We present and analyse a micro-macro acceleration method for the Monte Carlo simulation of stochastic differential equations with separation between the (fast) timescale of individual trajectories and the (slow) time-scale of the macroscopic function of interest. The algorithm combines short bursts of path simulations with extrapolation of a number of macroscopic state variables forward in time. The new microscopic state, consistent with the extrapolated variables, is obtained by a matching operator that minimises the perturbation caused by the extrapolation. We provide a proof of the convergence of this method, in the absence of statistical error, and we analyse various strategies for matching, as an operator on probability measures. Finally, we present numerical experiments that illustrate the effects of the different approximations on the resulting error in macroscopic predictions.

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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 [12], 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.

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.

Let Γ denote an uncountable set. We consider the questions if a Banach space X of the form C (K ) of a given class (1) has a complemented copy of c 0 (Γ ) or (2) for every c 0 (Γ ) ⊆ X has a complemented c 0 (E) for an uncountable E ⊆ Γ or (3) has a decomposition X = A ⊕ B where both A and B are nonseparable. The results concern a superclass of the class of nonmetrizable Eberlein compacts, namely K s such that C (K ) is Lindelöf in the weak topology and we restrict our attention to K s scattered of countable height. We show that the answers to all these questions for these C (K )s depend on additional combinatorial axioms which are independent of ZFC ± CH. If we assume the P -ideal dichotomy, for every c 0 (Γ ) ⊆ C (K ) there is a complemented c 0 (E) for an uncountable E ⊆ Γ , which yields the positive answer to the remaining questions. If we assume ♣, then we construct a nonseparable weakly Lindelöf C (K ) for K of height ω + 1 where every operator is of the form cI + S for c ∈ R and S with separable range and conclude from this that there are no decompositions as above which yields the negative answer to all the above questions.Since, in the case of a scattered compact K , the weak topology on C (K ) and the pointwise convergence topology coincide on bounded sets, and so the Lindelöf properties of these two topologies are equivalent, many results concern also the space C p (K ).