WIAS Preprint No. 2340, (2016)

A semismooth Newton method with analytical path-following for the $H^1$-projection onto the Gibbs simplex


  • Adam, Lukáš
    ORCID: 0000-0001-8748-4308
  • Hintermüller, Michael
    ORCID: 0000-0001-9471-2479
  • Surowiec, Thomas M.
    ORCID: 0000-0003-2473-4984

2010 Mathematics Subject Classification

  • 49M15 90C20


  • Gibbs simplex, metric projection, semismooth Newton, path-following, Ginzburg-Landau energy, multiphase field models, inpainting, data classification




An efficient, function-space-based second-order method for the $H^1$-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.

Appeared in

  • IMA J. Numer. Anal., 39 (2019), pp. 1276--1295 (published online on 07.06.2018), DOI 10.1093/imanum/dry034 .

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