Multilevel Tau preconditioners for symmetrized multilevel Toeplitz systems with applications to solving space fractional diffusion equations (2407.19386v2)

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.

• The paper introduces a novel technique that transforms non-symmetric Toeplitz matrices into symmetric Hankel systems.
• It employs a multilevel Tau preconditioner using discrete sine transforms, reducing computational overhead with mesh-independent convergence.
• Numerical tests confirm robust performance over varying fractional orders and spatial resolutions, outperforming traditional preconditioners.

Overview of Multilevel Tau Preconditioners for Space-Fractional Diffusion Equations

The discussed paper introduces a new preconditioning technique leveraging multilevel Tau preconditioners. This method specifically addresses the computational challenges associated with solving non-symmetric multilevel Toeplitz systems derived from space-fractional diffusion equations.

Space-fractional diffusion equations model anomalous diffusion or transport phenomena and often result in complex linear systems upon discretization. The paper outlines a method to transform these systems into symmetric multilevel Hankel systems, optimizing their solution using an innovative symmetric positive definite multilevel Tau preconditioner.

Key Methodological Contributions

1. Transformation to Symmetric Systems: The core advancement in this research is the transformation of non-symmetric multilevel Toeplitz matrices into symmetric multilevel Hankel matrices via symmetrization. This transformation paves the way for employing more efficient solution methods by leveraging the properties of symmetric matrices.
2. Tau Preconditioning: By using a multilevel Tau preconditioner, the paper proposes an effective strategy to address the computational inefficiencies commonly encountered with traditional preconditioners like circulant and band-Toeplitz. The Tau preconditioner can be efficiently implemented using discrete sine transforms, ensuring reduced computational overhead with mesh-independent convergence rates.
3. Validation and Theoretical Assurance: Theoretical analysis demonstrates that the eigenvalues of the preconditioned matrices are constrained within disjoint intervals including $\pm 1$. This property is crucial for ensuring effective preconditioning, which leads to the superior performance of the minimal residual (MINRES) method.
4. Robust Numerical Comparisons: Numerical experiments validate the proposed preconditioner’s efficacy compared to renowned alternatives, particularly highlighting stability in iteration numbers as problem sizes increase, irrespective of fractional orders.

Numerical Findings

Extensive numerical tests, encompassing variants of fractional orders and spatial resolutions, portray the preconditioner's superior performance in terms of both iterative efficiency and CPU time. The tests underscore its stability across varying spatial discretizations while maintaining robustness against diverse fractional derivatives.

Interestingly, as fractional orders approach their maximal limit, the system's preconditioning and convergence further improve, aligning with theoretical predictions. These improvements contrast with multigrid or band-Toeplitz preconditioners, which reveal limitations especially in higher-dimensional cases.

Implications and Future Directions

This innovative preconditioning approach holds significant promise for computational problems involving space-fractional differential equations, substantially reducing the computational effort and enabling handling of large-scale systems efficiently. The insights gained from this paper provide a foundation for extending preconditioning strategies to other complex systems characterized by similar mathematical structures.

Intriguingly, the paper's methods may be applicable beyond fractional diffusion equations, offering potential adaptation for other types of non-symmetric multilevel systems. Further research could expand on this work by exploring the integration of Tau preconditioners with emerging computational paradigms, such as parallel computing frameworks, to enhance real-time applications further.

In conclusion, this paper gives rise to a compelling and effective method for addressing the challenging task of solving space-fractional diffusion equations. It sets a benchmark for future studies in the domain of numerically solving complex partial differential equations with a focus on efficiency and scalability.