## Departmental Colloquium

Eitan Tadmor

University of Maryland

Images, PDEs and hierarchical construction of solutions with critical regularity

**Abstract:**Edges are noticeable features in images which can be extracted from noisy data using different variational models. The analysis of such models leads to the question of representing general L^2-data as the divergence of uniformly bounded vector fields. We use a multi-scale approach to construct uniformly bounded solutions of div U=f for general f’s in the critical regularity space L^2(T^2). The study of this equation and related problems was motivated by recent results of Bourgain & Brezis. The intriguing critical aspect here is that although the problems are linear, construction of their solution is not. These constructions are special cases of a rather general framework for solving linear equations in representations U=\sum_j u_j which we introduced earlier in the context of image processing, yielding a multi-scale decomposition of "image" U.

Friday October 6, 2017 at 3:00 PM in SEO 636