CDS/CIMMS Lunchtime Seminar: Modified Optimal Prediction and its Application to a Particle-Method Problem

Dr. Alina Chertock, Department of Mathematics, North Carolina State University

Many physical systems of scientific or engineering interest can have a large number of degrees of freedom. For example, a numerical simulation of a turbulent fluid flow, even at very modest Reynolds numbers, could require millions of variables and the computational resources required to work with such systems can be extreme. In many cases, however, we are not interested in all of the variables, but just want to know a handful of statistical quantities -- the time-averaged force on a wall, the average vorticity as a function of time, etc.  We, therefore, would like to find a dynamical system that has a substantially smaller number of degrees of freedom, yet still modeling interesting aspects of the original system with acceptable accuracy.

We are concerned with system reduction by statistical methods and, in particular, by the optimal prediction method. The method deals with problems that possess large and small scales and uses the conditional expectation to model the influence of the small scales on the large ones. In this talk, I will present different variants of the optimal prediction method as well as its  several approximations.  I will also demonstrate the performance of the method on a number of numerical examples, in which the original and modified optimal prediction methods have been applied to a system of ODEs obtained from a particle method discretization of a hyperbolic PDE.

This is joint work with D. Gottlieb and A. Solomonoff, Brown  University.