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Hybrid CGME and TCGME algorithms for large-scale general-form regularization

Yanfei Yang

TL;DR

The paper develops two inner-outer Krylov-based hybrids, hyb-CGME and hyb-TCGME, for large-scale general-form regularization by solving the underlying ill-posed problem with a Krylov solver and applying a general-form regularization to the projected subproblems. The inner linear least-squares problems are solved with LSQR, whose conditioning improves as the iteration index grows, enabling faster convergence. A key contribution is a stopping-tolerance analysis for LSQR that guarantees the computed regularized solution matches the accuracy of the exact solution under reasonable conditions, with a practical recommendation of $tol\approx 10^{-6}$. Numerical experiments show the hybrids often achieve equal or better accuracy than JBDQR while substantially reducing computational cost, making them attractive for large-scale problems where GSVD information is crucial.

Abstract

Two new hybrid algorithms are proposed for large-scale linear discrete ill-posed problems in general-form regularization. They are both based on Krylov subspace inner-outer iterative algorithms. At each iteration, they need to solve a linear least squares problem, which is the inner least squares problem. It is proved that inner linear least squares problems, solved by LSQR, become better conditioned as k increases, so LSQR converges faster. We also prove how to choose the stopping tolerance for LSQR to guarantee that the computed and exact best regularized solutions have the same accuracy. Numerical experiments are provided to demonstrate the effectiveness and efficiency of our new hybrid algorithms, along with comparisons to the existing algorithm.

Hybrid CGME and TCGME algorithms for large-scale general-form regularization

TL;DR

The paper develops two inner-outer Krylov-based hybrids, hyb-CGME and hyb-TCGME, for large-scale general-form regularization by solving the underlying ill-posed problem with a Krylov solver and applying a general-form regularization to the projected subproblems. The inner linear least-squares problems are solved with LSQR, whose conditioning improves as the iteration index grows, enabling faster convergence. A key contribution is a stopping-tolerance analysis for LSQR that guarantees the computed regularized solution matches the accuracy of the exact solution under reasonable conditions, with a practical recommendation of . Numerical experiments show the hybrids often achieve equal or better accuracy than JBDQR while substantially reducing computational cost, making them attractive for large-scale problems where GSVD information is crucial.

Abstract

Two new hybrid algorithms are proposed for large-scale linear discrete ill-posed problems in general-form regularization. They are both based on Krylov subspace inner-outer iterative algorithms. At each iteration, they need to solve a linear least squares problem, which is the inner least squares problem. It is proved that inner linear least squares problems, solved by LSQR, become better conditioned as k increases, so LSQR converges faster. We also prove how to choose the stopping tolerance for LSQR to guarantee that the computed and exact best regularized solutions have the same accuracy. Numerical experiments are provided to demonstrate the effectiveness and efficiency of our new hybrid algorithms, along with comparisons to the existing algorithm.
Paper Structure (5 sections, 6 theorems, 69 equations, 2 figures, 3 tables, 4 algorithms)

This paper contains 5 sections, 6 theorems, 69 equations, 2 figures, 3 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. CGME and truncated CGME algorithms
  3. New hybrid algorithms
  4. The stopping tolerance $tol$ of LSQR for solving the inner least squares
  5. Numerical experiments

Key Result

Theorem 1

For the rank-$k$ approximations $P_k{B}_kQ_k^T$ and $P_{k+1}{C}_kQ_{k+1}^T$ to $A$, $k=1,2,\dots,n-1,$ we have where ${\theta}_{k+1}^{k+1}$ is the smallest singular value of ${B}_{k+}$, $\gamma_k^{cgme}=\|A-P_k{B}_kQ_k^T\|, \gamma_k^{tcgme}=\|A-P_{k+1}{C}_kQ_{k+1}^T\|, \gamma_k^{lsqr}=\|A-P_{k+1}B_{k+}Q_k^T\|$, and $\gamma_0^{lsqr}=\|A\|$.

Figures (2)

  • Figure 1: The relative error of hyb-CGME, JBDQR and hyb-TCGME with $\varepsilon=10^{-2}$: (a) blur440; (b) grain; (c) mri; (d) satellite.
  • Figure 2: The exact images and the reconstructed images for the four two dimensional test problems with $\varepsilon=10^{-2}$ and $L$ defined in (\ref{['l3']}): (a) blur440; (b) mri; (c) grain; (d) satellite.

Theorems & Definitions (8)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Remark 2
  • Theorem 3
  • Lemma 1
  • Theorem 4
  • Theorem 5