A Fast and Memory Efficient Sparse Solver with Applications to Finite-Element Matrices

TL;DR

This paper presents a novel sparse matrix solver that combines a fast approximate multifrontal method with iterative refinement, achieving significant speed and memory improvements for finite element matrices from elliptic PDEs.

Contribution

The authors introduce a hybrid solver using hierarchical low-rank matrix representations to accelerate multifrontal elimination, outperforming traditional direct and preconditioned iterative methods.

Findings

Approximately 2x faster than conventional direct solvers.

Memory usage reduced by 2-3x compared to traditional methods.

Outperforms incomplete LU preconditioners in speed and effectiveness.

Abstract

In this article, we introduce a fast and memory efficient solver for sparse matrices arising from the finite element discretization of elliptic partial differential equations (PDEs). We use a fast direct (but approximate) multifrontal solver as a preconditioner, and use an iterative solver to achieve a desired accuracy. This approach combines the advantages of direct and iterative schemes to arrive at a fast, robust and accurate solver. We will show that this solver is faster ($\sim$ 2x) and more memory efficient ($\sim$ 2--3x) than a conventional direct multifrontal solver. Furthermore, we will demonstrate that the solver is both a faster and more effective preconditioner than other preconditioners such as the incomplete LU preconditioner. Specific speed-ups depend on the matrix size and improve as the size of the matrix increases. The solver can be applied to both structured and unstructured meshes in a similar manner. We build on our previous work and utilize the fact that dense frontal and update matrices, in the multifrontal algorithm, can be represented as hierarchically off-diagonal low-rank (HODLR) matrices. Using this idea, we replace all large dense matrix operations in the multifrontal elimination process with $O(N)$ HODLR operations to arrive at a faster and more memory efficient solver.