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arXiv · 1912.09240

Parareal computation of stochastic differential equations with time-scale separation: a numerical study

Tony Lelièvre·Frédéric Legoll·Keith Myerscough·Giovanni Samaey
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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 low-dimensional 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.

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Tony Lelièvre, Frédéric Legoll, Keith Myerscough, Giovanni Samaey. 2019-12-19. Parareal computation of stochastic differential equations with time-scale separation: a numerical study. https://arxiv.org/abs/1912.09240

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