Multi-scale construction of solutions to problems with critical regularity

Multi-scale construction of solutions to problems with critical regularity

###### Applied Math And Analysis Seminar

#### Eitan Tadmor (University of Maryland)

**Wednesday, February 14, 2018 -12:00pm to 1:00pm**

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 fs in the critical regularity space L^2(T^2). The study of this equation and related problems was motivated by 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 critical regularity spaces. The solutions are realized in terms of nonlinear hierarchical representations $U = \sum_j u_j$ which we introduced earlier in the context of image processing, yielding a multi-scale decomposition of images U.