### Carola-Bibiane Schonlieb (University of Cambridge)

#### Domain decomposition for total variation regularisation and applications

*Thursday 9 February 2012 at 13.00, JCMB 4312 - Joint with ACM*

##### Abstract

Domain decomposition methods were introduced as techniques for solving partial
differential equations based on a decomposition of the spatial domain of the
problem into several subdomains. The initial equation restricted to the
subdomains defines a sequence of new local problems. The main goal is to solve
the initial equation via the solution of the local problems. This procedure
induces a dimension reduction which is the major responsible of the success of
such a method. Indeed, one of the principal motivations is the formulation of
solvers which can be easily parallelized.

In this presentation we shall develop a domain decomposition algorithm to the
minimization of functionals with total variation constraints. In this case the
interesting solutions may be discontinuous, e.g., along curves in 2D. These
discontinuities may cross the interfaces of the domain decomposition patches.
Hence, the crucial difficulty is the correct treatment of interfaces, with the
preservation of crossing discontinuities and the correct matching where the
solution is continuous instead. I will present our domain decomposition
strategy, including convergence results for the algorithm and numerical
examples for its application in image inpainting and magnetic resonance
imaging.

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