Fast permutation preconditioning for fractional diffusion equations

Wang, Shengfeng; Huang, Ting-Zhu; Gu, Xian-Ming; Luo, Wei-Hua

doi:10.1186/s40064-016-2766-4

Cited by 3 publications

(4 citation statements)

References 33 publications

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“…One reason for characterizing the spectra of these flipped matrices relates to the solution of linear systems with Toeplitz coefficient matrices. Since Y n T n ( f ) is symmetric, the resulting linear system may be solved by the MINRES method [18,21] or by preconditioned MINRES [16,17], with its descriptive convergence rate bounds based on eigenvalues (see, e.g., [2,Chapters 2 and 4]). However, whilst there has been significant interest in relating the eigenvalues and singular values of Toeplitz sequences to generating functions, analogous results have not been proved for flipped Toeplitz sequences and corresponding preconditioned ones.…”

Section: Introductionmentioning

confidence: 99%

Spectral properties of flipped Toeplitz matrices and related preconditioning

Mazza

Pestana

2018

Bit Numer Math

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In this work, we investigate the spectra of "flipped" Toeplitz sequences, i.e., the asymptotic spectral behaviour of {Y n T n (f)} n , where T n (f) ∈ R n×n is a real Toeplitz matrix generated by a function f ∈ L 1 ([−π, π]), and Y n is the exchange matrix, with 1s on the main anti-diagonal. We show that the eigenvalues of Y n T n (f) are asymptotically described by a 2 × 2 matrix-valued function, whose eigenvalue functions are ± | f |. It turns out that roughly half of the eigenvalues of Y n T n (f) are well approximated by a uniform sampling of | f | over [− π, π], while the remaining are well approximated by a uniform sampling of − | f | over the same interval. When f vanishes only on a set of measure zero, this motivates that the spectrum is virtually half positive and half negative. Some insights on the spectral distribution of related preconditioned sequences are provided as well. Finally, a wide number of numerical results illustrate our theoretical findings.

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Section: Introductionmentioning

confidence: 99%

Spectral properties of flipped Toeplitz matrices and related preconditioning

Mazza

Pestana

2018

Bit Numer Math

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“…The test practical problem is from the reference [35]. We consider the fractional diffusion equation…”

Section: Examplementioning

confidence: 99%

“…, A is a nonsymmetric Toeplitz matrix. The permutation matrix J was introduced [35,36] to transform the non-symmetric Toeplitz matrix problem into symmetric Hankel matrix problem in order to use the properties of symmetric matrix. To analyze the eigenpairs of A is equal to solving the generalised Hankel eigenproblem Hx = λJx, where H = J T .…”

Section: Examplementioning

confidence: 99%

An inexact Krylov subspace method for large generalized Hankel eigenproblems

Huang,

Feng

2024

ScienceAsia

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Krylov subspace method is an effective method for large-scale eigenproblems. The shift-and-invert Arnoldi method is employed to compute a few eigenpairs of a large Hankel matrix pencil. However, a crucial step in the process is computing products between the inversion of a Hankel matrix and vectors. The inversion of the Hankel matrix can be obtained by solving two Hankel systems. By establishing a relationship between the errors of systems and the residuals of the Hankel eigenproblem, we provide a practical stopping criterion for solving Hankel systems and propose an inexact shift-and-invert Arnoldi method for the generalized Hankel eigenproblem. Numerical experiments are presented to demonstrate the efficiency of the new algorithm and our theoretical results.

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“…Fortunately, since G β is a Toeplitz matrix, it can be stored with only M entries [41]. The Krylov subspace methods with circulant preconditioners [45,61] can be used to solve Toeplitz-like linear systems with a fast convergence rate. In this case, we also remark that the computational complexity of preconditioned Krylov subspace methods is only in O(M log M) at each iteration step for implementing the implicit difference scheme.…”

Section: Fast Solution Techniques Based On Preconditioned Krylov Subsmentioning

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 permutation preconditioning for fractional diffusion equations

Cited by 3 publications

References 33 publications

Spectral properties of flipped Toeplitz matrices and related preconditioning

Spectral properties of flipped Toeplitz matrices and related preconditioning

An inexact Krylov subspace method for large generalized Hankel eigenproblems

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

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