable gains. The BLR format is compatible with numerical pivoting, and its simplicity and flexibility make it easy to use in the context of a general purpose, algebraic solver.