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BIT Numerical Mathematics

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24-02-2020

Study of micro–macro acceleration schemes for linear slow-fast stochastic differential equations with additive noise

Authors: Kristian Debrabant, Giovanni Samaey, Przemysław Zieliński

Published in: BIT Numerical Mathematics | Issue 4/2020

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Abstract

Computational multi-scale methods capitalize on a large time-scale separation to efficiently simulate slow dynamics over long time intervals. For stochastic systems, one often aims at resolving the statistics of the slowest dynamics. This paper looks at the efficiency of a micro–macro acceleration method that couples short bursts of stochastic path simulation with extrapolation of spatial averages forward in time. To have explicit derivations, we elicit an amenable linear test equation containing multiple time scales. We make derivations and perform numerical experiments in the Gaussian setting, where only the evolution of mean and variance matters. The analysis shows that, for this test model, the stability threshold on the extrapolation step is largely independent of the time-scale separation. In consequence, the micro–macro acceleration method increases the admissible time steps far beyond those for which a direct time discretization becomes unstable.

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Not to be confused with numerical stability (necessary for convergence) that measures the robustness of a numerical scheme with respect to perturbations, such as round-of errors, over finite time horizon as step-size tends to zero.

We use \(\varPi \) to unify the notation between this section and the subsequent one, in which we will project onto a lower-dimensional subspace. Here, one can think of \(\varPi \) as the projection from \(\mathbb {R}^d\) onto \(\mathbb {R}^d\).

More generally, the sufficient and necessary condition requires that the pair \((A,\overline{B})\) is controllable, see [24, pp. 355–356]. Assuming controllability only, we may not always be able to reduce (3.3) to (3.2). However, as (3.2) serves as a convenient test equation for asymptotic stability, this discrepancy is of minor importance.

The identity \(P = P^s\otimes P^{f|s}\) means that for every measurable rectangle \(U\times V\subseteq \mathbb {R}^{d_s}\oplus \mathbb {R}^{d_f}\) and Borel function \(g:U\times V\rightarrow \mathbb {R}\), it holds \(\int _{U\times V}g(x)\,P(\text { d}{x})=\int _{U}\int _{V}g(y,z)\,P^{f|s}(\text { d}{z}|y)\,P^s(\text { d}{y})\).

Stating that the matching of a prior \(\mathscr {N}_{\mu ,\varSigma }\) with a slow mean \(\overline{\mu }^s\) and a slow variance \(\overline{\varSigma }^s\) results in the normal distribution with the fast mean \(\overline{\mu }^f=\mu ^f+C^{\!\mathsf {T}}(\varSigma ^s)^{-1}(\overline{\mu }^s-\mu ^s)\) and fast variance \(\overline{\varSigma }^f=\varSigma ^f - C^{\!\mathsf {T}}(\varSigma ^s)^{-1}(C-\overline{C})\) where \(\overline{C}=C^{\!\mathsf {T}}(\varSigma ^s)^{-1}\overline{\varSigma }^s\).

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Title: Study of micro–macro acceleration schemes for linear slow-fast stochastic differential equations with additive noise
Authors: Kristian Debrabant
Giovanni Samaey
Przemysław Zieliński
Publication date: 24-02-2020
Publisher: Springer Netherlands
Published in: BIT Numerical Mathematics / Issue 4/2020
Print ISSN: 0006-3835
Electronic ISSN: 1572-9125
DOI: https://doi.org/10.1007/s10543-020-00804-5

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