[Paper Review] Analysis of a micro-macro acceleration method with minimum relative entropy moment matching

Tony Lelièvre, Giovanni Samaey+1 more|arXiv (Cornell University)|Jan 5, 2018

Fluid Dynamics and Turbulent Flows54 references4 citations

TL;DR

This paper analyzes a micro-macro acceleration method for stiff stochastic differential equations with time-scale separation, using minimum relative entropy to match microscopic states to extrapolated macroscopic variables. The method ensures numerical stability and weak convergence by minimizing Kullback-Leibler divergence between prior and matched distributions, with rigorous error bounds in terms of extrapolation time step and macroscopic variable count.

ABSTRACT

We analyse convergence of a micro-macro acceleration method for the Monte Carlo simulation of stochastic differential equations with time-scale separation between the (fast) evolution of individual trajectories and the (slow) evolution of the macroscopic function of interest. We consider a class of methods, presented in [Debrabant, K., Samaey, G., Zieli\\'nski, P. A micro-macro acceleration method for the Monte Carlo simulation of stochastic differential equations. SINUM, 55 (2017) no. 6, 2745-2786], that performs short bursts of path simulations, combined with the extrapolation of a few macroscopic state variables forward in time. After extrapolation, a new microscopic state is then constructed, consistent with the extrapolated variable and minimising the perturbation caused by the extrapolation. In the present paper, we study a specific method in which this perturbation is minimised in a relative entropy sense. We discuss why relative entropy is a useful metric, both from a theoretical and practical point of view, and rigorously study local errors and numerical stability of the resulting method as a function of the extrapolation time step and the number of macroscopic state variables. Using these results, we discuss convergence to the full microscopic dynamics, in the limit when the extrapolation time step tends to zero and the number of macroscopic state variables tends to infinity.

Motivation & Objective

• To develop a stable and accurate micro-macro acceleration method for multiscale stochastic differential equations with fast microscopic dynamics and slow macroscopic observables.
• To address the ill-posed nature of matching microscopic states to extrapolated macroscopic variables by using relative entropy minimization as a regularized inference procedure.
• To rigorously analyze local errors and numerical stability as functions of the extrapolation time step and number of macroscopic variables.
• To establish convergence of the method to the true microscopic dynamics in the limit of vanishing time step and increasing macroscopic variable count.
• To justify the use of relative entropy as a theoretically and practically sound metric for distribution matching in multiscale simulations.

Proposed method

• The method uses short bursts of microscopic simulation with time step δt to estimate macroscopic time derivatives of L observables E[R_l(X_t)].
• It extrapolates the macroscopic state variables m_l over a larger time step Δt ≫ δt using estimated derivatives.
• A new microscopic distribution ν is constructed by minimizing the relative entropy I(ν||μ) subject to moment constraints E[R_l(X)] = m_l, where μ is the prior distribution from the last microscopic state.
• The minimization problem (1.3) is solved via exponential family parameterization, leading to a Lagrange multiplier formulation involving the log-partition function A(λ, μ).
• The matching step is interpreted as a projection in the space of probability measures using relative entropy, ensuring minimal perturbation from the prior.
• Theoretical analysis relies on continuity and boundedness of the Lagrange multipliers and their dependence on the macroscopic state and prior measure.

Experimental results

Research questions

• RQ1 How does the relative entropy minimization strategy affect the stability and local error of micro-macro acceleration methods?
• RQ2 What is the dependence of the numerical error on the extrapolation time step Δt and the number L of macroscopic variables?
• RQ3 Under what conditions does the micro-macro method converge to the true microscopic dynamics as Δt → 0 and L → ∞?
• RQ4 Why is relative entropy a suitable metric for matching microscopic states to macroscopic constraints in this context?
• RQ5 How sensitive is the matching procedure to perturbations in the prior measure and the extrapolated macroscopic state?

Key findings

• The method exhibits local error that decreases as the extrapolation time step Δt decreases, with convergence rates dependent on the smoothness of the underlying dynamics.
• Numerical stability is guaranteed under boundedness and continuity assumptions on the Lagrange multipliers and the Hessian of the log-partition function.
• The mapping from prior measure and macroscopic state to the matched distribution is continuous in total variation norm, ensuring robustness to input perturbations.
• The inverse of the Hessian of the log-partition function, which governs the sensitivity of the Lagrange multipliers, is uniformly bounded on compact sets, ensuring well-posedness of the optimization.
• The relative entropy minimization framework ensures that the matched distribution ν is the least informative perturbation of the prior μ consistent with the extrapolated macroscopic state.
• In the limit Δt → 0 and L → ∞, the method converges weakly to the true microscopic dynamics, establishing its consistency for long-time simulation.