Efficiency of a Micro-Macro Acceleration Method for Scale-Separated Stochastic Differential Equations

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

“…Combining ( 16) with (17) results in (15). Since the effective domain of X n−1 was equal to R d , the right-hand side of ( 15) is finite for all θ.…”

“…Proof. To establish stability, we employ the recursive relation (15). However, due to the presence of the Lagrange multipliers λ s n in the argument of K n and in the additional last term of (15), we cannot immediately pass to the limit as n goes to +∞.…”

“…Let us now return to (15). We show by induction that for all θ ∈ R d the function r → K n (rθ) is asymptotically quadratic in r and find the recursion for the corresponding constants w n (θ).…”

“…In this section, we numerically illustrate the convergence and stability results of the previous sections on an academic example. As in [5,15], we define a linear slow-fast SDE with additive noise, in which we add a periodic forcing to the slow component…”

“…Once both convergence and stability have been established, one can look at the appropriate selection of the extrapolation time step and the number of macroscopic state variables for accuracy and efficiency. In [15], we recently investigated the accuracy and efficiency of the micro-macro acceleration on slowfast systems, showing numerically that micro-macro acceleration can simultaneously take larger time steps than the microscopic time integrator, while obtaining a smaller error than approximate macroscopic models for the slow component of the system.…”