| We consider the problem of predicting the long-term evolution of a large nonlinear system, when the system is too complex to be fully resolved in numerical computations. We assume that we have statistical information about the degrees of freedom that cannot be resolved. We discuss several methods of improving the underresolved computations by incorporating the statistical information about the unresolved degrees of freedom.; In this work, we build upon the optimal prediction methods developed by Chorin, Hald, Kast, Kupferman and Levy, which use statistical projections to evaluate the effect of the unresolved degrees of freedom on the small set of variables that are computed. The first-order optimal prediction method fails to reproduce the diminishing predictive value of partial initial data on the evolution of the system at later times. Higher order optimal prediction methods require a 'memory' term, which improves the accuracy of long-term computations. However, the evaluation of the memory term is often very expensive. We offer an alternative approach to the direct evaluation of the memory term by using a new stochastic model for the unresolved degrees of freedom, which generalize the widely used but often inaccurate Fokker-Plank equations without memory.; In particular, we explore the possibility of directly modelling the loss of information due to interaction between the computed variables and the unresolved degrees of freedom by adding an 'artificial dissipation' term to the first-order optimal prediction method. For a model problem based on the nonlinear Schroedinger equation, this term has a very simple form and remarkable scaling properties, allowing us to construct a fast method for underresolved computations. |
