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On the extended randomized multiple row method for solving linear least-squares problems

Nian-Ci Wu, Chengzhi Liu, Yatian Wang, Qian Zuo

TL;DR

An extended randomized multiple row-action method to solve a given overdetermined and inconsistent linear system and it is proved that the proposed method can linearly converge in the mean square to the least-squares solution with a minimum Euclidean norm.

Abstract

The randomized row method is a popular representative of the iterative algorithm because of its efficiency in solving the overdetermined and consistent systems of linear equations. In this paper, we present an extended randomized multiple row method to solve a given overdetermined and inconsistent linear system and analyze its computational complexities at each iteration. We prove that the proposed method can linearly converge in the mean square to the least-squares solution with a minimum Euclidean norm. Several numerical studies are presented to corroborate our theoretical findings. The real-world applications, such as image reconstruction and large noisy data fitting in computer-aided geometric design, are also presented for illustration purposes.

On the extended randomized multiple row method for solving linear least-squares problems

TL;DR

An extended randomized multiple row-action method to solve a given overdetermined and inconsistent linear system and it is proved that the proposed method can linearly converge in the mean square to the least-squares solution with a minimum Euclidean norm.

Abstract

The randomized row method is a popular representative of the iterative algorithm because of its efficiency in solving the overdetermined and consistent systems of linear equations. In this paper, we present an extended randomized multiple row method to solve a given overdetermined and inconsistent linear system and analyze its computational complexities at each iteration. We prove that the proposed method can linearly converge in the mean square to the least-squares solution with a minimum Euclidean norm. Several numerical studies are presented to corroborate our theoretical findings. The real-world applications, such as image reconstruction and large noisy data fitting in computer-aided geometric design, are also presented for illustration purposes.
Paper Structure (12 sections, 6 theorems, 64 equations, 5 figures, 4 tables, 3 algorithms)

This paper contains 12 sections, 6 theorems, 64 equations, 5 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preparation
  3. The extended randomized multiple row method
  4. The proposed method
  5. Convergence analysis
  6. Computational complexity
  7. Experimental results
  8. Synthetic data
  9. Image reconstruction
  10. Noisy data fitting
  11. Conclusions
  12. Appendix

Key Result

Theorem 2.1

\newlabelLemmaRMR For any given consistent linear system $A{\bm x}={\bm b}$, let ${\bm x}^{(k)}$ be the vector generated by Algorithm alg:RMR with ${\bm x}^{(0)}$ being in the space $\mathcal{R}(A^T)$. It holds that where the constant $\rho_1$ is defined by formula eq:two_para_rho.

Figures (5)

  • Figure 4.1: The convergence behaviors of RSE versus IT given by RMR for Example \ref{['ERMR:example1']} with $n=100$ (left) and $200$ (right) when the noise levels $\delta = 0$, $0.01$, and $0.1$.
  • Figure 4.2: The convergence behaviors of RSE versus IT (left) and Time(s) (right) given by ERMR, REABK, and GEK for Example \ref{['ERMR:example1']} with various $m$, $n$, $r$, and $\kappa$ when the noise level $\delta=0.1$.
  • Figure 4.3: Images of the exact phantom (a), and the approximate solutions obtained by REABK (b) and ERMR (c) for the seismic travel-time tomography.
  • Figure 4.4: Images of the exact phantom (a), and the approximate solutions obtained by REABK (b) and ERMR (c) for the spherical Radon transform tomography.
  • Figure 4.5: The initial data points (a) and the limit curves given by GEK(b), REABK(c), and ERMR(d) for Example \ref{['ERMR:example2']} when $n=200$ and $m=3000$.

Theorems & Definitions (16)

  • Theorem 2.1
  • Theorem 2.2
  • Remark 3.1
  • Remark 3.2
  • Theorem 3.1
  • proof
  • Remark 3.3
  • Theorem 3.2
  • proof
  • Example 4.1
  • ...and 6 more