@misc{WachsmuthHerrmannHerzogetal., author = {Wachsmuth, Gerd and Herrmann, Marc and Herzog, Roland and Schmidt, Stephan and Vidal-N{\´u}{\~n}ez, Jos{\´e}}, title = {Discrete Total Variation with Finite Elements and Applications to Imaging}, series = {Journal of Mathematical Imaging and Vision : (JMIV)}, volume = {61}, journal = {Journal of Mathematical Imaging and Vision : (JMIV)}, number = {4}, issn = {0924-9907}, doi = {10.1007/s10851-018-0852-7}, pages = {411 -- 431}, abstract = {The total variation (TV)-seminorm is considered for piecewise polynomial, globally discontinuous (DG) and continuous (CG) finite element functions on simplicial meshes. A novel, discrete variant (DTV) based on a nodal quadrature formula is defined. DTV has favorable properties, compared to the original TV-seminorm for finite element functions. These include a convenient dual representation in terms of the supremum over the space of Raviart-Thomas finite element functions, subject to a set of simple constraints. It can therefore be shown that a variety of algorithms for classical image reconstruction problems, including TV-L2 and TV-L1, can be implemented in low and higher-order finite element spaces with the same efficiency as their counterparts originally developed for images on Cartesian grids.}, language = {en} } @misc{CasasHerzogWachsmuth, author = {Casas, Eduardo and Herzog, Roland and Wachsmuth, Gerd}, title = {Analysis of Spatio-Temporally Sparse Optimal Control Problems of Semilinear Parabolic Equations}, series = {ESAIM: Control, Optimisation and Calculus of Variations (COCV)}, volume = {23}, journal = {ESAIM: Control, Optimisation and Calculus of Variations (COCV)}, number = {1}, issn = {1292-8119}, doi = {10.1051/cocv/2015048}, pages = {263 -- 295}, abstract = {Optimal control problems with semilinear parabolic state equations are considered. The objective features one out of three different terms promoting various spatio-temporal sparsity patterns of the control variable. For each problem, first-order necessary optimality conditions, as well as second-order necessary and sufficient optimality conditions are proved. The analysis includes the case in which the objective does not contain the squared norm of the control.}, language = {en} } @misc{HerzogObermeierWachsmuth, author = {Herzog, Roland and Obermeier, Johannes and Wachsmuth, Gerd}, title = {Annular and sectorial sparsity in optimal control of elliptic equations}, series = {Computational Optimization and Applications (COAP)}, volume = {62}, journal = {Computational Optimization and Applications (COAP)}, number = {1}, issn = {0926-6003}, doi = {10.1007/s10589-014-9721-5}, pages = {157 -- 180}, abstract = {Optimal control problems are considered with linear elliptic equations in polar coordinates. The objective contains L¹-type norms, which promote sparse optimal controls. The particular iterated structure of these norms gives rise to either annular or sectorial sparsity patterns. Optimality conditions and numerical solution approaches are developed.}, language = {en} } @misc{CasasHerzogWachsmuth, author = {Casas, Eduardo and Herzog, Roland and Wachsmuth, Gerd}, title = {Approximation of Sparse Controls in Semilinear Equations by Piecewise Linear Functions Numerische Mathematik}, series = {Numerische Mathematik}, volume = {122}, journal = {Numerische Mathematik}, number = {4}, issn = {0945-3245}, doi = {10.1007/s00211-012-0475-7}, pages = {645 -- 669}, abstract = {Semilinear elliptic optimal control problems involving the L¹ norm of the control in the objective are considered. A priori finite element error estimates for piecewise linear discretizations for the control and the state are proved. These are obtained by a new technique based on an appropriate discretization of the objective function. Numerical experiments confirm the convergence rates.}, language = {en} } @misc{CasasHerzogWachsmuth, author = {Casas, Eduardo and Herzog, Roland and Wachsmuth, Gerd}, title = {Optimality Conditions and Error Analysis of Semilinear Elliptic Control Problems with L¹ Cost Functional}, series = {SIAM Journal on Optimization}, volume = {22}, journal = {SIAM Journal on Optimization}, number = {3}, issn = {1052-6234}, doi = {10.1137/110834366}, pages = {795 -- 820}, abstract = {Semilinear elliptic optimal control problems involving the L¹ norm of the control in the objective are considered. Necessary and sufficient second-order optimality conditions are derived. A priori finite element error estimates for piecewise constant discretizations for the control and piecewise linear discretizations of the state are shown. Error estimates for the variational discretization of the problem in the sense of [M. Hinze, Comput. Optim. Appl., 30 (2005), pp. 45-61] are also obtained. Numerical experiments confirm the convergence rates.