Based on the Toeplitz structure contained in the banded preconditioner with shift compensation (BSC preconditioner), the asymptotic singular value distribution of the BSC-preconditioned matrix for solving the space fractional diffusion equations with nonequal diffusion coefficients is analyzed by exploiting the theory of the generalized locally Toeplitz (GLT) sequence. The theoretical analysis illustrates that the conditioning of the BSC-preconditioned matrix is bounded by <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal O(h^{-\beta (2-\beta )/2})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mi>h</mi> <mrow> <mo>-</mo> <mi>β</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo>-</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> when <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation> is sufficiently close to 2, and is bounded by <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal O(h^{\beta -1})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mi>h</mi> <mrow> <mi>β</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> when <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation> is sufficiently close to 1. Numerical computation also verifies that the BSC preconditioner is robust for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation> sufficiently approaching 1 or 2.

Asymptotic Singular Value Analysis of the BSC Preconditioning for Solving Space Fractional Diffusion Equations

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发表日期：

2025年3月20日

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Xiao-Yun Zhang,
Kang-Ya Lu,
Ying Sun

摘要

Based on the Toeplitz structure contained in the banded preconditioner with shift compensation (BSC preconditioner), the asymptotic singular value distribution of the BSC-preconditioned matrix for solving the space fractional diffusion equations with nonequal diffusion coefficients is analyzed by exploiting the theory of the generalized locally Toeplitz (GLT) sequence. The theoretical analysis illustrates that the conditioning of the BSC-preconditioned matrix is bounded by $\mathcal O(h^{-\beta (2-\beta )/2})$ $O (h^{- β (2 - β) / 2})$ when $\beta $ $β$ is sufficiently close to 2, and is bounded by $\mathcal O(h^{\beta -1})$ $O (h^{β - 1})$ when $\beta $ $β$ is sufficiently close to 1. Numerical computation also verifies that the BSC preconditioner is robust for $\beta $ $β$ sufficiently approaching 1 or 2.