Clement Weisbecker

Improving multifrontal methods by means of block low-rank representations

Matrices coming from elliptic Partial Differential Equations (PDEs) 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 to 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 algebra operations. Several strategies have been proposed to exploit this property. We propose a low-rank format called Block Low-Rank (BLR), and explain how it can be used to reduce the memory footprint and the complexity of direct solvers for sparse matrices based on the multifrontal method. We present experimental results that show how the BLR format delivers gains that are comparable to those obtained with hierarchical formats such as Hierarchical matrices (H matrices) and Hierarchically Semi-Separable (HSS matrices) but provides much greater flexibility and ease of use which are essential in the context of a general purpose, algebraic solver.

Shared-Memory Parallelism and Low-Rank Approximation Techniques Applied to Direct Solvers in FEM Simulation

In this paper, the performance of a parallel sparse direct solver on a shared-memory multicore system is presented. Large size test matrices arising from finite element simulation of induction heating industrial applications are used to evaluate the performance improvements due to low-rank representations and multicore parallelization.