@article{SchindeleBorzi2016,
  author    = {Schindele, Andreas and Borz{\`i}, Alfio},
  title     = {Proximal Methods for Elliptic Optimal Control Problems with Sparsity Cost Functional},
  series = {Applied Mathematics},
  volume    = {7},
  journal   = {Applied Mathematics},
  number    = {9},
  doi       = {10.4236/am.2016.79086},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-145850},
  pages     = {967-992},
  year      = {2016},
  abstract  = {First-order proximal methods that solve linear and bilinear elliptic optimal control problems with a sparsity cost functional are discussed. In particular, fast convergence of these methods is proved. For benchmarking purposes, inexact proximal schemes are compared to an inexact semismooth Newton method. Results of numerical experiments are presented to demonstrate the computational effectiveness of proximal schemes applied to infinite-dimensional elliptic optimal control problems and to validate the theoretical estimates.},
  language  = {en}
}
@article{GaviraghiSchindeleAnnunziatoetal.2016,
  author    = {Gaviraghi, Beatrice and Schindele, Andreas and Annunziato, Mario and Borz{\`i}, Alfio},
  title     = {On Optimal Sparse-Control Problems Governed by Jump-Diffusion Processes},
  series = {Applied Mathematics},
  volume    = {7},
  journal   = {Applied Mathematics},
  number    = {16},
  doi       = {10.4236/am.2016.716162},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-147819},
  pages     = {1978 -- 2004},
  year      = {2016},
  abstract  = {A framework for the optimal sparse-control of the probability density function of a jump-diffusion process is presented. This framework is based on the partial integro-differential Fokker-Planck (FP) equation that governs the time evolution of the probability density function of this process. In the stochastic process and, correspondingly, in the FP model the control function enters as a time-dependent coefficient. The objectives of the control are to minimize a discrete-in-time, resp. continuous-in-time, tracking functionals and its L2- and L1-costs, where the latter is considered to promote control sparsity. An efficient proximal scheme for solving these optimal control problems is considered. Results of numerical experiments are presented to validate the theoretical results and the computational effectiveness of the proposed control framework.},
  language  = {en}
}
@phdthesis{Schindele2016,
  author    = {Schindele, Andreas},
  title     = {Proximal methods in medical image reconstruction and in nonsmooth optimal control of partial differential equations},
  url       = {http://nbn-resolving.de/urn:nbn:de:bvb:20-opus-136569},
  school      = {Universit{\"a}t W{\"u}rzburg},
  year      = {2016},
  abstract  = {Proximal methods are iterative optimization techniques for functionals, J = J1 + J2, consisting of a differentiable part J2 and a possibly nondifferentiable part J1. In this thesis proximal methods for finite- and infinite-dimensional optimization problems are discussed. In finite dimensions, they solve l1- and TV-minimization problems that are effectively applied to image reconstruction in magnetic resonance imaging (MRI). Convergence of these methods in this setting is proved. The proposed proximal scheme is compared to a split proximal scheme and it achieves a better signal-to-noise ratio. In addition, an application that uses parallel imaging is presented. In infinite dimensions, these methods are discussed to solve nonsmooth linear and bilinear elliptic and parabolic optimal control problems. In particular, fast convergence of these methods is proved. Furthermore, for benchmarking purposes, truncated proximal schemes are compared to an inexact semismooth Newton method. Results of numerical experiments are presented to demonstrate the computational effectiveness of our proximal schemes that need less computation time than the semismooth Newton method in most cases. Results of numerical experiments are presented that successfully validate the theoretical estimates.},
  subject      = {Optimale Kontrolle},
  language  = {en}
}