Stochastic Optimal Prediction for the Kuramoto--Sivashinsky Equation

We examine the problem of predicting the evolution of solutions of the Kuramoto--Sivashinsky equation when initial data are missing. We use the optimal prediction method to construct equations for the reduced system. The resulting equations for the resolved components of the solution are random integrodifferential equations. The accuracy of the predictions depends on the type of projection used in the integral term of the optimal prediction equations and on the choice of resolved components. The novel features of our work include the first application of the optimal prediction formalism for the dimensional reduction of a nonlinear, non-Hamiltonian system of equations and the use of a noninvariant measure constructed through inference from empirical data.