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A $τ$-preconditioner for space fractional diffusion equation with non-separable variable coefficients

Xue-Lei Lin, Michael K. Ng

TL;DR

This work tackles unsteady space-fractional diffusion equations with non-separable coefficient functions, which yield Toeplitz-like linear systems at each time step. It introduces a $\tau$-matrix based preconditioner that replaces diagonal blocks by scalars and multi-level Toeplitz blocks by $\tau$-matrices, enabling fast inversion via the fast sine transform and ensuring GMRES convergence that is independent of discretization sizes. Theoretical results show linear convergence under mild, partially Lipschitz regularity of the coefficients, extended to multi-dimensional SFDEs, with a concrete bound on the rate. Numerical experiments in 2D and 3D confirm superior performance of the proposed preconditioner over circulant and Toeplitz alternatives, demonstrating near-size-independent iteration counts as the mesh is refined and substantial reductions in CPU time.

Abstract

In this paper, we study a $τ$-matrix approximation based preconditioner for the linear systems arising from discretization of unsteady state Riesz space fractional diffusion equation with non-separable variable coefficients. The structure of coefficient matrices of the linear systems is identity plus summation of diagonal-times-multilevel-Toeplitz matrices. In our preconditioning technique, the diagonal matrices are approximated by scalar identity matrices and the Toeplitz matrices are approximated by τ-matrices (a type of matrices diagonalizable by discrete sine transforms). The proposed preconditioner is fast invertible through the fast sine transform (FST) algorithm. Theoretically, we show that the GMRES solver for the preconditioned systems has an optimal convergence rate (a convergence rate independent of discretization stepsizes). To the best of our knowledge, this is the first preconditioning method with the optimal convergence rate for the variable-coefficients space fractional diffusion equation. Numerical results are reported to demonstrate the efficiency of the proposed method.

A $τ$-preconditioner for space fractional diffusion equation with non-separable variable coefficients

TL;DR

This work tackles unsteady space-fractional diffusion equations with non-separable coefficient functions, which yield Toeplitz-like linear systems at each time step. It introduces a -matrix based preconditioner that replaces diagonal blocks by scalars and multi-level Toeplitz blocks by -matrices, enabling fast inversion via the fast sine transform and ensuring GMRES convergence that is independent of discretization sizes. Theoretical results show linear convergence under mild, partially Lipschitz regularity of the coefficients, extended to multi-dimensional SFDEs, with a concrete bound on the rate. Numerical experiments in 2D and 3D confirm superior performance of the proposed preconditioner over circulant and Toeplitz alternatives, demonstrating near-size-independent iteration counts as the mesh is refined and substantial reductions in CPU time.

Abstract

In this paper, we study a -matrix approximation based preconditioner for the linear systems arising from discretization of unsteady state Riesz space fractional diffusion equation with non-separable variable coefficients. The structure of coefficient matrices of the linear systems is identity plus summation of diagonal-times-multilevel-Toeplitz matrices. In our preconditioning technique, the diagonal matrices are approximated by scalar identity matrices and the Toeplitz matrices are approximated by τ-matrices (a type of matrices diagonalizable by discrete sine transforms). The proposed preconditioner is fast invertible through the fast sine transform (FST) algorithm. Theoretically, we show that the GMRES solver for the preconditioned systems has an optimal convergence rate (a convergence rate independent of discretization stepsizes). To the best of our knowledge, this is the first preconditioning method with the optimal convergence rate for the variable-coefficients space fractional diffusion equation. Numerical results are reported to demonstrate the efficiency of the proposed method.
Paper Structure (17 sections, 13 theorems, 172 equations, 2 tables)

This paper contains 17 sections, 13 theorems, 172 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Discretization of \ref{['rsdiffusioneq']}--\ref{['initialcondition']} and the Toeplitz-like Linear Systems
  3. The $\tau$-preconditioner for the Toeplitz-like matrix and convergence of GMRES for the preconditioned system
  4. Convergence of GMRES solver for the preconditioned system \ref{['precsystem']}
  5. The $\tau$ Preconditioner for Multi-dimension SFDE with Variable Coefficient
  6. The multi-level Toeplitz like system arising from discretization of \ref{['mdrsdiffusioneq']}--\ref{['mdinitialcondition']}
  7. The $\tau$-preconditioner for the multi-level Toeplitz like system \ref{['mdtoeplikesys']} and the implementation
  8. Convergence of GMRES for the preconditioned multi-level Toeplitz like system \ref{['mdprecsys']}
  9. Numerical Results
  10. Conclusion
  11. Verification of $\{s_k^{(\gamma)}\}_{k\geq 0}$ arising from ccelik2012crank
  12. Verification of $\{s_k^{(\gamma)}\}_{k=0}^{\infty}$ arising from meerschaert2006finite
  13. Verification of $\{s_k^{(\gamma)}\}_{k=0}^{\infty}$ arising from sousaelic
  14. Proof of Lemma \ref{['taumatprecspectralm']}
  15. Proof of Lemma \ref{['resmatlemm']}
  16. ...and 2 more sections

Key Result

lemma 1

For any $\gamma\in(1,2)$ and any $m\in\mathbb{N}^{+}$, it holds that $\tau({\bf S}_{\gamma,m})\succ {\bf O}$.

Theorems & Definitions (29)

  • lemma 1
  • proof
  • lemma 2
  • lemma 3
  • proof
  • lemma 4
  • lemma 5
  • proof
  • lemma 6
  • proof
  • ...and 19 more