}, language = {en} } @misc{HerzogMeyerWachsmuth, author = {Herzog, Roland and Meyer, Christian and Wachsmuth, Gerd}, title = {B- and Strong Stationarity for Optimal Control of Static Plasticity with Hardening}, series = {SIAM Journal on Optimization}, volume = {23}, journal = {SIAM Journal on Optimization}, number = {1}, issn = {1052-6234}, doi = {10.1137/110821147}, pages = {321 -- 352}, abstract = {Optimal control problems for the variational inequality of static elastoplasticity with linear kinematic hardening are considered. The control-to-state map is shown to be weakly directionally differentiable, and local optimal controls are proved to verify an optimality system of B-stationary type. For a modified problem, local minimizers are shown to even satisfy an optimality system of strongly stationary type.}, language = {en} } @misc{HerzogMeyerWachsmuth, author = {Herzog, Roland and Meyer, Christian and Wachsmuth, Gerd}, title = {C-Stationarity for Optimal Control of Static Plasticity with Linear Kinematic Hardening}, series = {SIAM Journal on Optimization}, volume = {50}, journal = {SIAM Journal on Optimization}, number = {5}, issn = {0363-0129}, doi = {10.1137/100809325}, pages = {3052 -- 3082}, abstract = {An optimal control problem is considered for the variational inequality representing the stress-based (dual) formulation of static elastoplasticity. The linear kinematic hardening model and the von Mises yield condition are used. Existence and uniqueness of the plastic multiplier is rigorously proved, which allows for the reformulation of the forward system using a complementarity condition. In order to derive necessary optimality conditions, a family of regularized optimal control problems is analyzed, wherein the static plasticity problems are replaced by their viscoplastic approximations. By passing to the limit in the optimality conditions for the regularized problems, necessary optimality conditions of C-stationarity type are obtained.}, language = {en} } @misc{HerzogMeyerWachsmuth, author = {Herzog, Roland and Meyer, Christian and Wachsmuth, Gerd}, title = {Integrability of displacement and stresses in linear and nonlinear elasticity with mixed boundary conditions}, series = {Journal of Mathematical Analysis and Applications}, volume = {382}, journal = {Journal of Mathematical Analysis and Applications}, number = {2}, issn = {0022-247X}, doi = {10.1016/j.jmaa.2011.04.074}, pages = {802 -- 813}, abstract = {Equations of linear and nonlinear infinitesimal elasticity with mixed boundary conditionsare considered. The bounded domain is assumed to have a Lipschitz boundary and to satisfy additional regularity assumptions. W^(1,p) regularity for the displacements and L^p regularity for the stresses are proved for some p>2.}, language = {en} } @misc{EtlingHerzogLoayzaetal., author = {Etling, Tommy and Herzog, Roland and Loayza, Estefania and Wachsmuth, Gerd}, title = {First and Second Order Shape Optimization Based on Restricted Mesh Deformations}, series = {SIAM Journal on Scientific Computing}, volume = {42}, journal = {SIAM Journal on Scientific Computing}, number = {2}, issn = {1095-7197}, doi = {10.1137/19M1241465}, pages = {A1200 -- A1225}, abstract = {We consider shape optimization problems subject to elliptic partial differential equations. In the context of the finite element method, the geometry to be optimized is represented by the computational mesh, and the optimization proceeds by repeatedly updating the mesh node positions. It is well known that such a procedure eventually may lead to a deterioration of mesh quality, or even an invalidation of the mesh, when interior nodes penetrate neighboring cells. We examine this phenomenon, which can be traced back to the ineptness of the discretized objective when considered over the space of mesh node positions. As a remedy, we propose a restriction in the admissible mesh deformations, inspired by the Hadamard structure theorem. First and second order methods are considered in this setting. Numerical results show that mesh degeneracy can be overcome, avoiding the need for remeshing or other strategies. FEniCS code for the proposed methods is available on GitHub.}, language = {en} } @incollection{HerzogMeyerWachsmuth, author = {Herzog, Roland and Meyer, Christian and Wachsmuth, Gerd}, title = {Optimal Control of Elastoplastic Processes: Analysis, Algorithms, Numerical Analysis and Applications}, series = {Trends in PDE Constrained Optimization}, booktitle = {Trends in PDE Constrained Optimization}, editor = {Leugering, G{\"u}nter}, edition = {1. Auflage}, publisher = {Birkh{\"a}user}, address = {Cham}, isbn = {978-3-319-05082-9}, doi = {10.1007/978-3-319-05083-6_4}, pages = {27 -- 41}, abstract = {An optimal control problem is considered for the variational inequality representing the stress-based (dual) formulation of static elastoplasticity. The linear kinematic hardening model and the von Mises yield condition are used. The forward system is reformulated such that it involves the plastic multiplier and a complementarity condition. In order to derive necessary optimality conditions, a family of regularized optimal control problems is analyzed. C-stationarity type conditions are obtained by passing to the limit with the regularization. Numerical results are presented.}, language = {en} }