Fractional partial order diffusion equations are a generalization of classical partial differential equations, used to model anomalous diffusion phenomena. When using the implicit Euler formula and the shifted Grünwald formula, it has been shown that the related discretizations lead to a linear system whose coefficient matrix has a Toeplitz-like structure. In this paper we focus our attention on the case of variable diffusion coefficients. Under appropriate conditions, we show that the sequence of the coefficient matrices belongs to the Generalized Locally Toeplitz class and we compute the symbol describing its asymptotic eigenvalue/singular value distribution, as the matrix size diverges. We employ the spectral information for analyzing known methods of preconditioned Krylov and multigrid type, with both positive and negative results and with a look forward to the multidimensional setting. We also propose two new tridiagonal structure preserving preconditioners to solve the resulting linear system, with Krylov methods such as CGNR and GMRES. A number of numerical examples show that our proposal is more effective than recently used circulant preconditioners.

Spectral analysis and structure preserving preconditioners for fractional diffusion equations

DONATELLI, MARCO;Mazza, M.
;
SERRA CAPIZZANO, STEFANO
2016-01-01

Abstract

Fractional partial order diffusion equations are a generalization of classical partial differential equations, used to model anomalous diffusion phenomena. When using the implicit Euler formula and the shifted Grünwald formula, it has been shown that the related discretizations lead to a linear system whose coefficient matrix has a Toeplitz-like structure. In this paper we focus our attention on the case of variable diffusion coefficients. Under appropriate conditions, we show that the sequence of the coefficient matrices belongs to the Generalized Locally Toeplitz class and we compute the symbol describing its asymptotic eigenvalue/singular value distribution, as the matrix size diverges. We employ the spectral information for analyzing known methods of preconditioned Krylov and multigrid type, with both positive and negative results and with a look forward to the multidimensional setting. We also propose two new tridiagonal structure preserving preconditioners to solve the resulting linear system, with Krylov methods such as CGNR and GMRES. A number of numerical examples show that our proposal is more effective than recently used circulant preconditioners.
2016
2015
http://www.journals.elsevier.com/journal-of-computational-physics/
Fractional diffusion equations; Locally Toeplitz sequence of matrices; Preconditioning; Singular value/eigenvalue distribution; Toeplitz matrix; Computer Science Applications1707 Computer Vision and Pattern Recognition; Physics and Astronomy (miscellaneous)
Donatelli, Marco; Mazza, M.; SERRA CAPIZZANO, Stefano
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11383/2047966
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