In [McDonald, Pestana and Wathen, \textit{SIAM J. Sci. Comput.}, 40 (2018), pp. A1012--A1033], a block circulant preconditioner is proposed for all-at-once linear systems arising from evolutionary partial differential equations, in which the preconditioned matrix is proven to be diagonalizable and to have identity-plus-low-rank decomposition in the case of the heat equation. In this paper, we generalize the block circulant preconditioner by introducing a small parameter $ε>0$ into the top-right block of the block circulant preconditioner. The implementation of the generalized preconditioner requires the same computational complexity as that of the block circulant one.Theoretically, we prove that (i) the generalization preserves the diagonalizability and the identity-plus-low-rank decomposition; (ii) all eigenvalues of the new preconditioned matrix are clustered at 1 for sufficiently small $ε$; (iii) GMRES method for the preconditioned system has a linear convergence rate independent of size of the linear system when $ε$ is taken to be smaller than or comparable to square root of time-step size. Numerical results are reported to confirm the efficiency of the proposed preconditioner and to show that the generalization improves the performance of block circulant preconditioner.
In this paper we propose to use model reduction techniques for speeding up the diagonalization-based parallel-in-time (ParaDIAG) preconditioner, for iteratively solving all-at-once systems from evolutionary PDEs. In particular, we use the reduced basis method to seek a low-dimensional approximation to the sequence of complex-shifted systems arising from Step-(b) of the ParaDIAG preconditioning procedure. Different from the standard reduced order modeling that uses the separation of offline and online stages, we have to build the reduced order model (ROM) online for the considered systems at ea…
In this paper, we study a parallel-in-time (PinT) algorithm for all-at-once system from a non-local evolutionary equation with weakly singular kernel where the temporal term involves a non-local convolution with a weakly singular kernel and the spatial term is the usual Laplacian operator with variable coefficients. We propose to use a two-sided preconditioning technique for the all-at-once discretization of the equation. Our preconditioner is constructed by replacing the variable diffusion coefficients with a constant coefficient to obtain a constant-coefficient all-at-once matrix. We split a…
The $p$-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 $p$-step BDF ($p\geq 2$) 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 fractio…
An all-at-once linear system arising from the nonlinear tempered fractional diffusion equation with variable coefficients is studied. Firstly, the nonlinear and linearized implicit schemes are proposed to approximate such the nonlinear equation with continuous/discontinuous coefficients. The stabilities and convergences of the two schemes are proved under several suitable assumptions, and numerical examples show that the convergence orders of these two schemes are $1$ in both time and space. Secondly, a nonlinear all-at-once system is derived based on the nonlinear implicit scheme, which may s…
The Sinc-Nyström method is a high-order numerical method based on Sinc basis functions for discretizing evolutionary differential equations in time. But in this method we have to solve all the time steps in one-shot (i.e. all-at-once), which results in a large-scale nonsymmetric dense system that is expensive to handle. In this paper, we propose and analyze preconditioner for such dense system arising from both the parabolic and hyperbolic PDEs. The proposed preconditioner is a low-rank perturbation of the original matrix and has two advantages. First, we show that the eigenvalues of the preco…
This note proposes an efficient preconditioner for solving linear and semi-linear parabolic equations. With the Crank-Nicholson time stepping method, the algebraic system of equations at each time step is solved with the conjugate gradient method, preconditioned with hierarchical interpolative factorization. Stiffness matrices arising in the discretization of parabolic equations typically have large condition numbers, and therefore preconditioning becomes essential, especially for large time steps. We propose to use the hierarchical interpolative factorization as the preconditioning for the co…
Parallel-in-time methods have become increasingly popular in the simulation of time-dependent numerical PDEs, allowing for the efficient use of additional MPI processes when spatial parallelism saturates. Most methods treat the solution and parallelism in space and time separately. In contrast, all-at-once methods solve the full space-time system directly, largely treating time as simply another spatial dimension. All-at-once methods offer a number of benefits over separate treatment of space and time, most notably significantly increased parallelism and faster time-to-solution (when applicabl…
Matrix probing: a randomized preconditioner for the wave-equation Hessian
Department of Mathematics, MIT（麻省理工学院数学系）；Institute for Computational Mathematics and Engineering, Stanford（斯坦福大学计算数学与工程研究所）；Applied Mathematics Department, Université catholique de Louvain（天主教鲁汶大学应用数学系）；Total SA, Exploration & Production（道达尔公司勘探与生产部）；Courant Institute of Mathematical Sciences, NYU（纽约大学库朗数学科学研究所）
Within the framework of $ p $-adaptive flux reconstruction, we aim to construct efficient polynomial multigrid ($p$MG) preconditioners for implicit time integration of the Navier--Stokes equations using Jacobian-free Newton--Krylov (JFNK) methods. We hypothesise that in pseudo transient continuation (PTC), as the residual drops, the frequency of error modes that dictates the convergence rate gets higher and higher. We apply nonlinear $p$MG solvers to stiff steady problems at low Mach number ($\mathrm{Ma}=10^{-3}$) to verify our hypothesis. It is demonstrated that once the residual drops by a f…
We present and analyze a parallel implementation of a parallel-in-time collocation method based on $α$-circulant preconditioned Richardson iterations. While many papers explore this family of single-level, time-parallel "all-at-once" integrators from various perspectives, performance results of actual parallel runs are still scarce. This leaves a critical gap, because the efficiency and applicability of any parallel method heavily rely on the actual parallel performance, with only limited guidance from theoretical considerations. Further, challenges like selecting good parameters, finding suit…
In this article, we derive a new, fast, and robust preconditioned iterative solution strategy for the all-at-once solution of optimal control problems with time-dependent PDEs as constraints, including the heat equation and the non-steady convection--diffusion equation. After applying an optimize-then-discretize approach, one is faced with continuous first-order optimality conditions consisting of a coupled system of PDEs. As opposed to most work in preconditioning the resulting discretized systems, where a (first-order accurate) backward Euler method is used for the discretization of the time…
McMg: A Learned Phase-Space Multi-channel Multigrid Preconditioner for Helmholtz Equation
School of Mathematics, Jilin University（吉林大学数学学院）；Ocean University of China（中国海洋大学）；Shenzhen Loop Area Institute（深圳河套学院）；King Abdullah University of Science and Technology（卡布斯科学技术大学）
In this paper, we propose a direct parallel-in-time (PinT) algorithm for time-dependent problems with first- or second-order derivative. We use a second-order boundary value method as the time integrator that leads to a tridiagonal time discretization matrix. Instead of solving the corresponding all-at-once system iteratively, we diagonalize the time discretization matrix, which yields a direct parallel implementation across all time levels. A crucial issue on this methodology is how the condition number of the eigenvector matrix $V$ grows as $n$ is increased, where $n$ is the number of time l…
