research-article

Spectral Analysis and Multigrid Methods for Finite Volume Approximations of Space-Fractional Diffusion Equations

Authors:

Marco Donatelli,

Mariarosa Mazza,

Stefano Serra-CapizzanoAuthors Info & Claims

SIAM Journal on Scientific Computing, Volume 40, Issue 6

Pages A4007 - A4039

https://doi.org/10.1137/17M115164X

Published: 01 January 2018 Publication History

Abstract

We consider a boundary value problem in weak form of a steady-state Riesz space-fractional diffusion equation (FDE) of order $2-\alpha$ with $0<\alpha<1$. By using a finite volume approximation technique on uniform grids, we obtain a large linear system, whose coefficient matrix can be viewed as the sum of diagonal matrices times dense Toeplitz matrices. We study in detail the hidden nature of the resulting sequence of coefficient matrices, and we show that they fall in the class of generalized locally Toeplitz (GLT) sequences. The associated GLT symbol is obtained as the sum of products of functions, involving the Wiener generating functions of the Toeplitz components and the diffusion coefficients of the considered FDE. By exploiting a few analytical features of the GLT symbol, we obtain spectral information used for designing efficient preconditioners and multigrid methods. Several numerical experiments, both in the 1D and 2D cases, are reported and discussed, in order to show the optimality of the proposed algorithms.

References

[1]

A. Aricò and M. Donatelli, A V-cycle multigrid for multilevel matrix algebras: Proof of optimality, Numer. Math., 105 (2007), pp. 511--547.

[2]

A. Aricò, M. Donatelli, and S. Serra-Capizzano, V-cycle optimal convergence for certain (multilevel) structured linear systems, SIAM J. Matrix. Anal. Appl., 26 (2004), pp. 186--214, https://doi.org/10.1137/S0895479803421987.

[3]

O. Axelsson and G. Lindskög, On the rate of convergence of the preconditioned conjugate gradient method, Numer. Math., 48 (1986), pp. 499--523.

[4]

J. Bai and X.-C. Feng, Fractional-order anisotropic diffusion for image denoising, IEEE Trans. Image Process., 16 (2007), pp. 2492--2502.

[5]

B. Beckermann and S. Serra-Capizzano, On the asymptotic spectrum of finite element matrix sequences, SIAM J. Numer. Anal., 45 (2007), pp. 746--769, https://doi.org/10.1137/05063533X.

[6]

R. Bhatia, Matrix Analysis, Springer-Verlag, New York, 1997.

[7]

A. Böttcher and S. Grudsky, On the condition numbers of large semi-definite Toeplitz matrices, Linear Algebra Appl., 279 (1998), pp. 285--301.

[8]

T. Breiten, V. Simoncini, and M. Stoll, Low-rank solvers for fractional differential equations, Electron. Trans. Numer. Anal., 45 (2016), pp. 107--132.

[9]

R. H. Chan, Q.-S. Chang, and H.-W. Sun, Multigrid method for ill-conditioned symmetric Toeplitz systems, SIAM J. Sci. Comput., 19 (1998), pp. 516--529, https://doi.org/10.1137/S1064827595293831.

[10]

D. del Castillo-Negrete, B. Carreras, and V. Lynch, Fractional diffusion in plasma turbulence, Phys. Plasmas, 11 (2004), pp. 3854--3864.

[11]

W. Deng, B. Li, W. Tian, and P. Zhang, Boundary problems for the fractional and tempered fractional operators, Multiscale Model. Simul., 16 (2018), pp. 125--149, https://doi.org/10.1137/17M1116222.

[12]

M. Donatelli, An algebraic generalization of local Fourier analysis for grid transfer operators in multigrid based on Toeplitz matrices, Numer. Linear Algebra Appl., 17 (2010), pp. 179--197.

[13]

M. Donatelli, M. Mazza, and S. Serra-Capizzano, Spectral analysis and structure preserving preconditioners for fractional diffusion equations, J. Comput. Phys., 307 (2016), pp. 262--279.

[14]

G. Fiorentino and S. Serra-Capizzano, Multigrid methods for Toeplitz matrices, Calcolo, 28 (1991), pp. 283--305.

[15]

C. Garoni, C. Manni, F. Pelosi, S. Serra-Capizzano, and H. Speleers, On the spectrum of stiffness matrices arising from isogeometric analysis, Numer. Math., 127 (2014), pp. 751--799.

