johanna.mcbryde@icms.org.uk      0131 650 9816

Homogenisation II: Stochastic Problems

This course is run by MIGSAA. It is open to everyone, including non-MIGSAA students, as an SMSTC supplementary module.

HOMOGENISATION II: STOCHASTIC PROBLEMS

In this course, various concepts on stochastic homogenization theory are introduced with
a view on multiscale modelling of multiphase systems. Stochastic homogenisation is a reliable
and systematic theory for averaging partial di erential equations de ned on strongly
heterogeneous media and domains with random characteristics. It has a wide range of
applications from (reactive) transport in porous media, over to waves in heterogeneous
media up to material science such as energy storage/conversion devices. It allows to rigorously
derive e ective macroscopic properties of strongly heterogeneous random media such
as composite materials, the e ective macroscopic formulation of microscopic systems, as
well as the stable construction of multiscale computational schemes. We shall illustrate
these features by considering various examples from continuum mechanics and physics of
composite materials and porous media.

Structure of the course:
We begin with deriving e ective stochastic di erential equations (SDEs) based on the
asymptotic two-scale expansion method. This provides a systematic tool to derive e ective
di usion coecients for heterogeneous materials. We will brie
y discuss how numerically
solve SDEs and their e ective/upscaled formulation. We then introduce the two-scale
convergence methodology which is the basis for the stochastic two-scale convergence and
the stochastic two-scale convergence in the mean. As before, we will discuss how to compute
a numerical approximation of the resulting limit problems. The last topic of the course will
be on general concepts of percolation theory and investigate similarities and di erences to
homogenization. Again, we will also give the basic ideas how to computationally study
percolation problems. All these topics will be discussed based on examples and applications
such as interacting particle systems under uncertainty/randomness, and if time allows we
look also at the theory of fluctuations and correlations.
Assessment: (20 hrs give 15 credits)
 7 credits for active participation
 8 credits (for worked out lecture notes under guidance of instructor)

Recommended but optional prerequisites:
This course is a follow up on the course HOMOGENIZATION, but students and researchers
interested in stochastic averaging techniques will be able to follow it without having attended
the rst course. The following experience is helpful but not required: Advanced
PDEs 1 and basic knowledge in Probability Theory and stochastic di erential equations
and associated Kolmogorov equations. Useful are also basic knowledge about Measure
and Integration Theory, and Functional Analysis. Basic knowledge about Galerkin/Finite
Element approximations and Finite Di erence methods.

Syllabus:
 general concepts of stochastic di erential equations and percolation theory; two-scale
convergence, stochastic two-scale convergence, and stochastic two-scale convergence
in the mean for elliptic PDEs; stochastic modelling of a hard-sphere interacting
particle system; nite di erence approximations of homogenized equations; nite
element discretizations of the stochastic two-scale limit;

Selection of references:
 ALLAIRE, G., Shape optimization by the homogenization method, Springer Verlag,
New York (2002).
 BOURGEAT, A., MIKELIC, A., and WRIGHT, S.. "Stochastic two-scale convergence
in the mean and applications." Journal fur die reine und angewandte Mathematik
456 (1994).
 BENSOUSSAN, A., LIONS, J.L., PAPANICOLAOU, G., Asymptotic analysis for
periodic structures, North-Holland, Amsterdam (1978).
 PAVLIOTIS, G.A., STUART, A., Multiscale Methods: averaging and homogenization,
Springer (2008).