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Fast preconditioned iterative methods for fractional Sturm–Liouville equations
Author(s) -
Zhang Lei,
Zhang GuoFeng,
Liang ZhaoZheng
Publication year - 2021
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22704
Subject(s) - preconditioner , mathematics , krylov subspace , coefficient matrix , circulant matrix , discretization , generalized minimal residual method , iterative method , matrix (chemical analysis) , toeplitz matrix , linear system , tensor product , mathematical analysis , eigenvalues and eigenvectors , mathematical optimization , algorithm , pure mathematics , physics , materials science , quantum mechanics , composite material
In this paper, we have considered fast solutions of the linear system arising from the fractional Sturm–Liouville problem, whose coefficient matrix contains the product of Toeplitz‐like matrices. Based on suitable circulant approximations of the related coefficient matrix, we establish a matching preconditioner of matrix‐free form. In theory, the spectrum of the preconditioned matrix is shown to cluster around [1/2, 1) , which suggests the fast convergence speed of the proposed preconditioner within Krylov subspace acceleration. In addition, to reduce the computation time and storage, we consider an all‐at‐once discretized system and explore its low‐rank tensor structure and alternating iterative tensor algorithms. Numerical experiments are given to show the effectiveness of the proposed solution techniques compared with some existing methods.