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CNA Seminar/Colloquium/Joint Pitt-CNA Colloquium
Konstantinos Zygalakis
Oxford
Title: Qualitative behaviour of numerical methods for SDEs and application to homogenization

Abstract: In this talk we will focus on some analytical tools that can been used to explain the qualitative behaviour of numerical methods for stochastic differential equations (SDEs). These tools (backward error analysis, modified equations) are well developed in the case of numerical methods for ordinary differential equations (ODEs), but such a systematic theory is currently lacking in the case of SDEs. We will start by quickly recapping how to derive modified equations in the case of ODEs, and describe how these ideas can be generalized in the case of SDEs. Results will be presented for first order weak methods, such as the Euler-Maruyama and the Milstein method. We will then focus on a specific example relating with the homogenization of passive tracers in a cellular flow in the limit of vanishing molecular diffusion, and explain how the theory of modified equations can guide us towards choosing a suitable numerical method for such a problem. This method will then be used for calculating quantities of interest, such as the effective diffusivity and exit times. If there is some time, we will briefly discuss the use of modified equations as a tool for constructing higher order methods for SDEs.

Recording: http://vnc.math.cmu.edu/cna/CNA-Zygalakis-Oct-20-2011.avi
Pdf File: ZygalakisKonstantinos.pdf
Date: Thursday, October 20, 2011
Time: 2:30 pm
Location: Wean Hall 7218
Submitted by:  Gautam Iyer