# Meeting Details

Title: Localization and Uncertainties in Homogenization Applied Analysis Seminar Houman Owhadi, Caltech 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, with incomplete information on the probability distribution of the microstructure, uncertainties do not necessarily propagate across scales. Elements of the first part are joint work with Leonid Berlyand and Lei Zhang. Elements of the second part are joint work with C. Scovel, T. Sullivan, M. McKerns and M. Ortiz.