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Obtaining the pseudoinverse solution of singular range-symmetric linear systems with GMRES-type methods

Kui Du, Jia-Jun Fan, Fang Wang

TL;DR

The paper addresses solving singular range-symmetric linear systems $A x=b$ by targeting the pseudoinverse solution $A^\dag b$. It extends a lifting strategy to GMRES and introduces RSMAR, a new Krylov method that minimizes the $A$-residual $\|A r_k\|$ within the current Krylov subspace, and shows equivalence to MINARES in the symmetric case. It proves that GMRES-type methods like GMRES, RRGMRES, RSMAR, and DGMRES terminate at the same or the pseudoinverse solution under range-symmetry, with lifting providing a practical route to $A^\dag b$ when needed. Numerical experiments indicate that RSMAR-II offers superior accuracy and robustness for singular inconsistent range-symmetric systems, outperforming established GMRES-type methods, and that MINARES is effectively equivalent to RSMAR in the symmetric setting. These results advance Krylov techniques for ill-posed and singular problems, with practical implications and available MATLAB implementations.

Abstract

It is well known that for singular inconsistent range-symmetric linear systems, the generalized minimal residual (GMRES) method determines a least squares solution without breakdown. The reached least squares solution may be or not be the pseudoinverse solution. We show that a lift strategy can be used to obtain the pseudoinverse solution. In addition, we propose a new iterative method named RSMAR (minimum $\mathbf A$-residual) for range-symmetric linear systems $\mathbf A\mathbf x=\mathbf b$. At step $k$ RSMAR minimizes $\|\mathbf A\mathbf r_k\|$ in the $k$th Krylov subspace generated with $\{\mathbf A, \mathbf r_0\}$ rather than $\|\mathbf r_k\|$, where $\mathbf r_k$ is the $k$th residual vector and $\|\cdot\|$ denotes the Euclidean vector norm. We show that RSMAR and GMRES terminate with the same least squares solution when applied to range-symmetric linear systems. We provide two implementations for RSMAR. Our numerical experiments show that RSMAR is the most suitable method among GMRES-type methods for singular inconsistent range-symmetric linear systems.

Obtaining the pseudoinverse solution of singular range-symmetric linear systems with GMRES-type methods

TL;DR

The paper addresses solving singular range-symmetric linear systems by targeting the pseudoinverse solution . It extends a lifting strategy to GMRES and introduces RSMAR, a new Krylov method that minimizes the -residual within the current Krylov subspace, and shows equivalence to MINARES in the symmetric case. It proves that GMRES-type methods like GMRES, RRGMRES, RSMAR, and DGMRES terminate at the same or the pseudoinverse solution under range-symmetry, with lifting providing a practical route to when needed. Numerical experiments indicate that RSMAR-II offers superior accuracy and robustness for singular inconsistent range-symmetric systems, outperforming established GMRES-type methods, and that MINARES is effectively equivalent to RSMAR in the symmetric setting. These results advance Krylov techniques for ill-posed and singular problems, with practical implications and available MATLAB implementations.

Abstract

It is well known that for singular inconsistent range-symmetric linear systems, the generalized minimal residual (GMRES) method determines a least squares solution without breakdown. The reached least squares solution may be or not be the pseudoinverse solution. We show that a lift strategy can be used to obtain the pseudoinverse solution. In addition, we propose a new iterative method named RSMAR (minimum -residual) for range-symmetric linear systems . At step RSMAR minimizes in the th Krylov subspace generated with rather than , where is the th residual vector and denotes the Euclidean vector norm. We show that RSMAR and GMRES terminate with the same least squares solution when applied to range-symmetric linear systems. We provide two implementations for RSMAR. Our numerical experiments show that RSMAR is the most suitable method among GMRES-type methods for singular inconsistent range-symmetric linear systems.
Paper Structure (23 sections, 8 theorems, 79 equations, 4 figures, 2 tables)

This paper contains 23 sections, 8 theorems, 79 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Pseudoinverse solution and Drazin-inverse solution
  5. Range-symmetric matrix
  6. Krylov subspaces
  7. Summary of some scalars, vectors, and matrices
  8. GMRES-type methods for singular range-symmetric linear systems
  9. GMRES and a lifting strategy
  10. RRGMRES
  11. RSMAR: An iterative method for range-symmetric linear systems
  12. Implementation based on Arnoldi process for $\mathcal{K}_k(\mathbf A,\mathbf A\mathbf r_0)$
  13. Implementation based on Arnoldi process for $\mathcal{K}_k(\mathbf A,\mathbf r_0)$
  14. DGMRES
  15. Summary of GMRES-type methods for the pseudoinvese solution
  16. ...and 8 more sections

Key Result

Theorem 1

There is at most one least squares solution in $\mathbf x_0+\mathcal{K}_{\ell-1}(\mathbf A,\mathbf r_0)$ if $\mathbf b\notin{\rm range}(\mathbf A)$, and at most one solution in $\mathbf x_0+\mathcal{K}_\ell(\mathbf A,\mathbf r_0)$ if $\mathbf b\in{\rm range}(\mathbf A)$.

Figures (4)

  • Figure 1: Residual and $\mathbf A$-residual histories for GMRES, RRGMRES, RSMAR, and DGMRES on singular linear systems generated from the matrix arising in the finite difference discretization of the boundary value problem \ref{['bvpmodel']}. Left: consistent system with b = A*rand(m*m,1). Right: inconsistent system with $\mathbf b$ being a discretization of $f(x,y)=x+y$.
  • Figure 2: Residual and $\mathbf A$-residual histories for MINRES-QLP, MINARES, and RSMAR on singular linear systems generated from the matrix bcsstm36 ($n=23052$). Left: consistent system with $\mathbf b=\mathbf A\mathbf e$. Right: inconsistent system with $\mathbf b=\mathbf e$.
  • Figure 3: Residual and $\mathbf A$-residual histories for MINRES-QLP, MINARES, and RSMAR on singular linear systems generated from the matrix zenios ($n=2873$). Left: consistent system with $\mathbf b=\mathbf A\mathbf e$. Right: inconsistent system with $\mathbf b=\mathbf e$.
  • Figure 4: Residual and $\mathbf A$-residual histories for MINRES-QLP and MINARES on singular linear systems generated from the matrix laser ($n=3002$). Left: consistent system with $\mathbf b=\mathbf A\mathbf e$. Right: inconsistent system with $\mathbf b=\mathbf e$.

Theorems & Definitions (8)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • Lemma 5
  • Theorem 6
  • Theorem 7
  • Theorem 8