Analysis of a micro-macro acceleration method with minimum relative entropy moment matching

Lelièvre, Tony; Samaey, Giovanni; Zieliński, Przemysław

doi:10.48550/arxiv.1801.01740

Cited by 2 publications

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“…There are many possible matching strategies that come with a different choice for I, see [10] for more details. In this work, we will use matching in Kullback-Leibler divergence, or relative entropy, which is based on information-theoretic considerations [26]. The objective is to minimize…”

Section: The Continuous Matching Operatormentioning

confidence: 99%

“…As we mentioned in the introduction of this Section, a problem arises when the extrapolated macroscopic state variables m n+1 fall outside the domain of the matching operator M. In this case, there is no probability distribution that is consistent with the extrapolated states, and the micro-macro acceleration algorithm fails. We call such behavior a matching failure [10,26]. When the extrapolation step ∆t is smaller, the extrapolated macroscopic state variables will lie closer to the macroscopic state variables of the prior distribution, for which there exists a probability distribution consistent with the prior state variables, namely the prior distribution itself.…”

Section: An Adaptive Time-stepping Strategy For Matching Failuresmentioning

confidence: 99%

“…One possibility is minimizing the L 2norm between the matched and prior distribution, but this choice does not guarantee positivity of the matched distribution [10]. In this text, we consider matching in Kullback-Leibler divergence, which is based on notions from information theory [26]. We provide a more detailed mathematical description of the algorithm in Section 2.…”

Section: Introductionmentioning

confidence: 99%

“…It is shown that micro-macro acceleration can converge to the exact dynamics of the SDE, even when fixing a finite time-scale separation, for both matching in the L 2 -norm [10] and in Kullback-Leibler divergence [26]. The conditions under which convergence has been demonstrated are the following: (i) the number of state variables L must increase to infinity; and (ii) the time steps present (the microscopic time step δt and the extrapolation time step ∆t), must tend to zero [10,26]. Additionally, it was shown that the method is stable for the extrapolation step sizes independent of the fast time scales in the system [11].…”

Section: Introductionmentioning

confidence: 99%

“…The convergence analysis in [10,26] reveals that the errors in the micro-macro acceleration method that are caused by the combination of extrapolation and matching, can be viewed as a time discretization error, with a size that depends on the extrapolation time step. In contrast, equationfree [19,20] and heterogeneous multiscale methods [3,13] introduce a modeling error, because they compute the time evolution of an approximate macroscopic model in which a numerical closure relation is introduced.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Efficiency of a micro-macro acceleration method for scale-separated stochastic differential equations

Vandecasteele¹,

Zieliński²,

Samaey³

2019

Preprint

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We discuss through multiple numerical examples the accuracy and efficiency of a micromacro acceleration method for stiff stochastic differential equations (SDEs) with a time-scale separation between the fast microscopic dynamics and the evolution of some slow macroscopic state variables. The algorithm interleaves a short simulation of the stiff SDE with extrapolation of the macroscopic state variables over a longer time interval. After extrapolation, we obtain the reconstructed microscopic state via a matching procedure: we compute the probability distribution that is consistent with the extrapolated state variables, while minimally altering the microscopic distribution that was available just before the extrapolation. In this work, we numerically study the accuracy and efficiency of micro-macro acceleration as a function of the extrapolation time step and as a function of the chosen macroscopic state variables. Additionally, we compare the effect of different hierarchies of macroscopic state variables. We illustrate that the method can take significantly larger time steps than the inner microscopic integrator, while simultaneously being more accurate than approximate macroscopic models.

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Section: The Continuous Matching Operatormentioning

confidence: 99%

Section: An Adaptive Time-stepping Strategy For Matching Failuresmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Efficiency of a micro-macro acceleration method for scale-separated stochastic differential equations

Vandecasteele¹,

Zieliński²,

Samaey³

2019

Preprint

Self Cite

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Convergence and stability of a micro–macro acceleration method: Linear slow–fast stochastic differential equations with additive noise

Zieliński

Vandecasteele

Samaey

2021

Journal of Computational and Applied Mathematics

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We analyse the convergence and stability of a micro-macro acceleration algorithm for Monte Carlo simulations of stiff stochastic differential equations with a time-scale separation between the fast evolution of the individual stochastic realizations and some slow macroscopic state variables of the process. The micro-macro acceleration method performs a short simulation of a large ensemble of individual fast paths, before extrapolating the macroscopic state variables of interest over a larger time step. After extrapolation, the method constructs a new probability distribution that is consistent with the extrapolated macroscopic state variables, while minimizing Kullback-Leibler divergence with respect to the distribution available at the end of the Monte Carlo simulation. In the current work, we study the convergence and stability of this method on linear stochastic differential equations with additive noise, when only extrapolating the mean of the slow component. For this case, we prove convergence to the microscopic dynamics when the initial distribution is Gaussian and present a stability result for non-Gaussian initial laws.

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Analysis of a micro-macro acceleration method with minimum relative entropy moment matching

Cited by 2 publications

References 0 publications

Efficiency of a micro-macro acceleration method for scale-separated stochastic differential equations

Efficiency of a micro-macro acceleration method for scale-separated stochastic differential equations

Convergence and stability of a micro–macro acceleration method: Linear slow–fast stochastic differential equations with additive noise

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