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On maximum residual block Kaczmarz method for solving large consistent linear systems

Wen-Ning Sun, Mei Qin

TL;DR

The maximum residual block Kaczmarz method is proposed, which is designed to preferentially eliminate the largest block in the residual vector $r_{k}$ at each iteration, and the maximum residual average block Kaczmarz method is constructed.

Abstract

For solving large consistent linear systems by iteration methods, inspired by the maximum residual Kaczmarz method and the randomized block Kaczmarz method, we propose the maximum residual block Kaczmarz method, which is designed to preferentially eliminate the largest block in the residual vector $r_{k}$ at each iteration. At the same time, in order to further improve the convergence rate, we construct the maximum residual average block Kaczmarz method to avoid the calculation of pseudo-inverse in block iteration, which completes the iteration by projecting the iteration vector $x_{k}$ to each row of the constrained subset of $A$ and applying different extrapolation step sizes to average them. We prove the convergence of these two methods and give the upper bounds on their convergence rates, respectively. Numerical experiments validate our theory and show that our proposed methods are superior to some other block Kaczmarz methods.

On maximum residual block Kaczmarz method for solving large consistent linear systems

TL;DR

The maximum residual block Kaczmarz method is proposed, which is designed to preferentially eliminate the largest block in the residual vector at each iteration, and the maximum residual average block Kaczmarz method is constructed.

Abstract

For solving large consistent linear systems by iteration methods, inspired by the maximum residual Kaczmarz method and the randomized block Kaczmarz method, we propose the maximum residual block Kaczmarz method, which is designed to preferentially eliminate the largest block in the residual vector at each iteration. At the same time, in order to further improve the convergence rate, we construct the maximum residual average block Kaczmarz method to avoid the calculation of pseudo-inverse in block iteration, which completes the iteration by projecting the iteration vector to each row of the constrained subset of and applying different extrapolation step sizes to average them. We prove the convergence of these two methods and give the upper bounds on their convergence rates, respectively. Numerical experiments validate our theory and show that our proposed methods are superior to some other block Kaczmarz methods.
Paper Structure (5 sections, 2 theorems, 34 equations, 3 figures, 3 tables, 2 algorithms)

This paper contains 5 sections, 2 theorems, 34 equations, 3 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Maximum residual block Kaczmarz method
  3. Maximum residual average block Kaczmarz method
  4. Experimental results
  5. Conclusions

Key Result

Theorem 2.1

Let the linear system eq:1.1 be consistent, for a fixed partition $\mathcal{V}=\{\mathcal{V}_1,\mathcal{V}_2,\ldots,\mathcal{V}_t\}$ of $\mathrm[m]$, starting from any initial vector $x_{0}\in\mathcal{R}(A^{T})$ , the iteration sequence $\{x_{k}\}_{k=0}^{\infty}$ generated by the MRBK method, conver and

Figures (3)

  • Figure 1: RSE versus IT (a) and RSE versus CPU (b) for different block Kaczmrarz methods when the coefficient matrices are the $6000 \times 3000$ matrix in Table \ref{['tab:1']}.
  • Figure 2: RSE versus IT (a) and RSE versus CPU (b) for different block Kaczmrarz methods when the coefficient matrices are the $3000 \times 6000$ matrix in Table \ref{['tab:2']}.
  • Figure 3: RSE versus IT (a) and RSE versus CPU (b) for different block Kaczmrarz methods on matrix Trefethen_700.

Theorems & Definitions (7)

  • Definition 2.1
  • Theorem 2.1
  • proof
  • Remark 2.1
  • Theorem 3.1
  • proof
  • Remark 3.1