On the Complexity of the Block Low-Rank Multifrontal Factorization

Abstract

Abstract. Matrices coming from elliptic Partial Differential Equations have been shown to have a lowrank property: well defined off-diagonal blocks of their Schur complements can be approximated by low-rank products and this property can be efficiently exploited in multifrontal solvers to provide a substantial reduction of their complexity. Among the possible low-rank formats, the Block Low-Rank format (BLR) is easy to use in a general purpose multifrontal solver and has been shown to provide significant gains compared to full-rank on practical applications. However, unlike hierarchical formats, such as H and HSS, its theoretical complexity was unknown. In this paper, we extend the theoretical work done on hierarchical matrices in order to compute the theoretical complexity of the BLR multifrontal factorization. We then present several variants of the BLR multifrontal factorization, depending on the strategies used to perform the updates in the frontal matrices and on the constraints on how numerical pivoting is handled. We show how these variants can further reduce the complexity of the factorization. In the best case (3D, constant ranks), we obtain a complexity of the order of O(n 4/3 ). We provide an experimental study with numerical results to support our complexity bounds.

Large-scale 3-D EM modelling with a Block Low-Rank multifrontal direct solver

“…Earlier studies on 3-D Laplacians (Amestoy et al 2015) experimentally showed that the MUMPS-BLR solver has O(N 1.65 ) complexity in the number of flops, a significant improvement from the standard O(N 2 ) complexity for FR factorization, while the experimental complexity for 3-D seismic problems was found to be in O(N 1.78 ). All these results are quite close to the theoretical prediction for complexity of the BLR factorization, O(N 1.7 ), recently computed by Amestoy et al (2017). We also performed a power regression analysis of the flops data in Table 2: they show the expected N 2 trend for the FR data, while for BLR the dependence was N m with m =1.6 ± 0.1.…”

“…This could then be used to further reduce the memory footprint of the solver but is out of the scope of this paper. As suggested by theory (Amestoy et al 2017), the block size for the BLR format should depend on the matrix size. The block size was set to 256 on almost all matrices but was increased to 416 for our largest matrix S21.…”

Sparse supernodal solver using block low-rank compression: Design, performance and analysis

“…The strategy is described in [21] and a theoretical study of the complexity of the solver for regular meshes is presented in [33]. In this work, when a front is eliminated, different strategies are proposed to enhance the time-to-solution.…”

A new impedance accounting for short‐ and long‐range effects in mixed substructured formulations of nonlinear problems

“…In practice, to recover the structure of (10), while remaining tractable, we propose the local flexibility S neigh( ) −1 t to be the inverse of a sparse matrix and the long-range flexibility S far( ) −1 t to be low rank. The latter condition is also motivated by Bebendorf and Hackbusch 39 and Amestoy et al, 40 where it is shown that low-rank approximants of fully populated inverse operators, arising from finite element discretization of elliptic problems, can be derived from the hierarchical-matrices theory. Typically, we have…”

“…In practice, to recover the structure of , while remaining tractable, we propose the local flexibility to be the inverse of a sparse matrix and the long‐range flexibility to be low rank. The latter condition is also motivated by Bebendorf and Hackbusch and Amestoy et al, where it is shown that low‐rank approximants of fully populated inverse operators, arising from finite element discretization of elliptic problems, can be derived from the hierarchical‐matrices theory. Typically, we have where can refer, for instance, to expressions or , F is a small‐sized m × m square matrix, and V is an interface vectors basis of size n A × m .…”