@phdthesis{Herrmann2021,
  author    = {Herrmann, Marc},
  title     = {The Total Variation on Surfaces and of Surfaces},
  doi       = {10.25972/OPUS-24073},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-240736},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2021},
  abstract  = {This thesis is concerned with applying the total variation (TV) regularizer to surfaces and different types of shape optimization problems. The resulting problems are challenging since they suffer from the non-differentiability of the TV-seminorm, but unlike most other priors it favors piecewise constant solutions, which results in piecewise flat geometries for shape optimization problems.The first part of this thesis deals with an analogue of the TV image reconstruction approach [Rudin, Osher, Fatemi (Physica D, 1992)] for images on smooth surfaces. A rigorous analytical framework is developed for this model and its Fenchel predual, which is a quadratic optimization problem with pointwise inequality constraints on the surface. A function space interior point method is proposed to solve it. Afterwards, a discrete variant (DTV) based on a nodal quadrature formula is defined for piecewise polynomial, globally discontinuous and continuous finite element functions on triangulated surface meshes. DTV has favorable properties, which include a convenient dual representation. Next, an analogue of the total variation prior for the normal vector field along the boundary of smooth shapes in 3D is introduced. Its analysis is based on a differential geometric setting in which the unit normal vector is viewed as an element of the two-dimensional sphere manifold. Shape calculus is used to characterize the relevant derivatives and an variant of the split Bregman method for manifold valued functions is proposed. This is followed by an extension of the total variation prior for the normal vector field for piecewise flat surfaces and the previous variant of split Bregman method is adapted. Numerical experiments confirm that the new prior favours polyhedral shapes.},
  subject      = {Gestaltoptimierung},
  language  = {en}
}
@phdthesis{Klug2006,
  author    = {Klug, Andreas},
  title     = {Affine-Scaling Methods for Nonlinear Minimization Problems and Nonlinear Systems of Equations with Bound Constraints},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-18851},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2006},
  abstract  = {In this thesis affine-scaling-methods for two different types of mathematical problems are considered. The first type of problems are nonlinear optimization problems subject to bound constraints. A class of new affine-scaling Newton-type methods is introduced. The methods are shown to be locally quadratically convergent without assuming strict complementarity of the solution. The new methods differ from previous ones mainly in the choice of the scaling matrix. The second type of problems are semismooth system of equations with bound constraints. A new affine-scaling trust-region method for these problems is developed. The method is shown to have strong global and local convergence properties under suitable assumptions. Numerical results are presented for a number of problems arising from different areas.},
  subject      = {Skalierungsfunktion},
  language  = {en}
}