# Robust Optimal-Complexity Multilevel ILU for Predominantly Symmetric Systems

## TL;DR

This paper introduces PS-MILU, a multilevel incomplete LU factorization method optimized for predominantly symmetric systems, significantly reducing factorization time while maintaining robustness for large-scale sparse linear systems.

## Contribution

The paper presents PS-MILU, a novel multilevel ILU technique that exploits predominant symmetry to improve efficiency and robustness in solving large sparse systems from PDE discretizations.

## Key findings

- PS-MILU reduces factorization time by up to 50%.
- PS-MILU is robust for ill-conditioned systems.
- PS-MILU outperforms ILUPACK and SuperLU in speed and robustness.

## Abstract

Incomplete factorization is a powerful preconditioner for Krylov subspace methods for solving large-scale sparse linear systems. Existing incomplete factorization techniques, including incomplete Cholesky and incomplete LU factorizations, are typically designed for symmetric or nonsymmetric matrices. For some numerical discretizations of partial differential equations, the linear systems are often nonsymmetric but predominantly symmetric, in that they have a large symmetric block. In this work, we propose a multilevel incomplete LU factorization technique, called PS-MILU, which can take advantage of predominant symmetry to reduce the factorization time by up to half. PS-MILU delivers robustness for ill-conditioned linear systems by utilizing diagonal pivoting and deferred factorization. We take special care in its data structures and its updating and pivoting steps to ensure optimal time complexity in input size under some reasonable assumptions. We present numerical results with PS-MILU as a preconditioner for GMRES for a collection of predominantly symmetric linear systems from numerical PDEs with unstructured and structured meshes in 2D and 3D, and show that PS-MILU can speed up factorization by about a factor of 1.6 for most systems. In addition, we compare PS-MILU against the multilevel ILU in ILUPACK and the supernodal ILU in SuperLU to demonstrate its robustness and lower time complexity.