Multilevel Circulant Preconditioner for High-Dimensional Fractional Diffusion Equations

Lei, Siu‐Long; Chen, Xu; Zhang, Xinhe

doi:10.4208/eajam.060815.180116a

Cited by 30 publications

(27 citation statements)

References 58 publications

(70 reference statements)

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“…Unlike the preconditioner proposed in [6,17], the level-2 circulant preconditioner here employs circulant approximations not only on the discretized matrices of fractional derivatives, but also on the discretized matrices of the two compact operators. To make sure the preconditioner is well-defined, let us show that S is nonsingular.…”

Section: Gmres Methods With Level-2 Circulant Preconditionermentioning

confidence: 98%

“…Note that the total storage requirement of the preconditioned GMRES (PGMRES) method still keep O(M 1 M 2 ) and the total operation cost is still O(M 1 M 2 log(M 1 M 2 )) due to the calculation of S −1 v, for any real vector v, can be done via two-dimensional FFT [17].…”

Section: Gmres Methods With Level-2 Circulant Preconditionermentioning

confidence: 99%

“…One should notice that the scheme under consideration is a compact scheme. Preconditioners proposed in previous works [6,17] may not be applied directly. Further treatment are needed.…”

Section: Introductionmentioning

confidence: 96%

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High order finite difference method for time-space fractional differential equations with Caputo and Riemann-Liouville derivatives

et al. 2015

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We consider high order finite difference methods for two-dimensional fractional differential equations with temporal Caputo and spatial Riemann-Liouville derivatives in this paper. We propose a scheme and show that it converges with second order in time and fourth order in space. The accuracy of our proposed method can be improved by Richardson extrapolation. Approximate solution is obtained by the generalized minimal residual (GMRES) method. A preconditioner is proposed to improve the efficiency for the implementation of the GMRES method.Keywords Two-dimensional fractional differential equation · High order difference scheme · Discrete energy method · Preconditioned GMRES method Mathematics Subject Classification (2010) 35R11 · 65M06 · 65M12 · 65M15

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Section: Gmres Methods With Level-2 Circulant Preconditionermentioning

confidence: 98%

Section: Gmres Methods With Level-2 Circulant Preconditionermentioning

confidence: 99%

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High order finite difference method for time-space fractional differential equations with Caputo and Riemann-Liouville derivatives

et al. 2015

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“…Other testing preconditioners for (7) and (23) are listed as follows. The circulant preconditioner [17] and the multilevel circulant preconditioner [15] can be used to precondition (7) and (23), respectively. For convenience, we use C to denote the (multilevel) circulant preconditioner.…”

Section: Verification Of Scheme Frommentioning

confidence: 99%

A Splitting Preconditioner for Toeplitz-Like Linear Systems Arising from Fractional Diffusion Equations

Lin

Ng²,

Sun

2017

SIAM J. Matrix Anal. & Appl.

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In this paper, we study Toeplitz-like linear systems arising from time-dependent onedimensional and two-dimensional Riesz space-fractional diffusion equations with variable diffusion coefficients. The coefficient matrix is a sum of a scalar identity matrix and a diagonal-times-Toeplitz matrix which allows fast matrix-vector multiplication in iterative solvers. We propose and develop a splitting preconditioner for this kind of matrix and analyze the spectra of the preconditioned matrix. Under mild conditions on variable diffusion coefficients, we show that the singular values of the preconditioned matrix are bounded above and below by positive constants which are independent of temporal and spatial discretization step-sizes. When the preconditioned conjugate gradient squared method is employed to solve such preconditioned linear systems, the method converges linearly within an iteration number independent of the discretization step-sizes. Numerical examples are given to illustrate the theoretical results and demonstrate that the performance of the proposed preconditioner is better than other tested solvers.

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“…The circulant preconditioners [35,41,42] proved to be good choices to accelerate the convergence of Krylov subspace methods when solving the discretized linear systems. In high-dimensional cases, a nonsingular multilevel circulant preconditioner was proposed by Lei et al [43] to accelerate the convergence of Krylov subspace methods efficiently. In [44], Chou et al illustrated the efficiency of applying an approximate inverse preconditioner to the high dimensional FDEs when Krylov subspace methods are employed.…”

mentioning

confidence: 99%

Fast Second-Order Accurate Difference Schemes for Time Distributed-Order and Riesz Space Fractional Diffusion Equations

Jian¹,

Huang²,

Zhao³

et al. 2019

jaac

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The aim of this paper is to develop fast second-order accurate difference schemes for solving one-and two-dimensional time distributed-order and Riesz space fractional diffusion equations. We adopt the same measures for one-and two-dimensional problems as follows: we first transform the time distributed-order fractional diffusion problem into the multi-term time-space fractional diffusion problem with the composite trapezoid formula. Then, we propose a second-order accurate difference scheme based on the interpolation approximation on a special point to solve the resultant problem. Meanwhile, the unconditional stability and convergence of the new difference scheme in L 2 -norm are proved. Furthermore, we find that the discretizations lead to a series of Toeplitz systems which can be efficiently solved by Krylov subspace methods with suitable circulant preconditioners. Finally, numerical results are presented to show the effectiveness of the proposed difference methods and demonstrate the fast convergence of our preconditioned Krylov subspace methods.

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Multilevel Circulant Preconditioner for High-Dimensional Fractional Diffusion Equations

Cited by 30 publications

References 58 publications

High order finite difference method for time-space fractional differential equations with Caputo and Riemann-Liouville derivatives

High order finite difference method for time-space fractional differential equations with Caputo and Riemann-Liouville derivatives

A Splitting Preconditioner for Toeplitz-Like Linear Systems Arising from Fractional Diffusion Equations

Fast Second-Order Accurate Difference Schemes for Time Distributed-Order and Riesz Space Fractional Diffusion Equations

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