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- ItemA semismooth Newton method with analytical path-following for the H1-projection onto the Gibbs simplex(Oxford : Oxford Univ. Press, 2018) Adam, L.; Hintermüller, M.; Surowiec, T.M.An efficient, function-space-based second-order method for the H1-projection onto the Gibbs simplex is presented. The method makes use of the theory of semismooth Newton methods in function spaces as well as Moreau–Yosida regularization and techniques from parametric optimization. A path-following technique is considered for the regularization parameter updates. A rigorous first- and second-order sensitivity analysis of the value function for the regularized problem is provided to justify the update scheme. The viability of the algorithm is then demonstrated for two applications found in the literature: binary image inpainting and labeled data classification. In both cases, the algorithm exhibits mesh-independent behavior.
- ItemDensity of convex intersections and applications(London : Royal Society, 2017) Hintermüller, M.; Rautenberg, C.N.; Rösel, S.In this paper, we address density properties of intersections of convex sets in several function spaces. Using the concept of Γ-convergence, it is shown in a general framework, how these density issues naturally arise from the regularization, discretization or dualization of constrained optimization problems and from perturbed variational inequalities. A variety of density results (and counterexamples) for pointwise constraints in Sobolev spaces are presented and the corresponding regularity requirements on the upper bound are identified. The results are further discussed in the context of finite-element discretizations of sets associated with convex constraints. Finally, two applications are provided, which include elasto-plasticity and image restoration problems.