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Accelerating iterative solvers via a two-dimensional minimum residual technique

Fatemeh P. A. Beik, Michele Benzi, Mehdi Najafi-Kalyani

TL;DR

The paper introduces the Two-Step Two-Dimensional Minimum Residual (TSTMR) method, a parameter-free acceleration for a class of two-step iterative solvers that minimizes residuals over a two-dimensional subspace at each sub-step using two matrix splittings. It provides a convergence analysis based on field-of-values conditions and extends the approach to regularized augmented systems arising from ill-posed problems, including a practical gamma-parameter selection ensuring FoV-based convergence. The method is validated through comprehensive numerical experiments on well-posed convection–diffusion and Stokes–Darcy problems, as well as ill-posed problems from regularization and tomography, showing competitive or superior performance to MRHSS, FGMRES, CGW, and hybrid LSQR approaches, often with robust, parameter-free operation. The work highlights TSTMR as an effective iterative solver and iterative regularization tool with favorable memory, convergence, and accuracy properties, and points to future study of inexact inner solves and singular-consistent systems.

Abstract

This paper deals with speeding up the convergence of a class of two-step iterative methods for solving linear systems of equations. To implement the acceleration technique, the residual norm associated with computed approximations for each sub-iterate is minimized over a certain two-dimensional subspace. Convergence properties of the proposed method are studied in detail. The approach is further developed to solve (regularized) normal equations arising from the discretization of ill-posed problems. The results of numerical experiments are reported to illustrate the performance of exact and inexact variants of the method on several test problems from different application areas.

Accelerating iterative solvers via a two-dimensional minimum residual technique

TL;DR

The paper introduces the Two-Step Two-Dimensional Minimum Residual (TSTMR) method, a parameter-free acceleration for a class of two-step iterative solvers that minimizes residuals over a two-dimensional subspace at each sub-step using two matrix splittings. It provides a convergence analysis based on field-of-values conditions and extends the approach to regularized augmented systems arising from ill-posed problems, including a practical gamma-parameter selection ensuring FoV-based convergence. The method is validated through comprehensive numerical experiments on well-posed convection–diffusion and Stokes–Darcy problems, as well as ill-posed problems from regularization and tomography, showing competitive or superior performance to MRHSS, FGMRES, CGW, and hybrid LSQR approaches, often with robust, parameter-free operation. The work highlights TSTMR as an effective iterative solver and iterative regularization tool with favorable memory, convergence, and accuracy properties, and points to future study of inexact inner solves and singular-consistent systems.

Abstract

This paper deals with speeding up the convergence of a class of two-step iterative methods for solving linear systems of equations. To implement the acceleration technique, the residual norm associated with computed approximations for each sub-iterate is minimized over a certain two-dimensional subspace. Convergence properties of the proposed method are studied in detail. The approach is further developed to solve (regularized) normal equations arising from the discretization of ill-posed problems. The results of numerical experiments are reported to illustrate the performance of exact and inexact variants of the method on several test problems from different application areas.
Paper Structure (8 sections, 8 theorems, 113 equations, 4 tables)

This paper contains 8 sections, 8 theorems, 113 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Proposed method and its convergence analysis
  3. TSTMR for discrete ill-posed problems
  4. Problem reformulation
  5. TSTMR for solving the regularized problem
  6. Numerical experiments
  7. Experimental results for two well-posed test problems
  8. Experimental results for some ill-posed test problems

Key Result

Proposition 1.1

Let $A\in \mathbb{R}^{n\times n}$, and let ${\lambda _{\max }}$ and ${\lambda _{\min }}$ be the largest and smallest eigenvalues of $\mathcal{H}(A)$. If ${\lambda _{\max }}{\lambda _{\min }} >- {\lambda _{\min }} (\mathcal{S}^T(A)\mathcal{S}(A)),$ then there exists an $\eta$ for which $\|\tilde{\mat for which the value of ${\|\tilde{\mathcal{S}}^{-1}(A)\|}_2 {\|\tilde{\mathcal{H}}^{}(A)\|}_2$ is m

Theorems & Definitions (22)

  • Proposition 1.1
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • Theorem 2.5
  • proof
  • Remark 2.6
  • ...and 12 more