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

Abstract

Parareal and multigrid reduction in time (MGRiT) are two of the most popular parallel-in-time methods. The basic idea is to treat time integration in a parallel context by using a multigrid method in time. If Φ is the (fine-grid) time-stepping scheme of interest, such as RK4, then let Ψ denote a "coarse-grid" time-stepping scheme chosen to approximate k steps of Φ, where k ≥ 1. In particular, Ψ defines the coarse-grid correction, and evaluating Ψ should be (significantly) cheaper than evaluating Φ k . Parareal is a two-level method with a fixed relaxation scheme, and MGRiT is a generalization to the multilevel setting, with the additional option of a modified, stronger relaxation scheme.A number of papers have studied the convergence of Parareal and MGRiT. However, there have yet to be general conditions developed on the convergence of Parareal or MGRiT that answer simple questions such as, (i) for a given Φ and k, what is the best Ψ, or (ii) can Parareal/MGRiT converge for my problem? This work derives necessary and sufficient conditions for the convergence of Parareal and MGRiT applied to linear problems, along with tight two-level convergence bounds, under minimal additional assumptions on Φ and Ψ. Results all rest on the introduction of a temporal approximation property (TAP) that indicates how Φ k must approximate the action of Ψ on different vectors. Loosely, for unitarily diagonalizable operators, the TAP indicates that the fine-grid and coarse-grid time integration schemes must integrate geometrically smooth spatial components similarly, and less so for geometrically high frequency. In the (non-unitarily) diagonalizable setting, the conditioning of each eigenvector, v i , must also be reflected in how well Ψv i ∼ Φ k v i . In general, worst-case convergence bounds are exactly given by min ϕ < 1 such that an inequality along the lines of (Ψ − Φ k )v ≤ ϕ (I − Ψ)v holds for all v. Such inequalites are formalized as different realizations of the TAP in Section 2, and form the basis for convergence of MGRiT and Parareal applied to linear problems.

Convergence analysis for parallel‐in‐time solution of hyperbolic systems

“…Using the standard norm inequality for an operator E Δ , the convergence behavior of a two‐level method is then predicted by calculating the bounds that have since been shown to be accurate to order . Assuming that | μ n |≠1 for all n =1,…, N x , we obtain …”

“…We consider solving a time-stepping problem (19), arising from a scalar PDE, by two-level MGRIT. As above, let Φ and Φ c denote the two time-stepping operators on the fine time grid with N t time intervals and on the coarse time grid with N T = N t ∕m time intervals, respectively.…”

“…that have since been shown to be accurate to order (1∕N T ). 19 Assuming that | n | ≠ 1 for all n = 1, … , N x , we obtain 17…”

“…One possible generalization of the two-level reduction analysis to handle this case is to derive bounds for the coarse-grid iteration matrix in a mass matrix-induced norm instead of in the L 2 -norm. 17,19 Here, to enable a comparison of the different analysis techniques for the systems case, we derive bounds of the L 2 -norm of the coarse-grid iteration matrices. We consider q × q block time-stepping operators, Φ and Φ c , arising from a semidiscretization in space of a system of q scalar PDEs with q unknown functions; note that in the case of a PDE system with vector equations, we first have to break down the vectors into their scalar components.…”

“…In the context of space-time discretizations, three approaches to mode analysis have been discussed in the recent literature: space-time local Fourier analysis (LFA), 4,13,14 semi-algebraic mode analysis (SAMA), 9,15,16 and reduction-based theory. 8,[17][18][19] A fourth approach, Fourier-Laplace mode analysis, [20][21][22] can be seen as an intermediate between space-time LFA and the other two approaches and will be discussed only in that context. While not considered here, the recent analysis of Gander et al 23 extends the earlier result of Gander and Hairer 24 from the parareal context to MGRIT.…”