Department of Mathematics

Speaker: Guillaume Bal
Title: Equations with Random Coefficients and Theories of Random Fluctuations
Affiliation: Columbia University
Date: Tuesday, March 5, 2013
Place and Time: Room 204-B, Love Building, 3:35-4:30 pm

Abstract. Problems with small scale structures abound in applied sciences. Their detailed microscopic description is often not available or generates computationally intractable problems. Homogenization theory has then been developed to understand the influence of micro-structures at a macroscopic level. The assumptions on the micro-structure under which homogenization holds, such as, e.g., periodicity, quasi-periodicity, or stationarity and ergodicity, are often not satisfied in practice. However, the homogenization point of view proves to be very fruitful in, e.g., macroscopic parameter estimations and to assess how multi-scale algorithms fare in well-controlled settings.

Equally important in practice, but often much difficult to study, is the analysis of random fluctuations beyond the homogenization limit. These fluctuations model noise in parameter estimation measurements, which limits the reconstruction resolution. In this talk, I will review several recent results obtained on the random fluctuations of solutions of partial differential equations with random coefficients. We then analyze conditions under which multiscale algorithms correctly capture such random fluctuations.