使用Parareal和MGRIT在旋转球面上进行浅水方程的时间并行积分

Tight two-level convergence of Linear Parareal and MGRIT: Extensions and implications in practice

Two of the most popular parallel-in-time methods are Parareal and multigrid-reduction-in-time (MGRIT). Recently, a general convergence theory was developed in Southworth (2019) for linear two-level MGRIT/Parareal that provides necessary and sufficient conditions for convergence, with tight bounds on worst-case convergence factors. This paper starts by providing a new and simplified analysis of linear error and residual propagation of Parareal, wherein the norm of error or residual propagation is given by one over the minimum singular value of a certain block bidiagonal operator. New discussion…

On "Optimal" h-Independent Convergence of Parareal and MGRIT using Runge-Kutta Time Integration

Although convergence of the Parareal and multigrid-reduction-in-time (MGRIT) parallel-in-time algorithms is well studied, results on their optimality is limited. Appealling to recently derived tight bounds of two-level Parareal and MGRIT convergence, this paper proves (or disproves) $h_x$- and $h_t$-independent convergence of two-level Parareal and MGRIT, for linear problems of the form $\mathbf{u}'(t) + \mathcal{L}\mathbf{u}(t) = f(t)$, where $\mathcal{L}$ is symmetric positive definite and Runge-Kutta time integration is used. The theory presented in this paper also encompasses analysis of s…

Asynchronous Truncated Multigrid-reduction-in-time (AT-MGRIT)

In this paper, we present the new "asynchronous truncated multigrid-reduction-in-time" (AT-MGRIT) algorithm for introducing time parallelism to the solution of discretized time-dependent problems. The new algorithm is based on the multigrid-reduction-in-time (MGRIT) approach, which, in certain settings, is equivalent to another common multilevel parallel-in-time method, Parareal. In contrast to Parareal and MGRIT that both consider a global temporal grid over the entire time interval on the coarsest level, the AT-MGRIT algorithm uses truncated local time grids on the coarsest level, each grid …

On selecting coarse-grid operators for Parareal and MGRIT applied to linear advection

We consider the parallel time integration of the linear advection equation with the Parareal and two-level multigrid-reduction-in-time (MGRIT) algorithms. Our aim is to develop a better understanding of the convergence behaviour of these algorithms for this problem, which is known to be poor relative to the diffusion equation, its model parabolic counterpart. Using Fourier analysis, we derive new convergence estimates for these algorithms which, in conjunction with existing convergence theory, provide insight into the origins of this poor performance. We then use this theory to explore improve…

Optimizing multigrid reduction-in-time (MGRIT) and Parareal coarse-grid operators for linear advection

Parallel-in-time methods, such as multigrid reduction-in-time (MGRIT) and Parareal, provide an attractive option for increasing concurrency when simulating time-dependent PDEs in modern high-performance computing environments. While these techniques have been very successful for parabolic equations, it has often been observed that their performance suffers dramatically when applied to advection-dominated problems or purely hyperbolic PDEs using standard rediscretization approaches on coarse grids. In this paper, we apply MGRIT or Parareal to the constant-coefficient linear advection equation, …

并行时间方法Parareal和多网格时间还原的必要条件及紧双层收敛界

Necessary Conditions and Tight Two-level Convergence Bounds for Parareal and Multigrid Reduction in Time

本文研究了Parareal和MGRiT在解决线性问题时的收敛性条件，提出了紧双层收敛界，并引入了时间近似性质（TAP）来分析时间步进方案的逼近能力。