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<p>This paper introduces a sine-transform-based <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\tau \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>τ</mi> </math></EquationSource> </InlineEquation>-preconditioner for a specific class of two-level Toeplitz systems and rigorously analyzes its spectral properties. We prove that the eigenvalues of the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\tau \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>τ</mi> </math></EquationSource> </InlineEquation>-preconditioned matrices are uniformly bounded within the interval [1/4,&#xa0;9/4], ensuring the linear convergence of the preconditioned conjugate gradient method. This preconditioning strategy is applied to multidimensional nonlocal diffusion models discretized into ill-conditioned multilevel Toeplitz matrices, which belong to the class of matrices under consideration. Numerical experiments demonstrate that, compablue to the conjugate gradient method and multigrid methods, the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\tau \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>τ</mi> </math></EquationSource> </InlineEquation>-preconditioned conjugate gradient method achieves comparable accuracy and convergence rates while requiring fewer iterations and lower computational costs.</p>

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Spectral Analysis of \(\tau \)-Preconditioners for Two-Level Toeplitz Systems with Applications in Nonlocal Diffusion Models

  • 期刊论文

  • 发表日期：

    2025年12月5日

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  • Jiali Zhang,
  • Tao Sun,
  • Haiwei Sun,
  • Jiwei Zhang

摘要

This paper introduces a sine-transform-based \(\tau \) τ -preconditioner for a specific class of two-level Toeplitz systems and rigorously analyzes its spectral properties. We prove that the eigenvalues of the \(\tau \) τ -preconditioned matrices are uniformly bounded within the interval [1/4, 9/4], ensuring the linear convergence of the preconditioned conjugate gradient method. This preconditioning strategy is applied to multidimensional nonlocal diffusion models discretized into ill-conditioned multilevel Toeplitz matrices, which belong to the class of matrices under consideration. Numerical experiments demonstrate that, compablue to the conjugate gradient method and multigrid methods, the \(\tau \) τ -preconditioned conjugate gradient method achieves comparable accuracy and convergence rates while requiring fewer iterations and lower computational costs.