# EPSRC Symposium Capstone Conference

### Minisymposium: Numerical Methods for Multiscale Stochastic Systems Organizer: G.A. Pavliotis Friday 3 July 2009

#### Speakers:

C. Hartmann (FU Berlin) chartman@mi.fu-berlin.de Model reduction of partially-observed Langevin equations

We study balanced model reduction of a certain class of stochastic dif ferential equations. In doing so, we adopt ideas from large deviations theory and discuss notions of controllability and obervability for linear dissipative Hamiltonian systems with degenerate noise term, also known as Langevin equations. For partially observed Langevin equations, we illustrate model reduction by balanced truncation with an example from molecular dynamics and discuss aspects of structure-preservation.

M. Katsoulakis (U. Crete and U. Mass Amherst) markos@math.umass.edu Hierarchical and multi-level coarse-graining methods

We will discuss a variety of coarse-graining methods for many-body microscopic systems. We focus on mathematical, numerical and statistical methods allowing us to assess the parameter regimes where such approximations are valid. We also demonstrate, with direct comparisons between microscopic (DNS) and coarse-grained simulations, that the derived mesoscopic models can provide a substantial CPU reduction in the computational effort.
Furthermore, we discuss the feasibility of spatiotemporal adaptivity methods for the coarse-graining of microscopic simulations, having the capacity of automatically adjusting during the simulation if substantial deviations are detected in a suitable error indicator. Here we will show that in some cases the adaptivity criterion can be based on a posteriori estimates on the loss of information in the transition from a microscopic to a coarse-grained system.
Finally, motivated by related problems in the simulation of macromolecular systems, we discuss mathematical strategies for reversing the coarse-graining procedure. The principal purpose of such a task is recovering local microscopic information in a large system by first employing inexpensive coarse-grained solvers.

F. Legoll (ENPC) legoll@lami.enpc.fr Effective dynamics using conditional expectations

We consider a system described by its position $X_t$, that evolves according to the overdamped Langevin equation, and a scalar function $\xi(X)$ of the state variable $X$. Our aim is to design a one-dimensional dynamics that approximates the evolution of $\xi(X_t)$. Using conditional expectations, we build an original dynamics, whose accuracy is supported by error estimates between the laws of the two processes, at any fixed time. Simple numerical simulations illustrate the efficiency of the approach as well as the accuracy of the proposed dynamics according to various criteria, including residence times in potential energy wells.

K.C. Zygalakis (Oxford) zygalakis@maths.ox.ac.uk On the existence and applications of modified equations for stochastic differential equations

In this talk we describe a general framework for deriving modified equations for stochastic differential equations with respect to weak convergence. Results are presented for first order methods such as the Euler-Maruyama and the Milstein method. In the case of linear SDEs, using the Gaussianity of the underlying solutions, we derive a SDE which the numerical method solves exactly in the weak sense. Applications of modified equations in the numerical study of Langevin equations and in the calculation of effective diffusivities are also discussed.

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