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EFFICIENT PRECONDITIONING for TIME FRACTIONAL DIFFUSION INVERSE SOURCE PROBLEMS
On the bilateral preconditioning for an L2-type all-at-once system arising from time-space fractional Bloch-Torrey equations
Time-space fractional Bloch-Torrey equations (TSFBTEs) are developed by some
researchers to investigate the relationship between diffusion and
fractional-order dynamics. In this paper, we first propose a second-order
implicit difference scheme for TSFBTEs by employing the recently proposed
L2-type formula [A. A. Alikhanov, C. Huang, Appl. Math. Comput. (2021) 126545].
Then, we prove the stability and the convergence of the proposed scheme. Based
on such a numerical scheme, an L2-type all-at-once system is derived. In order
to solve this system in a parallel-in-time pattern, a bilateral preconditioning
technique is designed to accelerate the convergence of Krylov subspace solvers
according to the special structure of the coefficient matrix of the system. We
theoretically show that the condition number of the preconditioned matrix is
uniformly bounded by a constant for the time fractional order . Numerical results are reported to show the efficiency of our
method.Comment: 24 pages, 6 tables, 4 figure
A note on parallel preconditioning for the all-at-once solution of Riesz fractional diffusion equations
The -step backwards difference formula (BDF) for solving the system of
ODEs can result in a kind of all-at-once linear systems, which are solved via
the parallel-in-time preconditioned Krylov subspace solvers (see McDonald,
Pestana, and Wathen [SIAM J. Sci. Comput., 40(2) (2018): A1012-A1033] and Lin
and Ng [arXiv:2002.01108, 17 pages]. However, these studies ignored that the
-step BDF () is not selfstarting, when they are exploited to solve
time-dependent PDEs. In this note, we focus on the 2-step BDF which is often
superior to the trapezoidal rule for solving the Riesz fractional diffusion
equations, but its resultant all-at-once discretized system is a block
triangular Toeplitz system with a low-rank perturbation. Meanwhile, we first
give an estimation of the condition number of the all-at-once systems and then
adapt the previous work to construct two block circulant (BC) preconditioners.
Both the invertibility of these two BC preconditioners and the eigenvalue
distributions of preconditioned matrices are discussed in details. The
efficient implementation of these BC preconditioners is also presented
especially for handling the computation of dense structured Jacobi matrices.
Finally, numerical experiments involving both the one- and two-dimensional
Riesz fractional diffusion equations are reported to support our theoretical
findings.Comment: 18 pages. 2 figures. 6 Table. Tech. Rep.: Institute of Mathematics,
Southwestern University of Finance and Economics. Revised-1: refine/shorten
the contexts and correct some typos; Revised-2: correct some reference
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