Toeplitz-like preconditioner for linear systems from spatial fractional diffusion equations

Document Type : Research Article


School of mathematics and computer science, Damghan university, Damghan, Iran.


‎The article deals with constructing Toeplitz-like preconditioner for linear systems arising from finite difference discretization of the spatial fractional diffusion equations‎. ‎The coefficient matrices of these linear systems have an $S+L$ structure‎, ‎where $S$ is a symmetric positive definite (SPD) matrix and $L$ satisfies $\mbox{rank}(L)\leq 2$‎. ‎We introduce an approximation for the SPD part $S$‎, ‎which is called $P_S$‎, ‎and then we show that the preconditioner $P=P_S+L$ has the Toeplitz-like structure and its displacement rank is 6‎. The analysis shows that the eigenvalues of the corresponding preconditioned matrix are clustered around 1. Numerical experiments exhibit that the Toeplitz-like preconditioner can significantly improve the convergence properties of the applied iteration method.


1. Agrawal, O.P., Solution for a fractional diffusion-wave equation defined in a bounded domain, Nonlinear Dynam. 29(1-4) (2002) 145–155.
2. Akhoundi N., 2n-by-2n circulant preconditioner for a kind of spatial fractional diffusion equations, J. Math. Model. 8 (2020), 207–218.
3. Bouchaud, J.P. and Georges A., Anomalous diffusion in disordered me dia: statistical mechanisms, models and physical applications, Phys. Rep. 195 (1990), 127–293.
4. Fang, Z.W., Ng, M.K. and Sun, H.W., Circulant preconditioners for a kind of spatial fractional diffusion equations, Numer. Algorithms 82(2019), 729–747.
5. Favat, P., Lotti, G. and Menchi, O., Stability of the Levinson algorithm for Toeplitz-like systems, SIAM J. Matrix Anal. Appl. 31 (2010), 2531–2552.
6. Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J., Theory and applications of fractional differential equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.
7. Metzler, R. and Klafter, J., The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339 (2000), 77 pp.
8. Podlubny, I., Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Elsevier; 1998.
9. Rudolf, H., Editor. Applications of fractional calculus in physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000.
10. Saad, Y., Preconditioned Krylov subspace methods for the numerical solution of Markov chains, In: Stewart W.J. (eds) Computations with Markov Chains. Springer, Boston, MA.