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Preprocessed GMRES for fast solution of linear equations

Juan Zhang, Yiyi Luo

TL;DR

This work tackles fast solution of linear systems by embedding inner preconditioned iterations (RPCG, ADI, or Kaczmarz) inside BA-GMRES as an outer Krylov solver. By generating preprocessing matrices $B^{\ell_k}$ through these inner schemes, BA-GMRES achieves accelerated convergence for ill-conditioned or large-scale problems. The paper provides convergence analyses for each inner method, a complexity discussion, and extensive numerical experiments showing improved efficiency over standard GMRES in various test cases. This preprocessing-based approach offers a practical pathway to faster, more robust solutions in applications where large sparse linear systems arise.

Abstract

The article mainly introduces preprocessing algorithms for solving linear equation systems. This algorithm uses three algorithms as inner iterations, namely RPCG algorithm, ADI algorithm, and Kaczmarz algorithm. Then, it uses BA-GMRES as an outer iteration to solve the linear equation system. These three algorithms can indirectly generate preprocessing matrices, which are used for solving equation systems. In addition, we provide corresponding convergence analysis and numerical examples. Through numerical examples, we demonstrate the effectiveness and feasibility of these preprocessing methods. Furthermore, in the Kaczmarz algorithm, we introduce both constant step size and adaptive step size, and extend the parameter range of the Kaczmarz algorithm to $α\in(0,\infty)$. We also study the solution rate of linear equation systems using different step sizes. Numerical examples show that both constant step size and adaptive step size have higher solution efficiency than the solving algorithm without preprocessing.

Preprocessed GMRES for fast solution of linear equations

TL;DR

This work tackles fast solution of linear systems by embedding inner preconditioned iterations (RPCG, ADI, or Kaczmarz) inside BA-GMRES as an outer Krylov solver. By generating preprocessing matrices through these inner schemes, BA-GMRES achieves accelerated convergence for ill-conditioned or large-scale problems. The paper provides convergence analyses for each inner method, a complexity discussion, and extensive numerical experiments showing improved efficiency over standard GMRES in various test cases. This preprocessing-based approach offers a practical pathway to faster, more robust solutions in applications where large sparse linear systems arise.

Abstract

The article mainly introduces preprocessing algorithms for solving linear equation systems. This algorithm uses three algorithms as inner iterations, namely RPCG algorithm, ADI algorithm, and Kaczmarz algorithm. Then, it uses BA-GMRES as an outer iteration to solve the linear equation system. These three algorithms can indirectly generate preprocessing matrices, which are used for solving equation systems. In addition, we provide corresponding convergence analysis and numerical examples. Through numerical examples, we demonstrate the effectiveness and feasibility of these preprocessing methods. Furthermore, in the Kaczmarz algorithm, we introduce both constant step size and adaptive step size, and extend the parameter range of the Kaczmarz algorithm to . We also study the solution rate of linear equation systems using different step sizes. Numerical examples show that both constant step size and adaptive step size have higher solution efficiency than the solving algorithm without preprocessing.
Paper Structure (20 sections, 7 theorems, 63 equations, 5 figures, 6 algorithms)

This paper contains 20 sections, 7 theorems, 63 equations, 5 figures, 6 algorithms.

Table of Contents

  1. Introduction
  2. RPCG-BA-GMRES Method
  3. Introduction to BA-GMRES
  4. Convergence Analysis of BA-GMRES
  5. Introduction to RPCG Method
  6. Convergence Analysis of RPCG Method
  7. ADI-BA-GMRES Method
  8. Introduction to ADI Method
  9. Convergence Analysis of the ADI Method
  10. Kaczmarz-BA-GMRES Method
  11. Introduction to Kaczmarz Method
  12. Constant Step Size for Random Kaczmarz Method
  13. Adaptive Step of Random Kaczmarz Method
  14. Complexity Analysis
  15. Numerical Example
  16. ...and 5 more sections

Key Result

Theorem 2.1

If $A \in \mathbb{R}^{n \times n}$ and $b \in \mathbb{R}^n$ , the preconditioning matrix $B$ of $Ax=b$ is where $A = M-N$, and $k$ is any positive integer.

Figures (5)

  • Figure 1: Error comparison plot of the sparse matrix 'dwt_198'
  • Figure 2: Error comparison plot of the sparse matrix can__229
  • Figure 3: Error comparison plot of the sparse matrix 494_bus
  • Figure 4: Error comparison plot of the sparse matrix bcspwr02
  • Figure 5: Error comparison plot of the sparse matrix three diagonal matrices

Theorems & Definitions (8)

  • Theorem 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 3.1
  • Theorem 4.1
  • Theorem 4.2