Multilevel Tau preconditioners for symmetrized multilevel Toeplitz systems with applications to solving space fractional diffusion equations

TL;DR

This paper develops a novel multilevel Tau-based preconditioning strategy for symmetrized multilevel Toeplitz systems arising from space Riemann–Liouville fractional diffusion equations. By transforming the nonsymmetric system to a symmetric multilevel Hankel form and employing SPD Tau preconditioners, the authors establish mesh-independent MINRES convergence and provide efficient DST-based implementations. They rigorously prove spectral clustering of the preconditioned operators into disjoint intervals containing ±1 for both second- and first-order time discretizations, and validate the theory with numerical experiments showing iteration counts that do not grow with mesh refinement. The work offers a practical, fast, and robust approach to solvers for fractional diffusion problems and lays groundwork for extensions to more general symmetrized Toeplitz systems and GMRES-based schemes.

Abstract

In this work, we develop a novel multilevel Tau matrix-based preconditioned method for a class of non-symmetric multilevel Toeplitz systems. This method not only accounts for but also improves upon an ideal preconditioner pioneered by [J. Pestana. Preconditioners for symmetrized Toeplitz and multilevel Toeplitz matrices. SIAM J. Matrix Anal. Appl., 40(3):870-887, 2019]. The ideal preconditioning approach was primarily examined numerically in that study, and an effective implementation was not included. To address these issues, we first rigorously show in this study that this ideal preconditioner can indeed achieve optimal convergence when employing the MINRES method, with a convergence rate is that independent of the mesh size. Then, building on this preconditioner, we develop a practical and optimal preconditioned MINRES method. To further illustrate its applicability and develop a fast implementation strategy, we consider solving Riemann-Liouville fractional diffusion equations as an application. Specifically, following standard discretization on the equation, the resultant linear system is a non-symmetric multilevel Toeplitz system, affirming the applicability of our preconditioning method. Through a simple symmetrization strategy, we transform the original linear system into a symmetric multilevel Hankel system. Subsequently, we propose a symmetric positive definite multilevel Tau preconditioner for the symmetrized system, which can be efficiently implemented using discrete sine transforms. Theoretically, we demonstrate that mesh-independent convergence can be achieved. In particular, we prove that the eigenvalues of the preconditioned matrix are bounded within disjoint intervals containing $\pm 1$, without any outliers. Numerical examples are provided to critically discuss the results, showcase the spectral distribution, and support the efficacy of our strategy.

Figures (6)

• Figure 1: Eigenvalues of $Y_n{\color{black} \widetilde{A}_n}$ with $\alpha_1=\alpha_2=1.1$ and $n=225$.
• Figure 2: Eigenvalues of $Y_n{\color{black} \widetilde{A}_n}$ with $\alpha_1=\alpha_2=1.1$ and $n=961$.
• Figure 3: Eigenvalues of $Y_n{\color{black} \widetilde{A}_n}$ with $\alpha_1=\alpha_2=1.5$ and $n=225$.
• Figure 4: Eigenvalues of $Y_n{\color{black} \widetilde{A}_n}$ with $\alpha_1=\alpha_2=1.5$ and $n=961$.
• Figure 5: Eigenvalues of $Y_n{\color{black} \widetilde{A}_n}$ with $\alpha_1=\alpha_2=1.9$ and $n=225$.

Theorems & Definitions (34)

• Lemma 4.1
• Corollary 1
• Lemma 4.2
• Theorem 4.1
• proof
• Corollary 2
• Lemma 4.3
• Lemma 4.4
• Lemma 4.5
• proof