[16]

C. Garoni and S. Serra-Capizzano, Generalized Locally Toeplitz Sequences: Theory and Applications, Vol. 1, Springer, Cham, 2017.

[17]

L. Golinskii and S. Serra-Capizzano, The asymptotic properties of the spectrum of nonsymmetrically perturbed Jacobi matrix sequences, J. Approx. Theory, 144 (2007), pp. 84--102.

[18]

R. Gorenflo and F. Mainardi, Random walk models for space-fractional diffusion processes, Fract. Calc. Appl. Anal., 1 (1998), pp. 167--191.

[19]

U. Grenander and G. Szegö, Toeplitz Forms and Their Applications, 2nd ed., Chelsea, New York, 1984.

[20]

P. W. Hemker, On the order of prolongations and restrictions in multigrid procedures, J. Comput. Appl. Math., 32 (1990), pp. 423--429.

[21]

Y. Jiang and X. Xu, Multigrid methods for space fractional partial differential equations, J. Comput. Phys., 302 (2015), pp. 374--392.

[22]

T. Langlands, B. Henry, and S. Wearne, Fractional cable equation models for anomalous electrodiffusion in nerve cells: Infinite domain solutions, J. Math. Biol., 59 (2009), pp. 761--808.

[23]

S.-L. Lei and H.-W. Sun, A circulant preconditioner for fractional diffusion equations, J. Comput. Phys., 242 (2013), pp. 715--725.

[24]

X.-L. Lin, M. K. Ng, and H.-W. Sun, A multigrid method for linear systems arising from time-dependent two-dimensional space-fractional diffusion equations, J. Comput. Phys., 336 (2017), pp. 69--86.

[25]

X.-L. Lin, M. K. Ng, and H.-W. Sun, A splitting preconditioner for Toeplitz-like linear systems arising from fractional diffusion equations, SIAM J. Matrix Anal. Appl., 38 (2017), pp. 1580--1614, https://doi.org/10.1137/17M1115447.

[26]

M. M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection--dispersion flow equations, J. Comput. Appl. Math., 172 (2004), pp. 65--77.

[27]

H. Moghaderi, M. Dehghan, M. Donatelli, and M. Mazza, Spectral analysis and multigrid preconditioners for two-dimensional space-fractional diffusion equations, J. Comput. Phys., 350 (2017), pp. 992--1011.

[28]

J. Pan, R. Ke, M. K. Ng, and H.-W. Sun, Preconditioning techniques for diagonal-times-Toeplitz matrices in fractional diffusion equations, SIAM J. Sci. Comput., 36 (2014), pp. A2698--A2719, https://doi.org/10.1137/130931795.

[29]

J. Pan, M. K. Ng, and H. Wang, Fast iterative solvers for linear systems arising from time-dependent space-fractional diffusion equations, SIAM J. Sci. Comput., 38 (2016), pp. A2806--A2826, https://doi.org/10.1137/15M1030273.

[30]

J. Pan, M. K. Ng, and H. Wang, Fast preconditioned iterative methods for finite volume discretization of steady-state space-fractional diffusion equations, Numer. Algorithms, 74 (2017), pp. 153--173.

[31]

H.-K. Pang and H.-W. Sun, Multigrid method for fractional diffusion equations, J. Comput. Phys., 231 (2012), pp. 693--703.

[32]

I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Math. Sci. Engrg. 198, Academic Press, San Diego, CA, 1999.

[33]

M. Raberto, E. Scalas, and F. Mainardi, Waiting-times and returns in high-frequency financial data: An empirical study, Phys. A, 314 (2002), pp. 749--755.

[34]

J. W. Ruge and K. Stüben, Algebraic multigrid, in Multigrid Methods, S. F. McCormick, ed., Frontiers Appl. Math. 3, SIAM, Philadelphia, 1987, pp. 73--130, https://doi.org/10.1137/1.9781611971057.ch4.

[35]

A. I. Saichev and G. M. Zaslavsky, Fractional kinetic equations: Solutions and applications, Chaos, 7 (1997), pp. 753--764.

