Convergence of Optimal Prediction for Nonlinear Hamiltonian Systems

Optimal prediction is a computational method for systems that cannot be properly resolved, in which the unresolved variables are viewed as random. This paper presents a first analysis of optimal prediction as a numerical method. We prove the convergence of the scheme for a class of equations of Schrödinger type and derive error bounds for the mean error between the optimal prediction solution and the set of exact solutions with random initial data. It is shown that optimal prediction is the scheme that minimizes the mean truncation error.