2pm - 3pm
Friday 25 November 2016
On long time approximation of ergodic stochastic differential equations; Examples from homogenization and molecular dynamics
This is a joint DSPDE and FMS seminar. Kostas Zygalakis (Edinburgh) will be speaking.
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In this talk we will discuss a variety of different problems related to questions related to the long time behaviour of solutions of stochastic differential equations.
This is an important question with relevance in many fields of applied mathematics, such as modelling turbulent diffusion and molecular dynamics and can be addressed with a variety of different techniques such as for example homogenization.
Once the limiting behaviour has been established, one needs to design appropriate numerical algorithms that are able to capture it correctly. This is a very delicate problem and in addressing this, we will discuss how the recently developed theory of modified equations/backward error analysis for SDEs can explain the behaviour of existing stochastic numerical methods and guide the construction of more efficient ones.
G.A. Pavliotis, A.M. Stuart, and K. C Zygalakis, Calculating effective diffusivities in the limit of vanishing molecular diffusion. J. Comp. Phys. 228(4) 1030-1055, (2009).
K.C. Zygalakis, On the Existence and Applications of Modified Equations for Stochastic Differential Equations. SIAM J. Sci. Comput. 33(1), 102-130, (2011).
G. Iyer and K. C. Zygalakis. Numerical studies of homogenization under a fast cellular flow. Multiscale Model. Simul., 10(3):1046-1058, (2012).
A. Abdulle, G. Villmart, K. C. Zygalakis. High order numerical approximation of the invariant measure of ergodic SDEs. SIAM J. Num. Anal. 52(4):1600-1622, (2014).
A. Abdulle, G. Vilmart, and K.C. Zygalakis, Long time accuracy of Lie-Trotter splitting methods for Langevin dynamics . SIAM J. Numer. Anal., 53(1):1-16, (2015).