Parareal computation of stochastic differential equations with time-scale separation: a numerical convergence study
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PINT 2019
Parareal computation of stochastic differential equations with time-scale separation: a numerical convergence study Frédéric Legoll1 · Tony Lelièvre1 · Keith Myerscough2 · Giovanni Samaey2 Received: 19 December 2019 / Accepted: 15 July 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The parareal algorithm is known to allow for a significant reduction in wall clock time for accurate numerical solutions by parallelising across the time dimension. We present and test a micro-macro version of parareal, in which the fine propagator is based on a (high-dimensional, slow-fast) stochastic microscopic model, and the coarse propagator is based on a lowdimensional approximate effective dynamics at slow time scales. At the microscopic level, we use an ensemble of Monte Carlo particles, whereas the approximate coarse propagator uses the (deterministic) Fokker–Planck equation for the slow degrees of freedom. The required coupling between microscopic and macroscopic representations of the system introduces several design options, specifically on how to generate a microscopic probability distribution consistent with a required macroscopic probability distribution and how to perform the coarse-level updating of the macroscopic probability distribution in a meaningful manner. We numerically study how these design options affect the efficiency of the algorithm in a number of situations. The choice of the coarse-level updating operator strongly impacts the result, with a superior performance if addition and subtraction of the quantile function (inverse cumulative distribution) is used. How microscopic states are generated has a less pronounced impact, provided a suitable prior microscopic state is used. Keywords Parallel method · Stochastic differential equations · Time-scale separation
1 Introduction In many applications, a system is modelled using a highdimensional system of stochastic differential equations (SDEs) that capture phenomena occurring at multiple time scales. Usually the quantities of interest are macroscopic observables of the system, lower-dimensional than the full (microscopic) state, and evolving on a slower time scale Communicated by Daniel Ruprecht.
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Giovanni Samaey [email protected] Frédéric Legoll [email protected] Tony Lelièvre [email protected] Keith Myerscough [email protected]
1
Ecole des Ponts ParisTech and Inria, 6-8 avenue Blaise Pascal, Cité Descartes, 77455 Marne la Vallée, France
2
Department of Computer Science, KU Leuven, Celestijnenlaan 200A, 3001 Leuven, Belgium
than the fine-scale, microscopic dynamics. In such a setting, the computational cost of simulating the microscopic system over macroscopic time intervals may be prohibitive, due to both time step restrictions and the large number of degrees of freedom. One then often resorts to simulations using low-dimensional (coarse-grained, effective) models, in which the fast degrees of freedom are eliminated. Typically, this is done by taking a limit in which the time-scale separ
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