## Abstract

This article presents a multilevel parallel preconditioning technique for solving general large sparse linear systems of equations. Subdomain coloring is invoked to reorder the coefficient matrix by multicoloring the adjacency graph of the subdomains, resulting in a two-level block diagonal structure. A full binary tree structure (Formula presented.) is then built to facilitate the construction of the preconditioner. A key property that is exploited is the observation that the difference between the inverse of the original matrix and that of its block diagonal approximation is often well approximated by a low-rank matrix. This property and the block diagonal structure of the reordered matrix lead to a multicolor low-rank (MCLR) preconditioner. The construction procedure of the MCLR preconditioner follows a bottom-up traversal of the tree (Formula presented.). All irregular matrix computations, such as ILU factorizations and related triangular solves, are restricted to leaf nodes where these operations can be performed independently. Computations in nonleaf nodes only involve easy-to-optimize dense matrix operations. In order to further reduce the number of iteration of the Preconditioned Krylov subspace procedure, we combine MCLR with a few classical block-relaxation techniques. Numerical experiments on various test problems are proposed to illustrate the robustness and efficiency of the proposed approach for solving large sparse symmetric and nonsymmetric linear systems.

Original language | English (US) |
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Article number | e2316 |

Journal | Numerical Linear Algebra with Applications |

Volume | 27 |

Issue number | 4 |

DOIs | |

State | Published - Aug 1 2020 |

### Bibliographical note

Funding Information:information US National Science Foundation, DMS-1521573; DMS-1912048This research was supported in part by the US National Science Foundation under grants DMS-1521573 and DMS-1912048, Shuimu Scholar of Tsinghua University and the Minnesota Supercomputing Institute.

Funding Information:

This research was supported in part by the US National Science Foundation under grants DMS‐1521573 and DMS‐1912048, Shuimu Scholar of Tsinghua University and the Minnesota Supercomputing Institute.

Publisher Copyright:

© 2020 John Wiley & Sons, Ltd.

## Keywords

- Krylov subspace methods
- domain decomposition
- low-rank approximation
- parallel preconditioners
- recursive multilevel methods