[36]

S. Serra, How to choose the best iterative strategy for symmetric Toeplitz systems, SIAM J. Numer. Anal., 36 (1999), pp. 1078--1103, https://doi.org/10.1137/S0036142996311866.

[37]

S. Serra Capizzano and E. Tyrtyshnikov, Any circulant-like preconditioner for multilevel Toeplitz matrices is not superlinear, SIAM J. Matrix Anal. Appl., 21 (1999), pp. 431--439, https://doi.org/10.1137/S0895479897331941.

[38]

S. Serra-Capizzano, On the extreme eigenvalues of Hermitian (block) Toeplitz matrices, Linear Algebra Appl., 270 (1998), pp. 109--129.

[39]

S. Serra-Capizzano, Convergence analysis of two-grid methods for elliptic Toeplitz and PDEs matrix-sequences, Numer. Math., 92 (2002), pp. 433--465.

[40]

S. Serra-Capizzano, Generalized locally Toeplitz sequences: Spectral analysis and applications to discretized partial differential equations, Linear Algebra Appl., 366 (2003), pp. 371--402.

[41]

S. Serra-Capizzano, The GLT class as a generalized Fourier analysis and applications, Linear Algebra Appl., 419 (2006), pp. 180--233.

[42]

W. Tian, H. Zhou, and W. Deng, A class of second order difference approximations for solving space fractional diffusion equations, Math. Comp., 84 (2015), pp. 1703--1727.

[43]

P. Tilli, Locally Toeplitz sequences: Spectral properties and applications, Linear Algebra Appl., 278 (1998), pp. 91--120.

[44]

P. Tilli, A note on the spectral distribution of Toeplitz matrices, Linear and Multilinear Algebra, 45 (1998), pp. 147--159.

[45]

U. Trottenberg, C. W. Oosterlee, and A. Schuller, Multigrid, Academic Press, San Diego, CA, 2000.

[46]

E. E. Tyrtyshnikov and N. L. Zamarashkin, Spectra of multilevel Toeplitz matrices: Advanced theory via simple matrix relationships, Linear Algebra Appl., 270 (1998), pp. 15--27.

[47]

H. Wang and N. Du, A superfast-preconditioned iterative method for steady-state space-fractional diffusion equations, J. Comput. Phys., 240 (2013), pp. 49--57.

[48]

H. Wang, K. Wang, and T. Sircar, A direct ${O}({N}\log^2{N})$ finite difference method for fractional diffusion equations, J. Comput. Phys., 229 (2010), pp. 8095--8104.

[49]

I. Yavneh, Coarse-grid correction for nonelliptic and singular pertubation problems, SIAM J. Sci. Comput., 19 (1998), pp. 1682--1699, https://doi.org/10.1137/S1064827596310998.

Cited By

Pang HQin HNi S(2024)Sine Transform Based Preconditioning for an Inverse Source Problem of Time-Space Fractional Diffusion EquationsJournal of Scientific Computing10.1007/s10915-024-02634-x100:3Online publication date: 25-Jul-2024
https://dl.acm.org/doi/10.1007/s10915-024-02634-x
Chen MCao RSerra-Capizzano S(2024)Fast algebraic multigrid for block-structured dense systems arising from nonlocal diffusion problemsCalcolo: a quarterly on numerical analysis and theory of computation10.1007/s10092-024-00612-161:4Online publication date: 13-Sep-2024
https://dl.acm.org/doi/10.1007/s10092-024-00612-1
Ngondiep E(2023)A high-order numerical scheme for multidimensional convection-diffusion-reaction equation with time-fractional derivativeNumerical Algorithms10.1007/s11075-023-01516-x94:2(681-700)Online publication date: 4-Mar-2023
https://dl.acm.org/doi/10.1007/s11075-023-01516-x
Show More Cited By

Recommendations

Multigrid preconditioners for anisotropic space-fractional diffusion equations
Abstract
We focus on a two-dimensional time-space diffusion equation with fractional derivatives in space. The use of Crank-Nicolson in time and finite differences in space leads to dense Toeplitz-like linear systems. Multigrid strategies that exploit such ...
Spectral analysis and structure preserving preconditioners for fractional diffusion equations

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 ...
Band-Toeplitz preconditioners for ill-conditioned Toeplitz systems
Abstract
Preconditioning for Toeplitz systems has been an active research area over the past few decades. Along this line of research, circulant preconditioners have been recently proposed for the Toeplitz-like system arising from discretizing fractional ...

