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

Przemysław Zieliński, Hannes Vandecasteele, Giovanni Samaey

TL;DR

The paper studies a micro-macro acceleration method to accelerate Monte Carlo simulations of stiff linear SDEs with time-scale separation, by interleaving short microscopic runs with extrapolation of slow macroscopic variables and KL-based matching to rebuild a consistent microscopic distribution. It proves convergence to the exact microscopic dynamics when extrapolating only the slow mean for Gaussian initial conditions, and establishes stability for non-Gaussian tails via cumulant generating function analysis. The theoretical results are complemented by numerical experiments on a periodically forced slow-fast system, demonstrating that smaller extrapolation steps reduce error and that the method remains stable even with large extrapolation steps. The work advances efficient simulation of slow-fast stochastic systems and provides a rigorous foundation for using KL-based matching in this coarse-grained framework.

Abstract

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.

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

TL;DR

The paper studies a micro-macro acceleration method to accelerate Monte Carlo simulations of stiff linear SDEs with time-scale separation, by interleaving short microscopic runs with extrapolation of slow macroscopic variables and KL-based matching to rebuild a consistent microscopic distribution. It proves convergence to the exact microscopic dynamics when extrapolating only the slow mean for Gaussian initial conditions, and establishes stability for non-Gaussian tails via cumulant generating function analysis. The theoretical results are complemented by numerical experiments on a periodically forced slow-fast system, demonstrating that smaller extrapolation steps reduce error and that the method remains stable even with large extrapolation steps. The work advances efficient simulation of slow-fast stochastic systems and provides a rigorous foundation for using KL-based matching in this coarse-grained framework.

Abstract

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.

Paper Structure

This paper contains 21 sections, 11 theorems, 74 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Slow-fast linear SDEs with additive noise
  3. Outline of the paper
  4. The micro-macro acceleration algorithm
  5. Matching with the slow mean
  6. The complete micro-macro acceleration method
  7. Step 1: Microscopic time integration
  8. Step 2: Restriction
  9. Step 3: Extrapolation
  10. Step 4: Matching
  11. Convergence of the micro-macro acceleration method with slow mean extrapolation
  12. An iterative formula for slow mean-only extrapolation
  13. Convergence theorem
  14. Stability of micro-macro acceleration with initial condition with Gaussian tails
  15. Micro-macro step in terms of the cumulant generating function
  16. ...and 6 more sections

Key Result

Proposition 1

Let $\bar{Q}=\mathcal{M}(\bar{\mu}^s, P)$ be the solution to eq:slowmeanmatching and $\bar{Q}^s=\mathcal{M}(\overline{\mu}^s, P^s)$ be the solution to the matching of slow prior marginal $P^s$. Then, the matching densities satisfy and, in particular, all slow observables of $\bar{Q}$ equal the corresponding observables of $\bar{Q}^s$. Moreover, for any function $f$ on $\mathbb{R}^{d_f}$, the fast

Figures (2)

  • Figure 1: The error in the slow mean of micro-macro acceleration as a function of the extrapolation step size $\Delta t$, computed against the analytic solution \ref{['eq:lineardrivensolution']} (blue) and against the numerical result obtained by the Euler-Maruyama method with time step $\delta t$ (orange) for $\varepsilon = 0.5$ (left) and $\varepsilon=0.05$ (right). We clearly see that the micro-macro acceleration error decreases when $\Delta t$ decreases, as given by Theorem \ref{['thm:linearslowconvergence']}. Moreover, for small $\varepsilon$ there is almost no difference between the error computed against the analytic solution and the microscopic time integrator as the latter is very accurate. This difference is higher for larger $\varepsilon$.
  • Figure 2: The $(\delta t, \Delta t)$ stability plane of micro-macro acceleration on the linear driven system \ref{['eq:lineardrivensde']}. A blue dot indicates stability and a red dot instability. When at least one matching failure occurs, we mark an instability at the corresponding time step sizes. The stability domain has a V-shaped domain, and for every microscopic time step, the maximal extrapolation time step is larger than the deterministic stability bound of 0.3.

Theorems & Definitions (23)

  • Proposition 1
  • proof
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • Definition 1
  • Example 1
  • Remark 1: What does $\Theta_0=\mathbb{R}^d$ mean?
  • ...and 13 more