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Robust Optimal-Complexity Multilevel ILU for Predominantly Symmetric Systems

Aditi Ghai, Xiangmin Jiao

TL;DR

PS-MILU introduces a robust, multilevel incomplete LU preconditioner tailored for predominantly symmetric systems by unifying LDL$^{T}$ and LDU factorizations with diagonal pivoting, inverse-based dropping, and a hybrid Schur complement. It employs an augmented sparse storage scheme and a Crout-update-based factorization to achieve near-linear time in the input size under typical PDE discretization sparsity, while delivering substantial speedups over existing MILU variants and maintaining strong robustness. The methodology is validated on large PDE-derived matrices, showing linear-ish scaling, improved factorization speed when symmetry is exploited, and effective preconditioning for GMRES with competitive iteration counts. The work highlights PS-MILU as a practical, scalable alternative for solving large sparse linear systems arising from PDE discretizations, with avenues for extending to non-PDE applications and parallel implementations.

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.

Robust Optimal-Complexity Multilevel ILU for Predominantly Symmetric Systems

TL;DR

PS-MILU introduces a robust, multilevel incomplete LU preconditioner tailored for predominantly symmetric systems by unifying LDL and LDU factorizations with diagonal pivoting, inverse-based dropping, and a hybrid Schur complement. It employs an augmented sparse storage scheme and a Crout-update-based factorization to achieve near-linear time in the input size under typical PDE discretization sparsity, while delivering substantial speedups over existing MILU variants and maintaining strong robustness. The methodology is validated on large PDE-derived matrices, showing linear-ish scaling, improved factorization speed when symmetry is exploited, and effective preconditioning for GMRES with competitive iteration counts. The work highlights PS-MILU as a practical, scalable alternative for solving large sparse linear systems arising from PDE discretizations, with avenues for extending to non-PDE applications and parallel implementations.

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.

Paper Structure

This paper contains 20 sections, 2 theorems, 38 equations, 8 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Predominantly Symmetric Multilevel Incomplete LU-Factorization
  4. Preprocessing for PS-MILU
  5. Incomplete LDL$^{\text{T}}$ and LDU Factorization
  6. Diagonal Pivoting
  7. Inverse-Based Thresholding
  8. Hybrid Schur Complement
  9. Implementation Details and Complexity Analysis
  10. An Augmented Sparse Matrix Storage
  11. Algorithm Details and Pseudocode
  12. Overall PS-MILU Algorithm
  13. Multilevel Triangular Solves
  14. Time Complexity Analysis
  15. Numerical Study
  16. ...and 5 more sections

Key Result

Lemma 1

Given a preprocessed matrix $\hat{\boldsymbol{A}}\in\mathbb{R}^{n\times n}$, let $\boldsymbol{P}\in\mathbb{N}^{n\times n}$, $\boldsymbol{Q}\in\mathbb{N}^{n\times n}$, $\boldsymbol{L}\in\mathbb{R}^{n\times m}$, $\boldsymbol{D}\in\mathbb{R}^{m\times m}$ and $\boldsymbol{U}\in\mathbb{R}^{m\times n}$ be so is the overall time complexity of ilupd_factor, where $\boldsymbol{a}_{i}^{T}$ and $\boldsymbol{

Figures (8)

  • Figure 1: The overall workflow of PS-MILU.
  • Figure 2: Sparsity pattern of a predominantly symmetric example matrix from FEM 2D before (left) and after MC64 matching and AMD reordering.
  • Figure 3: Illustration of updating $\boldsymbol{\ell}_{k}$ in Crout update.
  • Figure 4: Illustration of diagonal pivoting for level-$1$ symmetric block. It removes "problematic" rows and columns from the current level and merge them with the block for more robust processing in the next level.
  • Figure 5: Definition of data structure Prec for each level of PS-MILU.
  • ...and 3 more figures

Theorems & Definitions (2)

  • Lemma 1
  • Proposition 2