18/10/2001, 17:00 — 18:00 — Amphitheatre Pa1, Mathematics Building
Alexandre Chorin, University of California at Berkeley
Prediction from incomplete calculations
There are many nonlinear initial value problems whose solutions are too complex to be fully resolved numerically; typically, these are problems where one cares only about the behavior of the solutions on one scale, while the dynamics involve many scales. The problem is to find effective equations on the scale of interest and average properly on the other scales.
In optimal prediction methods one assumes that initial data for the degrees of freedom that cannot be resolved are drawn from a known probability distribution; this is a reasonable assumption in many cases. The problem is to make optimal predictions about the behavior of the solutions at later times given partial initial data and partial numerical resolution. An exact equation of motion for the resolved degrees of freedom can be written down, but its solution requires the solution of an auxiliary equation, the orthogonal dynamics equation. This is in general as difficult to solve as the original problem, but in special cases the solution of the orthogonal dynamics equation can be approximated in a straightforward way. The formalism is a generalization of the fluctuation/dissipation theory of irreversible statistical mechanics.
I shall give examples and explain some of the difficulties of these methods.