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

arXiv: https://arxiv.org/abs/1510.07363 · HTML: https://mlsanity.com/paper/1510.07363/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Oct 2015.

## Built on by (most cited first, 1 of 1)

- [Factor Fitting, Rank Allocation, and Partitioning in Multilevel Low Rank Matrices](https://mlsanity.com/paper/2310.19214/) (2023)

## Abstract (arXiv)

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.