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

Abstract

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.

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

“…We also present a stability result for non-Gaussian initial laws that complements the previous findings in the Gaussian framework [14]. For this part, we consider a class of (non-Gaussian) initial conditions having Gaussian tails.…”

“…For this part, we consider a class of (non-Gaussian) initial conditions having Gaussian tails. Inspired by the case of Gaussian initial conditions analyzed in [14], where the explicit formulas are available, we prove that the stability of the micro-macro acceleration method hinges on the stability of the mean obtained from the matching procedure. In this case, however, due to the non-Gaussianity of distributions, we have to look at the propagation of all higher moments throughout the method.…”

“…It is then shown that convergence not only depends on taking the micro and extrapolation time steps to zero, but also on extrapolating an increasing number of macroscopic state variables as the extrapolation time step decreases. Second, in [14], the asymptotic numerical stability of the micromacro acceleration method was studied on a linear system of SDEs with Gaussian initial conditions. The stability criterion that was employed checks if the distributions obtained from the micro-macro acceleration method reach the equilibrium Gaussian distribution of the underlying microscopic integrator as the number of micro-macro time-steps tends to infinity.…”

“…Second, Ohrnstein-Uhlenbeck processes have an invariant distribution. We can then investigate whether micro-macro acceleration converges in distribution to the correct equilibrium distribution, as was done in [14] as a function of the time-scale separation and the extrapolation step size. Third, linear systems are popular in the context of ordinary differential equations to determine the stability of deterministic methods by an eigenvalue analysis of the drift matrix A.…”

“…Using the spectral decomposition theorem, we introduce the orthogonal projections Π s : R d → R ds and Π f : R d → R d−ds that map the full state space onto the 'slow' R ds and 'fast' R d−ds state spaces, respectively, which correspond to the gap in the spectrum of A. Such a procedure is also called 'coarse-graining' [14]. The decomposition is such that we can express the full state space and the drift matrix as…”