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

doi:10.1007/s42967-025-00478-2

Communications on Applied Mathematics and Computation ›› 2026, Vol. 8 ›› Issue (3): 893-908.doi: 10.1007/s42967-025-00478-2

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

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

School of Applied Science, Beijing Information Science and Technology University, Beijing, 100192, China

收稿日期:2024-08-15 修回日期:2024-12-07 出版日期:2026-06-20 发布日期:2026-05-29
通讯作者: Kang-Ya Lu, Email: lukangya@bistu.edu.cn E-mail:lukangya@bistu.edu.cn
作者简介:Xiao-Yun Zhang, Email: zhangxy361990@163.com;Ying Sun, Email: 2023021074@bistu.edu.cn
基金资助:
K.-Y. Lu was supported by the National Natural Science Foundation of China (No. 12001048).

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

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

School of Applied Science, Beijing Information Science and Technology University, Beijing, 100192, China

Received:2024-08-15 Revised:2024-12-07 Online:2026-06-20 Published:2026-05-29
Contact: Kang-Ya Lu, Email: lukangya@bistu.edu.cn E-mail:lukangya@bistu.edu.cn
Supported by:
This research is supported by Guizhou Provincial Science and Technology Projects, China (20191161), the Natural Science Research Project of Department of Education of Guizhou Province, China (QJJ2023012), and the Research Foundation of Guizhou Minzu University, China (GZMUZK[2023]YB10).

摘要/Abstract

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 $\mathcal{O}\left(h^{-\beta(2-\beta) / 2}\right)$ when $\beta$ is sufficiently close to 2, and is bounded by $\mathcal{O}\left(h^{\beta-1}\right)$ when $\beta$ is sufficiently close to 1. Numerical computation also verifies that the BSC preconditioner is robust for $\beta$ sufficiently approaching 1 or 2.

关键词: Banded preconditioner with shift compensation (BSC preconditioner), Generalized locally Toeplitz (GLT), Asymptotic singular value distribution

Abstract: 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 $\mathcal{O}\left(h^{-\beta(2-\beta) / 2}\right)$ when $\beta$ is sufficiently close to 2, and is bounded by $\mathcal{O}\left(h^{\beta-1}\right)$ when $\beta$ is sufficiently close to 1. Numerical computation also verifies that the BSC preconditioner is robust for $\beta$ sufficiently approaching 1 or 2.

Key words: Banded preconditioner with shift compensation (BSC preconditioner), Generalized locally Toeplitz (GLT), Asymptotic singular value distribution

中图分类号:

Xiao-Yun Zhang, Kang-Ya Lu, Ying Sun. Asymptotic Singular Value Analysis of the BSC Preconditioning for Solving Space Fractional Diffusion Equations[J]. Communications on Applied Mathematics and Computation, 2026, 8(3): 893-908.

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

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

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