Comments

Information & Contributors

Information

Published In

cover image SIAM Journal on Scientific Computing

SIAM Journal on Scientific Computing Volume 40, Issue 6

DOI:10.1137/sjoce3.40.6

Issue’s Table of Contents

© 2018, Society for Industrial and Applied Mathematics.

Publisher

Society for Industrial and Applied Mathematics

United States

Publication History

Published: 01 January 2018

Author Tags

Author Tags

Qualifiers

Research-article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

10
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 03 Feb 2025

Other Metrics

View Author Metrics

Citations

Cited By

Pang HQin HNi S(2024)Sine Transform Based Preconditioning for an Inverse Source Problem of Time-Space Fractional Diffusion EquationsJournal of Scientific Computing10.1007/s10915-024-02634-x100:3Online publication date: 25-Jul-2024
https://dl.acm.org/doi/10.1007/s10915-024-02634-x
Chen MCao RSerra-Capizzano S(2024)Fast algebraic multigrid for block-structured dense systems arising from nonlocal diffusion problemsCalcolo: a quarterly on numerical analysis and theory of computation10.1007/s10092-024-00612-161:4Online publication date: 13-Sep-2024
https://dl.acm.org/doi/10.1007/s10092-024-00612-1
Ngondiep E(2023)A high-order numerical scheme for multidimensional convection-diffusion-reaction equation with time-fractional derivativeNumerical Algorithms10.1007/s11075-023-01516-x94:2(681-700)Online publication date: 4-Mar-2023
https://dl.acm.org/doi/10.1007/s11075-023-01516-x
Cai YSun HTam S(2023)Numerical Study of a Fast Two-Level Strang Splitting Method for Spatial Fractional Allen–Cahn EquationsJournal of Scientific Computing10.1007/s10915-023-02196-495:3Online publication date: 18-Apr-2023
https://dl.acm.org/doi/10.1007/s10915-023-02196-4
Alyusof RAlyusof SIqbal NArefin M(2022)Novel Evaluation of the Fractional Acoustic Wave Model with the Exponential-Decay KernelComplexity10.1155/2022/97123882022Online publication date: 1-Jan-2022
https://dl.acm.org/doi/10.1155/2022/9712388
Zhang CYu JWang X(2022)A fast second-order scheme for nonlinear Riesz space-fractional diffusion equationsNumerical Algorithms10.1007/s11075-022-01367-y92:3(1813-1836)Online publication date: 22-Jul-2022
https://dl.acm.org/doi/10.1007/s11075-022-01367-y
Bogoya MGrudsky SSerra–Capizzano STablino–Possio C(2022)Fine spectral estimates with applications to the optimally fast solution of large FDE linear systemsBIT10.1007/s10543-022-00916-062:4(1417-1431)Online publication date: 1-Dec-2022
https://dl.acm.org/doi/10.1007/s10543-022-00916-0
Donatelli MKrause RMazza MTrotti K(2021)All-at-once multigrid approaches for one-dimensional space-fractional diffusion equationsCalcolo: a quarterly on numerical analysis and theory of computation10.1007/s10092-021-00436-358:4Online publication date: 1-Dec-2021
https://dl.acm.org/doi/10.1007/s10092-021-00436-3
Donatelli MKrause RMazza MTrotti K(2020)Multigrid preconditioners for anisotropic space-fractional diffusion equationsAdvances in Computational Mathematics10.1007/s10444-020-09790-246:3Online publication date: 3-Jun-2020
https://dl.acm.org/doi/10.1007/s10444-020-09790-2
Durastante F(2019)Efficient solution of time-fractional differential equations with a new adaptive multi-term discretization of the generalized Caputo–Dzherbashyan derivativeCalcolo: a quarterly on numerical analysis and theory of computation10.1007/s10092-019-0329-056:4(1-24)Online publication date: 1-Dec-2019
https://dl.acm.org/doi/10.1007/s10092-019-0329-0

View Options

View options

Figures

Tables

Media

View Issue’s Table of Contents