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The GMRES method for solving the large indefinite least squares problem via an accelerated preconditioner

Jun Li, Lingsheng Meng

TL;DR

The paper tackles large-indefinite least squares (ILS) problems by transforming the normal equations into sparse block $3\times3$ systems and solving them with GMRES under a novel accelerated block preconditioner. A new block splitting $\mathcal{A}=\mathcal{P}-\mathcal{Q}$ with a parameter $\alpha>0$ yields an iteration that converges for $0<\alpha<\tfrac{1}{2}\lambda_{\min}(A_1^TA_1-A_2^TA_2)$, and its preconditioned operator $\mathcal{P}^{-1}\mathcal{A}$ has real eigenvalues clustering at $1$ as $\alpha\to0_+$. Theoretical results are supported by numerical experiments showing that the proposed preconditioner outperforms existing options (BS$_2$ and BUT) in terms of iteration counts and CPU time across Tolosa-based and TLS-like tests. This work enhances the practicality of Krylov subspace methods for large sparse ILS problems by achieving faster convergence through spectral clustering of the preconditioned system.

Abstract

In this research, to solve the large indefinite least squares problem, we firstly transform its normal equation into a sparse block three-by-three linear systems, then use GMRES method with an accelerated preconditioner to solve it. The construction idea of the preconditioner comes from the thought of Luo et.al [Luo, WH., Gu, XM., Carpentieri, B., BIT 62, 1983-2004(2022)], and the advantage of this is that the preconditioner is closer to the coefficient matrix of the block three-by-three linear systems when the parameter approachs zero. Theoretically, the iteration method under the preconditioner satisfies the conditional convergence, and all eigenvalues of the preconditioned matrix are real numbers and gathered at point $(1,0)$ as parameter is close to $0$. In the end, numerical results reflect that the theoretical results is correct and the proposed preconditioner is effective by comparing with serval existing preconditioners.

The GMRES method for solving the large indefinite least squares problem via an accelerated preconditioner

TL;DR

The paper tackles large-indefinite least squares (ILS) problems by transforming the normal equations into sparse block systems and solving them with GMRES under a novel accelerated block preconditioner. A new block splitting with a parameter yields an iteration that converges for , and its preconditioned operator has real eigenvalues clustering at as . Theoretical results are supported by numerical experiments showing that the proposed preconditioner outperforms existing options (BS and BUT) in terms of iteration counts and CPU time across Tolosa-based and TLS-like tests. This work enhances the practicality of Krylov subspace methods for large sparse ILS problems by achieving faster convergence through spectral clustering of the preconditioned system.

Abstract

In this research, to solve the large indefinite least squares problem, we firstly transform its normal equation into a sparse block three-by-three linear systems, then use GMRES method with an accelerated preconditioner to solve it. The construction idea of the preconditioner comes from the thought of Luo et.al [Luo, WH., Gu, XM., Carpentieri, B., BIT 62, 1983-2004(2022)], and the advantage of this is that the preconditioner is closer to the coefficient matrix of the block three-by-three linear systems when the parameter approachs zero. Theoretically, the iteration method under the preconditioner satisfies the conditional convergence, and all eigenvalues of the preconditioned matrix are real numbers and gathered at point as parameter is close to . In the end, numerical results reflect that the theoretical results is correct and the proposed preconditioner is effective by comparing with serval existing preconditioners.

Paper Structure

This paper contains 5 sections, 2 theorems, 29 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. A new block splitting method
  3. Theoretical results
  4. Numerical experiments
  5. Conclusions

Key Result

Theorem 3.1

Solving the block linear systems (1.3) to obtain the solution of the ILS problem (1.1). The new iteration method denotes in (2.2). Then the iteration method is convergent if where $S=A_1^{T}A_1-A_2^{T}A_2 \succ 0$.

Figures (2)

  • Figure 1: $\alpha$ vs IT and CPU of $\mathcal{P}$ preconditioned GMRES method under TOLS340 in Example 4.1.
  • Figure 2: The eigenvalue distribution of the initial coefficient matrices $\mathcal{\hat{A}}$, $\mathcal{A}$ of (\ref{['1.3']}), (\ref{['1.4']}) and the preconditioned matrices under different preconditioners in example 4.2 with $p=524,q=500,n=512$.

Theorems & Definitions (8)

  • Remark 2.1
  • Theorem 3.1
  • Remark 3.1
  • Theorem 3.2
  • Remark 3.2
  • Remark 3.3
  • Example 4.1
  • Example 4.2