A τ Matrix Based Approximate Inverse Preconditioning for Tempered Fractional Diffusion Equations

TL;DR

The paper tackles efficient solution of linear systems arising from Crank-Nicolson discretization of tempered fractional diffusion equations by exploiting the SPD Toeplitz structure of the discretization matrix $A=I+DG$. It introduces a $\tau$-matrix based approximate inverse preconditioner in a row-by-row framework, with DST-based fast application and interpolation to reduce cost to $O(lN\log N)$. Spectral analysis demonstrates that the preconditioned matrix $P_3^{-1}A$ has eigenvalues clustered near $1$, leading to rapid Krylov convergence. Numerical experiments show superior performance of the proposed preconditioner over existing approaches for both smooth and singular diffusion coefficients, validating its robustness and scalability.

Abstract

Tempered fractional diffusion equations are a crucial class of equations widely applied in many physical fields. In this paper, the Crank-Nicolson method and the tempered weighted and shifts Grünwald formula are firstly applied to discretize the tempered fractional diffusion equations. We then obtain that the coefficient matrix of the discretized system has the structure of the sum of the identity matrix and a diagonal matrix multiplied by a symmetric positive definite(SPD) Toeplitz matrix. Based on the properties of SPD Toeplitz matrices, we use $τ$ matrix approximate it and then propose a novel approximate inverse preconditioner to approximate the coefficient matrix. The $τ$ matrix based approximate inverse preconditioner can be efficiently computed using the discrete sine transform(DST). In spectral analysis, the eigenvalues of the preconditioned coefficient matrix are clustered around 1, ensuring fast convergence of Krylov subspace methods with the new preconditioner. Finally, numerical experiments demonstrate the effectiveness of the proposed preconditioner.

Figures (6)

• Figure : Fig. 5.1. Comparison of the exact solutions and $P_{TAI}^{-1}\left( 10 \right)$ numerical solutions($d=d_1\left( x \right)$, $\lambda =1.5$, $\beta =1.2$, $\gamma _1=0.75$)
• Figure : Fig. 5.2. Comparison of average iterations and computation time for $P_{TAI}^{-1}\left( l \right)$, $P_{CAI}^{-1}\left( l \right)$, $P_{DCS}^{-1}$, $P_{SDTAS_{\tau}}^{-1}$($d=d_1\left( x \right)$, $\lambda =1.5$, $\beta =1.2$, $\gamma _1=0.75$)
• Figure : Fig. 5.3. The eigenvalue distributions of the preconditioned matrices $AP_{TAI}^{-1}\left( l \right)$, $AP_{CAI}^{-1}\left( l \right)$, $AP_{DCS}^{-1}$, $AP_{SDTAS_{\tau}}^{-1}$ and the original matrix $A$($d=d_1\left( x \right)$, $\lambda =1.5$, $\beta =1.2$, $\gamma _1=0.75$)
• Figure : Fig. 5.4. Comparison of the exact solutions and $P_{TAI}^{-1}\left( l \right)$ numerical solutions($d=d_2\left( x \right)$, $\lambda =1.5$, $\beta =1.2$, $\gamma _1=0.75$)
• Figure : Fig. 5.5. Comparison of average iterations and computation time for $P_{TAI}^{-1}\left( l \right)$, $P_{CAI}^{-1}\left( l \right)$, $P_{DCS}^{-1}$, $P_{SDTAS_{\tau}}^{-1}$($d=d_1\left( x \right)$, $\lambda =1.5$, $\beta =1.2$, $\gamma _1=0.75$)

Theorems & Definitions (12)

• Lemma 2.1
• Definition 4.1
• Lemma 4.1
• Lemma 4.2
• Lemma 4.3
• Lemma 4.4
• Lemma 4.5
• Lemma 4.6
• Theorem 4.1
• Theorem 4.2