Fast hierarchical solvers for sparse matrices using extended sparsification and low-rank approximation

TL;DR

The paper introduces a novel algebraic sparse matrix solver that combines extended sparsification and low-rank approximation, achieving linear complexity and versatile use as a direct solver or preconditioner.

Contribution

It presents a new fully algebraic hierarchical solver based on Gauss elimination with low-rank approximations, offering linear complexity and tunable accuracy.

Findings

Achieves linear complexity in sparse matrix solving.

Can be used as a stand-alone solver or as a preconditioner.

Provides controlled accuracy through tunable parameters.

Abstract

Inversion of sparse matrices with standard direct solve schemes is robust, but computationally expensive. Iterative solvers, on the other hand, demonstrate better scalability; but, need to be used with an appropriate preconditioner (e.g., ILU, AMG, Gauss-Seidel, etc.) for proper convergence. The choice of an effective preconditioner is highly problem dependent. We propose a novel fully algebraic sparse matrix solve algorithm, which has linear complexity with the problem size. Our scheme is based on the Gauss elimination. For a given matrix, we approximate the LU factorization with a tunable accuracy determined a priori. This method can be used as a stand-alone direct solver with linear complexity and tunable accuracy, or it can be used as a black-box preconditioner in conjunction with iterative methods such as GMRES. The proposed solver is based on the low-rank approximation of fill-ins generated during the elimination. Similar to H-matrices, fill-ins corresponding to blocks that are well-separated in the adjacency graph are represented via a hierarchical structure. The linear complexity of the algorithm is guaranteed if the blocks corresponding to well-separated clusters of variables are numerically low-rank.