On<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si43.gif" display="inline" overflow="scroll"><mml:mi>k</mml:mi></mml:math>-step CSCS-based polynomial preconditioners for Toeplitz linear systems with application to fractional diffusion equations

Gu, Xian-Ming; Huang, Ting-Zhu; Li, Hong; Li, Liang; Luo, Wei-Hua

doi:10.1016/j.aml.2014.11.005

Cited by 23 publications

(12 citation statements)

References 19 publications

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“…The conventional time-stepping schemes utilizing the Gaussian elimination require the computational cost of O(N 3 ) and storage of O(N 2 ) at each time step, where N is the spatial grid number. For the purpose of optimizing the computational complexity, numerous fast algorithms [6,8,15,[17][18][19][20][21] are designed.…”

Section: Introductionmentioning

confidence: 99%

A limited-memory block bi-diagonal Toeplitz preconditioner for block lower triangular Toeplitz system from time–space fractional diffusion equation

Zhao

Zhu

et al. 2019

Journal of Computational and Applied Mathematics

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A block lower triangular Toeplitz system arising from the time-space fractional diffusion equation is discussed. For efficient solutions of such the linear system, the preconditioned biconjugate gradient stabilized method and the flexible general minimal residual method are exploited. The main contribution of this paper has two aspects: (i) A block bi-diagonal Toeplitz preconditioner is developed for the block lower triangular Toeplitz system, whose storage is of O(N ) with N being the spatial grid number; (ii) A new skew-circulant preconditioner is designed to accelerate the inverse of the block bi-diagonal Toeplitz preconditioner multiplying a vector. Numerical experiments are given to demonstrate the effectiveness of our two proposed preconditioners.

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Section: Introductionmentioning

confidence: 99%

A limited-memory block bi-diagonal Toeplitz preconditioner for block lower triangular Toeplitz system from time–space fractional diffusion equation

Zhao

Zhu

et al. 2019

Journal of Computational and Applied Mathematics

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“…On the other hand, we show that the discretizations of TDFDEs lead to a nonsymmetric Toeplitz-like system of linear equations. The linear system can be solved efficiently by using Krylov subspace methods with suitable circulant preconditioners [52][53][54]. It can greatly reduce the memory and computational costs; the memory requirement and computational complexity are only O(M) and O(M log M) in each iteration step, respectively.…”

Section: Introductionmentioning

confidence: 99%

A fast implicit difference scheme for a new class of time distributed-order and space fractional diffusion equations with variable coefficients

et al. 2018

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Recently, several problems in mathematics, physics, and engineering have been modeled via distributed-order fractional diffusion equations. In this paper, a new class of time distributed-order and space fractional diffusion equations with variable coefficients on bounded domains and Dirichlet boundary conditions is considered. By performing numerical integration we transform the time distributed-order fractional diffusion equations into multiterm time-space fractional diffusion equations. An implicit difference scheme for the multiterm time-space fractional diffusion equations is proposed along with a discussion about the unconditional stability and convergence. Then, the fast Krylov subspace methods with suitable circulant preconditioners are developed to solve the resultant linear system in light of their Toeplitz-like structures. The aforementioned methods are proved to acquire the capability to reduce the memory storage of the proposed implicit difference scheme from O(M 2) to O(M) and the computational cost from O(M 3) to O(M log M) during iteration procedures, where M is the number of grid nodes. Finally, numerical experiments are employed to support the theoretical findings and show the efficiency of the proposed methods.

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“…Fast methods Lei and Sun ( 2013 ), Popolizio ( 2015 ), Gu et al. ( 2015 , 2015 ) have been developed to solve FDEs with the shifted Grünwald formula. Wang and Wang ( 2011 ) proposed a conjugate gradient normal residual (CGNR) to solve the discretized system by Meerschact and Tadjeran’s method with the computational cost of .…”

Section: Introductionmentioning

confidence: 99%

Fast permutation preconditioning for fractional diffusion equations

Wang

Huang

et al. 2016

SpringerPlus

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In this paper, an implicit finite difference scheme with the shifted Grünwald formula, which is unconditionally stable, is used to discretize the fractional diffusion equations with constant diffusion coefficients. The coefficient matrix possesses the Toeplitz structure and the fast Toeplitz matrix-vector product can be utilized to reduce the computational complexity from to , where N is the number of grid points. Two preconditioned iterative methods, named bi-conjugate gradient method for Toeplitz matrix and bi-conjugate residual method for Toeplitz matrix, are proposed to solve the relevant discretized systems. Finally, numerical experiments are reported to show the effectiveness of our preconditioners.

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Onk-step CSCS-based polynomial preconditioners for Toeplitz linear systems with application to fractional diffusion equations

Cited by 23 publications

References 19 publications

A limited-memory block bi-diagonal Toeplitz preconditioner for block lower triangular Toeplitz system from time–space fractional diffusion equation

A limited-memory block bi-diagonal Toeplitz preconditioner for block lower triangular Toeplitz system from time–space fractional diffusion equation

A fast implicit difference scheme for a new class of time distributed-order and space fractional diffusion equations with variable coefficients

Fast permutation preconditioning for fractional diffusion equations

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