146 research outputs found
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
Recommended from our members
EFFICIENT PRECONDITIONING for TIME FRACTIONAL DIFFUSION INVERSE SOURCE PROBLEMS
PRECONDITIONERS AND TENSOR PRODUCT SOLVERS FOR OPTIMAL CONTROL PROBLEMS FROM CHEMOTAXIS
In this paper, we consider the fast numerical solution of an optimal control
formulation of the Keller--Segel model for bacterial chemotaxis. Upon
discretization, this problem requires the solution of huge-scale saddle point
systems to guarantee accurate solutions. We consider the derivation of
effective preconditioners for these matrix systems, which may be embedded
within suitable iterative methods to accelerate their convergence. We also
construct low-rank tensor-train techniques which enable us to present efficient
and feasible algorithms for problems that are finely discretized in the space
and time variables. Numerical results demonstrate that the number of
preconditioned GMRES iterations depends mildly on the model parameters.
Moreover, the low-rank solver makes the computing time and memory costs
sublinear in the original problem size.Comment: 23 page
Preconditioned fast solvers for large linear systems with specific sparse and/or Toeplitz-like structures and applications
In this thesis, the design of the preconditioners we propose starts from applications instead of treating the problem in a completely general way. The reason is that not all types of linear systems can be addressed with the same tools. In this sense, the techniques for designing efficient iterative solvers depends mostly on properties inherited from the continuous problem, that has originated the discretized sequence of matrices. Classical examples are locality, isotropy in the PDE context, whose discrete counterparts are sparsity and matrices constant along the diagonals, respectively.
Therefore, it is often important to take into account the properties of the originating continuous model for obtaining better performances and for providing an accurate convergence analysis. We consider linear systems that arise in the solution of both linear and nonlinear partial differential equation of both integer and fractional type. For the latter case, an introduction to both the theory and the numerical treatment is given.
All the algorithms and the strategies presented in this thesis are developed having in mind their parallel implementation. In particular, we consider the processor-co-processor framework, in which the main part of the computation is performed on a Graphics Processing Unit (GPU) accelerator.
In Part I we introduce our proposal for sparse approximate inverse preconditioners for either the solution of time-dependent Partial Differential Equations (PDEs), Chapter 3, and Fractional Differential Equations (FDEs), containing both classical and fractional terms, Chapter 5. More precisely, we propose a new technique for updating preconditioners for dealing with sequences of linear systems for PDEs and FDEs, that can be used also to compute matrix functions of large matrices via quadrature formula in Chapter 4 and for optimal control of FDEs in Chapter 6. At last, in Part II, we consider structured preconditioners for quasi-Toeplitz systems. The focus is towards the numerical treatment of discretized convection-diffusion equations in Chapter 7 and on the solution of FDEs with linear multistep formula in boundary value form in Chapter 8
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