SEMINAR
Stochastic Partial Differential Equations: Motivating Examples and Numerical Issues
Michael Eckhoff
Program in Scientific Computing
University of Southern Mississippi
ABSTRACT
Stochastic (ordinary) differential equations (SDEs) were introduced in the early 1900s in the context of Brownian motion B. The historical develpment of SDEs does not parallel that of its deterministic counterpart. In fact, SDEs and ODEs require fundamentally different methods of analysis. The central theme of stochastic analysis is "white noise", the derivative of B.
Stochastic PDEs made their appearance in the 1970s, following the development of Brownian motion in Banach and Hilbert spaces. While stochasticians proceeded to study "SDEs in infinite-dimensional space," physicists were busy solving "SPDEs on a lattice". More recently, engineers have extended their traditional Ritz-Galerkin methods by introducing "stochastic finite elements," for random (uncertain) media.
The engineering community has not yet looked into SPDEs involving white noise. In this talk, it is argued that the Galerkin approach provides a natural implementation for SPDEs with white-noise forcing. Up until now, all such computer simulations have employed rather unsophisticated finite difference schemes. There are valid reasons for this, due to the difficulties posed by white noise. These problems are discussed, along with suggestions for handling the discretization procedure.
Given that white noise has several representations (all of which are non-trivial), one is easily discouraged when considering the use of stochastic models. "Is it worth it?" is a legitimate question. Examples from physics, meterology and biology are presented to show some interesting effects which only noise seems to produce.
WHERE: TEC 251
WHEN(day): November 5th, 1999
WHEN(time): 2:00 PM
EVERYBODY IS INVITED