Optimal Prediction for Non-Linear Problems

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

The method of optimal prediction has been developed recently. It gives the best approximation to non-linear equations assuming that the higher modes are random. It is suited for computing complicated flows, when the information about the initial data is incomplete. The proposer will study the convergence of the optimal prediction method applied to the cubic Schrodinger equation. Such equations occur in quantum mechanics. In addition the method will be used to study the flow of incompressible and inviscid fluids in three dimensions. The complicated behavior of such fluids is part of the problem of turbulence. Turbulent flows typically consist of a slow, large scale motion, and a lot of small, but fast, eddies. Instead of computing the flow by a single numerical algorithm, the investigator will treat the small scale motion as a random component, and incorporate its influence on the large scale motion approximately. The goal is to develope faster and more robust computational schemes for turbulent flows.