Stochastic Optimal Prediction with Application to Averaged Euler Equations

TL;DR

TheOptimal prediction methods compensate for a lack of resolution in the numerical solution of complex problems through the use of an invariant measure as a prior measure in the Bayesian sense are explained.

Abstract

Optimal prediction (OP) methods compensate for a lack of resolution in the numerical solution of complex problems through the use of an invariant measure as a prior measure in the Bayesian sense. In first-order OP, unresolved information is approximated by its conditional expectation with respect to the invariant measure. In higher-order OP, unresolved information is approximated by a stochastic estimator, leading to a system of random or stochastic differential equations. We explain the ideas through a simple example, and then apply them to the solution of Averaged Euler equations in two space dimensions.

Figures (2)

• Figure 1: Comparison of autocorrelation widths $\sigma({\bf k})$ in the invariant measure and in a specific run with prescribed partial data. Widths are scaled by multiplying by the magnitude of $\bf k$.
• Figure 2: Comparison of decay of the mean A-enstrophy. The physical domain is $[0,2\pi]\times[0,2\pi]$. The resolved region in wave-number space is $[-5,5]\times[-5,5]$. The parameter $a$ in the Average Euler equation was taken to be 1. First curve is the true decay as calculated by Monte Carlo. Second curve results from approximating system as stochastic differential equation.