1,090 research outputs found
Multigrid waveform relaxation for the time-fractional heat equation
In this work, we propose an efficient and robust multigrid method for solving
the time-fractional heat equation. Due to the nonlocal property of fractional
differential operators, numerical methods usually generate systems of equations
for which the coefficient matrix is dense. Therefore, the design of efficient
solvers for the numerical simulation of these problems is a difficult task. We
develop a parallel-in-time multigrid algorithm based on the waveform relaxation
approach, whose application to time-fractional problems seems very natural due
to the fact that the fractional derivative at each spatial point depends on the
values of the function at this point at all earlier times. Exploiting the
Toeplitz-like structure of the coefficient matrix, the proposed multigrid
waveform relaxation method has a computational cost of
operations, where is the number of time steps and is the number of
spatial grid points. A semi-algebraic mode analysis is also developed to
theoretically confirm the good results obtained. Several numerical experiments,
including examples with non-smooth solutions and a nonlinear problem with
applications in porous media, are presented
A nested divide-and-conquer method for tensor Sylvester equations with positive definite hierarchically semiseparable coefficients
Linear systems with a tensor product structure arise naturally when
considering the discretization of Laplace type differential equations or, more
generally, multidimensional operators with separable coefficients. In this
work, we focus on the numerical solution of linear systems of the form where the matrices are
symmetric positive definite and belong to the class of hierarchically
semiseparable matrices.
We propose and analyze a nested divide-and-conquer scheme, based on the
technology of low-rank updates, that attains the quasi-optimal computational
cost where is the condition number of the linear
system, and the target accuracy. Our theoretical analysis highlights
the role of inexactness in the nested calls of our algorithm and provides worst
case estimates for the amplification of the residual norm. The performances are
validated on 2D and 3D case studies
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
- β¦