Parallel High-Order Integrators

Abstract

Abstract. In this work we discuss a class of defect correction methods which is easily adapted to create parallel time integrators for multi-core architectures and is ideally suited for developing methods which can be order adaptive in time. The method is based on Integral Deferred Correction (IDC), which was itself motivated by Spectral Deferred Correction by Dutt, Greengard and Rokhlin (BIT-2000).The method presented here is a revised formulation of explicit IDC, dubbed Revisionist IDC, which can achieve p th -order accuracy in "wall-clock time" comparable to a single forward Euler simulation on problems where the time to evaluate the right-hand side of a system of differential equations is greater than latency costs of inter-processor communication, such as in the case of the N -body problem. The key idea is to re-write the defect correction framework so that, after initial startup costs, each correction loop can be lagged behind the previous correction loop in a manner that facilitates running the predictor and M = p − 1 correctors in parallel on an interval which has K steps, where K p. We prove that given an r th -order Runge-Kutta method in both the prediction and M correction loops of RIDC, then the method is order r × (M + 1).The parallelization in Revisionist IDC uses a small number of cores (the number of processors used is limited by the order one wants to achieve). Using a four-core CPU, it is natural to think about fourth-order RIDC built with forward Euler, or eighth-order RIDC constructed with secondorder Runge-Kutta. Numerical tests on an N -body simulation show that RIDC methods can be significantly faster than popular Runge-Kutta methods such as the classical fourth-order RungeKutta scheme.In a PDE setting, one can imagine coupling RIDC time integrators with parallel spatial evaluators, thereby increasing the parallelization. The ideas behind RIDC extend to implicit and semiimplicit IDC and have high potential in this area.

50 Years of Time Parallel Time Integration

“…The integral deferred correction methods we have already seen in Subsection 4.4 in the context of PFASST can also be used to create naturally small scale parallelism [14]:…”

“…18 Classical application of integral deferred correction, picture taken from [14] deferred correction the quadrature formula for (33) at time t j,m+1 , using quadrature points at time t j,0 ,t j,1 , . .…”

Domain Decomposition Methods in Science and Engineering XXIII

“…In the relatively recent field of time parallelization, there are four main algorithmic techniques that have been investigated: methods based on multiple shooting (Chartier and Philippe 1993), like the parareal algorithm (Lions et al 2001) for which a detailed convergence analysis can be found in Gander and Vandewalle (2007) for the linear case and in Gander and Hairer (2008) for the nonlinear case; methods based on space-time decomposition, like classical Schwarz waveform relaxation (Bjørhus 1995;Gander and Stuart 1998;Giladi and Keller 2002) and optimized variants (Bennequin et al 2009;Halpern 2005, 2007;, and Dirichlet-Neumann and Neumann-Neumann waveform relaxation (Gander et al 2016b;Kwok 2014;Mandal 2014); space-time multigrid methods (Emmett and Minion 2012;Gander and Neumüller 2016;Hackbusch 1984;Horton and Vandewalle 1995); and direct time parallelization methods like tensor product methods (Maday and Rønquist 2008), RIDC (Christlieb et al 2010), and ParaExp (Gander and Güttel 2013); for an up to date overview and a historical perspective of these approaches, see Gander (2015). This is because modern supercomputers have now so many processors that often space parallelization strategies for evolution problems saturate before all available processors can be used.…”

Integrating an N-Body Problem with SDC and PFASST

“…We are not aware of any other studies that investigate SDC integration methods in conjunction with particle-based spatial solvers aside from a small, one-dimensional Nbody example in [2] for Revisionist Integral Deferred Corrections (RIDC).…”