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SUMMARY:Sparse solutions for dynamic inverse problems with Optimal Transpo
 rt regularisers - Marcello Carioni\, University of Graz
DTSTART:20200122T130000Z
DTEND:20200122T140000Z
UID:TALK138553@talks.cam.ac.uk
CONTACT:Yury Korolev
DESCRIPTION:The aim of the first part of this talk is to provide a charact
 erization for sparse solutions of abstract variational inverse problems wi
 th finite dimensional data. We consider the minimization of functionals th
 at are the sum of two terms: a convex regularizer and a finite dimensional
  soft constraint. It was observed for specific examples that minimizers of
  variational problems of this type are sparse in a suitable sense. We form
 alise this fact proving the existence of a minimizer that is represented a
 s a finite linear combination of extremal points of the unit ball of the r
 egularizer. This finding provides a natural notion of sparsity for abstrac
 t variational inverse problems. \nWe apply this abstract result to relevan
 t examples as TV denoising and higher order scalar regularizers. Then\, we
  consider the framework of dynamic inverse problems with the Benamou-Breni
 er energy as a regularizer. Using the classical theory of Optimal Transpor
 t\, we provide a characterisation for sparse solutions in this specific ca
 se. Then\, in the last part of the talk\, we show how to construct a varia
 nt of the Alternating Descent Conditional Gradient Method that relies on t
 he structure of sparse solutions for dynamic inverse problems.
LOCATION:MR 14\, Centre for Mathematical Sciences
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