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A single-sided all-at-once preconditioning for linear system from a non-local evolutionary equation with weakly singular kernels

Xuelei Lin, Jiamei Dong, Sean Hon

TL;DR

This work addresses the computational challenge of solving all-at-once linear systems arising from non-local evolutionary equations with weakly singular kernels. It introduces a single-sided preconditioner based on a τ-matrix spatial proxy while keeping the temporal operator intact, enabling faster matrix-vector products and simpler implementation than two-sided schemes. The authors prove that GMRES convergence with the single-sided preconditioner is no slower than the auxiliary two-sided system and establish a uniform bound κ2 ≤ 3 for the two-sided preconditioned operator, with optimal scaling at η = √3/2. Numerical experiments in 2D corroborate the theoretical findings, showing that the single-sided approach achieves similar iteration counts but reduced CPU time and robustness across discretizations and parameters.

Abstract

{In [X. L. Lin, M. K. Ng, and Y. Zhi. {\it J. Comput. Phys.}, 434 (2021), pp. 110221] and [Y. L. Zhao, J. Wu, X. M. Gu, and H. Li. {\it Comput. Math. Appl.}, 148(2023), pp. 200--210]}, two-sided preconditioning techniques are proposed for non-local evolutionary equations, which possesses (i) mesh-size independent theoretical bound of condition number of the two-sided preconditioned matrix; (ii) small and stable iteration numbers in numerical tests. In this paper, we modify the two-sided preconditioning by multiplying the left-sided and the right-sided preconditioners together as a single-sided preconditioner. Such a single-sided preconditioner essentially derives from approximating the spatial matrix with a fast diagonalizable matrix and keeping the temporal matrix unchanged. Clearly, the matrix-vector multiplication of the single-sided preconditioning is faster to compute than that of the two-sided one, since the single-sided preconditioned matrix has a simpler structure. More importantly, we show theoretically that the single-sided preconditioned generalized minimal residual (GMRES) method has a convergence rate no worse than the two-sided preconditioned one. As a result, the one-sided preconditioned GMRES solver requires less computational time than the two-sided preconditioned GMRES solver in total. Numerical results are reported to show the efficiency of the proposed single-sided preconditioning technique.

A single-sided all-at-once preconditioning for linear system from a non-local evolutionary equation with weakly singular kernels

TL;DR

This work addresses the computational challenge of solving all-at-once linear systems arising from non-local evolutionary equations with weakly singular kernels. It introduces a single-sided preconditioner based on a τ-matrix spatial proxy while keeping the temporal operator intact, enabling faster matrix-vector products and simpler implementation than two-sided schemes. The authors prove that GMRES convergence with the single-sided preconditioner is no slower than the auxiliary two-sided system and establish a uniform bound κ2 ≤ 3 for the two-sided preconditioned operator, with optimal scaling at η = √3/2. Numerical experiments in 2D corroborate the theoretical findings, showing that the single-sided approach achieves similar iteration counts but reduced CPU time and robustness across discretizations and parameters.

Abstract

{In [X. L. Lin, M. K. Ng, and Y. Zhi. {\it J. Comput. Phys.}, 434 (2021), pp. 110221] and [Y. L. Zhao, J. Wu, X. M. Gu, and H. Li. {\it Comput. Math. Appl.}, 148(2023), pp. 200--210]}, two-sided preconditioning techniques are proposed for non-local evolutionary equations, which possesses (i) mesh-size independent theoretical bound of condition number of the two-sided preconditioned matrix; (ii) small and stable iteration numbers in numerical tests. In this paper, we modify the two-sided preconditioning by multiplying the left-sided and the right-sided preconditioners together as a single-sided preconditioner. Such a single-sided preconditioner essentially derives from approximating the spatial matrix with a fast diagonalizable matrix and keeping the temporal matrix unchanged. Clearly, the matrix-vector multiplication of the single-sided preconditioning is faster to compute than that of the two-sided one, since the single-sided preconditioned matrix has a simpler structure. More importantly, we show theoretically that the single-sided preconditioned generalized minimal residual (GMRES) method has a convergence rate no worse than the two-sided preconditioned one. As a result, the one-sided preconditioned GMRES solver requires less computational time than the two-sided preconditioned GMRES solver in total. Numerical results are reported to show the efficiency of the proposed single-sided preconditioning technique.
Paper Structure (13 sections, 10 theorems, 100 equations, 2 tables)

This paper contains 13 sections, 10 theorems, 100 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Discretization of Eqn \ref{['tsfde']} and the time-space linear system
  3. Temporal discretization
  4. Space discretization
  5. All-at-once system
  6. The $\tau$ preconditioner and the convergence behavior of GMRES solver for the preconditioned system
  7. Convergence behavior of GMRES solver for system \ref{['prewhole']}
  8. The Implementation
  9. Numerical examples
  10. Conclusion
  11. Verification of $\{w_k\}_{k\geq 0}^{(\beta)}$ arising from ccelik2012crank
  12. Verification of $\{w_k^{(\beta)}\}_{k=0}^{\infty}$ arising from meerschaert2006finite
  13. Verification of $\{w_k^{(\beta)}\}_{k=0}^{\infty}$ arising from sousaelic

Key Result

Lemma 1

For any $\beta\in(1,2)$ and any $m\geq 1$, ${\bf W}_{m}^{(\beta)}$ is a positive definite matrix.

Theorems & Definitions (14)

  • Lemma 1
  • Proposition 1
  • Lemma 2
  • Theorem 3
  • Lemma 4
  • Remark 1
  • Proposition 2
  • Lemma 5
  • Lemma 6
  • Proposition 3
  • ...and 4 more