Non-Markovian Stochastic Reduced Equations on the Fly

Abstract

In this chapter, we consider a new stochastic PM candidate obtained as the pullback limit associated with the two-layer backward-forward system of Chap. 4; and we illustrate that the corresponding reduced system, when applied to the stochastic Burgers-type equation introduced in Chap. 6, provides better performances in modeling the dynamics of the resolved modes when compared with those achieved by the reduced system based on $\widehat{h}^{(1)}_\lambda$. Methodological aspects are presented in Sects. 7.1 and 7.2, where a numerical procedure is described to determine “on the fly” the reduced random vector field (based on $\widehat{h}^{(2)}_\lambda$) along a trajectory $\xi (t,\omega )$ generated by the latter as the time is advanced. This method is particularly useful when no analytic formulas of $\widehat{h}^{(2)}_\lambda$ are available. The substitutive cornerstone in this case, is the pullback characterization of $\widehat{h}^{(2)}_\lambda$ given by (4.45), which allows us to update the reduced vector field once $\xi (t,\omega )$ is known at a particular time instance $t$. As illustrated on the stochastic Burgers-type equation, it is shown in Sect. 7.5 that the statistics of the large excursions present in the SPDE dynamics projected onto the resolved modes are reproduced with a very good accuracy from simulations of the reduced dynamics based on $\widehat{h}^{(2)}_\lambda$.