On “Optimal” h‐independent convergence of Parareal and multigrid‐reduction‐in‐time using Runge‐Kutta time integration

Abstract

Parareal and multigrid-reduction-in-time (MGRIT) are two popular parallel-in-time algorithms. The idea of both algorithms is to combine the (fine-grid) time-stepping scheme of interest with a "coarse-grid" time-integration scheme that approximates several steps of the fine-grid time-stepping method. Convergence of Parareal and MGRIT has been studied in a number of papers. Research on the optimality of both methods, however, is limited, with results existing only for specific time-integration schemes. This paper focuses on analytically showing ℎ -and ℎ -independent convergence of two-level Parareal and MGRIT, for linear problems of the form ′ ( ) +  ( ) = ( ), where  is symmetric positive definite and Runge-Kutta time integration is used. The analysis is based on recently derived tight bounds of two-level Parareal and MGRIT convergence that allow for analyzing arbitrary coarse-and finegrid time integration schemes, coarsening factors, and time-step sizes. The theory presented in this paper shows that not all Runge-Kutta schemes are equal from the perspective of parallel-in-time. Some schemes, particularly L-stable methods, offer significantly better convergence than others. On the other hand, some schemes do not obtain ℎ-optimal convergence, and two-level convergence is restricted to certain parameter regimes. In certain cases, an (1) factor change in time step ℎ can be the difference between convergence factors ≈ 0.02 and divergence! Numerical results confirm the analysis in the practical setting and, in particular, emphasize the importance of a priori analysis in choosing an effective coarse-grid scheme and coarsening factor. A Mathematica notebook to perform a priori two-grid analysis is available at https://github.com/XBraid/xbraid-convergence-est. Recently, two-level theory was developed by Southworth 15 that provides tight bounds on linear Parareal and two-level MGRIT for arbitrary time-propagation operators. The central aim of this paper is to use these tight bounds to analytically show ℎindependent convergence of two-level Parareal and MGRIT for linear problems of the form ′ ( )+ ( ) = ( ), where  is SPD and Runge-Kutta time integration is used. Thus, this paper highlights the practical implications of the new theory, particularly for parabolic-type PDEs with SPD spatial operators. The relatively clean formulae derived in 15 allow for straightforward convergence analysis of arbitrary coarse-and fine-grid propagators, coarsening factors, and time-step sizes. Therefore, this paper can be seen as a substantial generalization of previous works by Mathew et al. 25 and by Wu and Zhou 26, 28 on ℎ-independent convergence of Parareal. Our theoretical framework also encompasses analysis of various modified Parareal algorithms, such as the (scalar) -Parareal method 29 and modified A-/L-stable fine-grid propagators introduced in 30 .Because the underlying two-grid bounds used in this work are tight, they allow us to easily demonstrate that not all Runge-Kutta schemes are equal from the perspective of parallel-in-...

Weighted Relaxation for Multigrid Reduction in Time

“…Then, two-level convergence bounds are derived as a function of relaxation weight, providing insight on choosing the weight in practice. Although MGRIT uses full approximation storage (FAS) nonlinear multigrid cycling 25 to solve nonlinear problems, the linear two-grid setting makes analysis more tractable (e.g., [26][27][28][29][30] ), and MGRIT behavior for linear problems is often indicative of MGRIT behavior for related nonlinear problems 27 . Thus, consider a linear system of ordinary differential equations (ODEs) with spatial degrees of freedom, = ( ) + ( ), (0) = 0 , ∈ [0, ],…”

“…where the Runge-Kutta matrix 0 = ( , ) and weight vector 0 = ( 1 , ..., ) are taken from the Butcher tableau of an s-stage Runge-Kutta method 30 .…”

“…Here, we consider A-stable two-stage third-order SDIRK-23, L-stable two-stage second-order SDIRK-22, and L-stable threestage third-order SDIRK-33 methods (see Appendix of 30 for coefficients), where SDIRK refers to singly diagonally implicit Runge-Kutta. Figures 5 -7 depict the convergence bound (18) in the complex plane as a function of ℎ over various for these methods, respectively.…”

Multigrid methods

“…The special issue for this years conference consists of the nine papers 1‐9 . In Reference 1, Murray and Weinzierl develop a stabilized asynchronous FAC Multigrid solver for spacetrees, that is, meshes as they are constructed from octrees and quadtrees.…”

“…In Reference 8, Mitchell et al study the parallel performance of algebraic multigrid domain decomposition methods on multicore architectures. The paper, 9 by Friedhoff and Southworth, analyzes the optimal (h‐Independent) convergence of Parareal and multigrid reduction in time using Runge–Kutta time integration.…”