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Convex regularization of discrete-valued inverse problems

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VMVW01 - Variational methods, new optimisation techniques and new fast numerical algorithms

We consider inverse problems where where a distributed parameter is known a priori to only take on values from a given discrete set. This property can be promoted in Tikhonov regularization with the aid of a suitable convex but nondifferentiable regularization term. This allows applying standard approaches to show well-posedness and convergence rates in Bregman distance. Using the specific properties of the regularization term, it can be shown that convergence (albeit without rates) actually holds pointwise. Furthermore, the resulting Tikhonov functional can be minimized efficiently using a semi-smooth Newton method. Numerical examples illustrate the properties of the regularization term and the numerical solution.

This is joint work with Thi Bich Tram Do, Florian Kruse, and Karl Kunisch.

This talk is part of the Isaac Newton Institute Seminar Series series.

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