Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A preconditioned iteration method for solving saddle point problems

Juan Zhang, Yiyi Luo

TL;DR

The paper tackles efficient solution of large-scale saddle point problems by introducing the PSLR preconditioner, which blends a power-series based approximate inverse of the Schur complement with a low-rank correction to avoid nested dissection. It establishes convergence criteria for the power-series approach, provides error bounds for the approximate inverse, and analyzes the computational complexity of the PSLR-GMRES framework. Through extensive numerical experiments on symmetric and asymmetric systems (including Stokes-type discretizations), the results show improved convergence rates and reduced computational effort compared to standard Krylov methods like CG, PCG, and ADI in many settings. The work demonstrates a practical, scalable approach for preconditioning saddle-point systems, with potential impact on constrained optimization and fluid dynamics simulations.

Abstract

This paper introduces a preconditioned method designed to comprehensively address the saddle point system with the aim of improving convergence efficiency. In the preprocessor construction phase, a technical approach for solving the approximate inverse matrix of sparse matrices is presented. The effectiveness of the proposed method is demonstrated through numerical examples, emphasizing its efficacy in approximating the inverse matrix. Furthermore, the preprocessing technology includes a low-rank processing step, effectively reducing algorithmic complexity. Numerical experiments validate the effectiveness and feasibility of PSLR-GMRES in solving the saddle point system.

A preconditioned iteration method for solving saddle point problems

TL;DR

The paper tackles efficient solution of large-scale saddle point problems by introducing the PSLR preconditioner, which blends a power-series based approximate inverse of the Schur complement with a low-rank correction to avoid nested dissection. It establishes convergence criteria for the power-series approach, provides error bounds for the approximate inverse, and analyzes the computational complexity of the PSLR-GMRES framework. Through extensive numerical experiments on symmetric and asymmetric systems (including Stokes-type discretizations), the results show improved convergence rates and reduced computational effort compared to standard Krylov methods like CG, PCG, and ADI in many settings. The work demonstrates a practical, scalable approach for preconditioning saddle-point systems, with potential impact on constrained optimization and fluid dynamics simulations.

Abstract

This paper introduces a preconditioned method designed to comprehensively address the saddle point system with the aim of improving convergence efficiency. In the preprocessor construction phase, a technical approach for solving the approximate inverse matrix of sparse matrices is presented. The effectiveness of the proposed method is demonstrated through numerical examples, emphasizing its efficacy in approximating the inverse matrix. Furthermore, the preprocessing technology includes a low-rank processing step, effectively reducing algorithmic complexity. Numerical experiments validate the effectiveness and feasibility of PSLR-GMRES in solving the saddle point system.
Paper Structure (22 sections, 6 theorems, 55 equations, 5 figures, 10 tables, 3 algorithms)

This paper contains 22 sections, 6 theorems, 55 equations, 5 figures, 10 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Overview of PSLR pretreatment methods
  3. Low-rank Matrix Correction
  4. Error Analysis of Approximate Inverse Matrix
  5. Pinv method
  6. Analysis of the Result of Power Series Expansion
  7. Convergence Analysis of Power Series
  8. Complexity Analysis of PSLR-GMRES Method
  9. Establishing the PSLR preprocessor
  10. Complexity of PSLR-GMRES Method
  11. Experimental results
  12. Example 1
  13. The Impact of Different Types of Initial Values
  14. The Influence of Different Orders on Solution Efficiency
  15. Example 2
  16. ...and 7 more sections

Key Result

Theorem 2.2

For any matrix norm $\|\cdot\|$, the approximate accuracy of the $S^{-1}$ application program satisfies the following inequality, where

Figures (5)

  • Figure 1: whether or no low-rank processed eigenvalue plots
  • Figure 2: Graph of the change of matrix eigenvalues with the number of unfolded items
  • Figure 3: The effect of different orders on CPU time
  • Figure 4: Influence of different power series expansion terms $m$ on saddle point matrix solving
  • Figure 5: Three methods error trend chart

Theorems & Definitions (8)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Definition 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Theorem 3.5