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

Lelièvre, Tony; Samaey, Giovanni; Zieliński, Przemysław

doi:10.1016/j.spa.2019.10.008

Cited by 3 publications

(14 citation statements)

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“…Such a prior is naturally available as the final distribution from the simulation step of the micro-macro acceleration method. In this work, we use matching introduced in [12,13] and based on minimizing the Kullback-Leibler divergence (also called relative entropy)…”

Section: The Micro-macro Acceleration Algorithmmentioning

confidence: 99%

“…In this paper, we are particularly interested in the matching procedure that reconstructs a full microscopic distribution based only on the slow mean. For more general matching operators, see [12,13]. In this case, denoting byμ s the mean of the slow component, the matching reads…”

Section: Matching With the Slow Meanmentioning

confidence: 99%

“…There are many ways of measuring the difference between probability distributions. Following [13], we choose relative entropy or Kullback-Leibler divergence in the present paper, based on information theoretic considerations; for some other strategies see [12].…”

Section: Introductionmentioning

confidence: 99%

“…A few fundamental properties of micro-macro acceleration have already been investigated. First, in [13], convergence was studied for general SDEs and for any time-scale separation. The analysis requires fixing an infinite hierarchy of macroscopic state variables that are used for extrapolation.…”

Section: Introductionmentioning

confidence: 99%

“…Dragging the microscopic time step to zero, we further obtain convergence to the exact dynamics of the slow-fast SDE. In contrast to the convergence result from [13], this analysis does not require any hierarchy of macroscopic state variables; both the mean of the fast component and the variance of the SDE are never extrapolated.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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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Section: The Micro-macro Acceleration Algorithmmentioning

confidence: 99%

Section: Matching With the Slow Meanmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Legoll

Lelièvre

Myerscough

et al. 2020

Comput. Visual Sci.

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

2020

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Computational multi-scale methods capitalize on large separation between different time scales in a model to efficiently simulate its slow dynamics over long time intervals. For stochastic systems, the focus lies often on the statistics of the slowest dynamics and most methods rely on an approximate closed model for slow scale or a coupling strategy that alternates between the scales. 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 first elicit an amenable linear model that can accommodate multiple time scales. For this test model, we show that the stability threshold on the extrapolation step, above which the simulation breaks down, is largely independent from the time-scale separation parameter of the linear model, which severely restricts the time step of direct path simulation. We make derivations and perform numerical experiments in the Gaussian setting, where only the evolution of mean and variance matters. Our results demonstrate that the micro-macro acceleration method increases the admissible time step for multi-scale systems beyond step sizes for which a direct time discretization becomes unstable.

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

Cited by 3 publications

References 51 publications

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

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

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

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

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