Localization and Uncertainties in Homogenization

Houman Owhadi
California Institute of Technology

In the first part of this talk we show how to construct localized elliptic cell problems for homogenization with non-separated scales, high-contrast and arbitrary deterministic coefficients. Randomness, scale separation, mixing or ``epsilon-sequences'' are not required because the proposed method solely relies on the compactness of the solution space. The support of cell problems can be localized to arbitrarily small subsets of the whole domain and explicit approximation error estimates are obtained as a function of the size of those subsets. In the second part this talk we consider the situation where coefficients (corresponding to microstructure and source terms) are random and have an imperfectly known probability distributions. Treating those distributions as optimization variables (in an infinite dimensional, non separable space) we obtain optimal bounds on probabilities of deviation of solutions. Surprisingly, explicit and optimal bounds show that uncertainties do not necessarily propagate across scales. This first part of the talk is a joint work with Lei Zhang (University of Oxford). Elements of the second part are joint work with C. Scovel, T. Sullivan, M. McKerns and M. Ortiz.

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