TY - GEN A1 - Wachsmuth, Gerd A1 - Herrmann, Marc A1 - Herzog, Roland A1 - Schmidt, Stephan A1 - Vidal-Núñez, José T1 - Discrete Total Variation with Finite Elements and Applications to Imaging T2 - Journal of Mathematical Imaging and Vision : (JMIV) N2 - 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. KW - Discrete total variaton KW - Dual problem KW - Image reconstruction KW - Numerical algorithms Y1 - 2019 U6 - https://doi.org/10.1007/s10851-018-0852-7 SN - 0924-9907 SN - 1573-7683 VL - 61 IS - 4 SP - 411 EP - 431 ER - TY - GEN A1 - Casas, Eduardo A1 - Herzog, Roland A1 - Wachsmuth, Gerd T1 - Analysis of Spatio-Temporally Sparse Optimal Control Problems of Semilinear Parabolic Equations T2 - ESAIM: Control, Optimisation and Calculus of Variations (COCV) N2 - 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. KW - Optimal control KW - directional sparsity KW - second-order optimality conditions KW - semilinear parabolic equations Y1 - 2017 U6 - https://doi.org/10.1051/cocv/2015048 SN - 1292-8119 SN - 1262-3377 VL - 23 IS - 1 SP - 263 EP - 295 ER - TY - GEN A1 - Herzog, Roland A1 - Obermeier, Johannes A1 - Wachsmuth, Gerd T1 - Annular and sectorial sparsity in optimal control of elliptic equations T2 - Computational Optimization and Applications (COAP) N2 - 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. KW - Directional sparsity KW - Polar coordinates KW - Coordinate transformation Y1 - 2015 U6 - https://doi.org/10.1007/s10589-014-9721-5 SN - 0926-6003 SN - 1573-2894 VL - 62 IS - 1 SP - 157 EP - 180 ER - TY - GEN A1 - Casas, Eduardo A1 - Herzog, Roland A1 - Wachsmuth, Gerd T1 - Approximation of Sparse Controls in Semilinear Equations by Piecewise Linear Functions Numerische Mathematik T2 - Numerische Mathematik N2 - 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. Y1 - 2012 U6 - https://doi.org/10.1007/s00211-012-0475-7 SN - 0945-3245 SN - 0029-599X VL - 122 IS - 4 SP - 645 EP - 669 ER - TY - GEN A1 - Casas, Eduardo A1 - Herzog, Roland A1 - Wachsmuth, Gerd T1 - Optimality Conditions and Error Analysis of Semilinear Elliptic Control Problems with L¹ Cost Functional T2 - SIAM Journal on Optimization N2 - 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. KW - optimal control of partial differential equations KW - nondifferentiable objective KW - sparse controls KW - finite element discretization KW - a priori error estimates Y1 - 2012 U6 - https://doi.org/10.1137/110834366 SN - 1052-6234 SN - 1095-7189 VL - 22 IS - 3 SP - 795 EP - 820 ER - TY - GEN A1 - Herzog, Roland A1 - Meyer, Christian A1 - Wachsmuth, Gerd T1 - B- and Strong Stationarity for Optimal Control of Static Plasticity with Hardening T2 - SIAM Journal on Optimization N2 - 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. KW - mathematical programs with complementarity constraints in function space KW - variational inequalities of first kind KW - elastoplasticity KW - Bouligand and strong stationarity Y1 - 2013 U6 - https://doi.org/10.1137/110821147 SN - 1052-6234 SN - 1095-7189 VL - 23 IS - 1 SP - 321 EP - 352 ER - TY - GEN A1 - Herzog, Roland A1 - Meyer, Christian A1 - Wachsmuth, Gerd T1 - C-Stationarity for Optimal Control of Static Plasticity with Linear Kinematic Hardening T2 - SIAM Journal on Optimization N2 - 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. KW - mathematical programs with complementarity constraints KW - static plasticity KW - optimal control KW - optimality conditions Y1 - 2012 U6 - https://doi.org/10.1137/100809325 SN - 0363-0129 SN - 1095-7138 VL - 50 IS - 5 SP - 3052 EP - 3082 ER - TY - GEN A1 - Herzog, Roland A1 - Meyer, Christian A1 - Wachsmuth, Gerd T1 - Integrability of displacement and stresses in linear and nonlinear elasticity with mixed boundary conditions T2 - Journal of Mathematical Analysis and Applications N2 - 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. KW - Nonlinear elasticity KW - Integrability KW - Mixed boundary conditions KW - Discontinuous coefficients Y1 - 2011 U6 - https://doi.org/10.1016/j.jmaa.2011.04.074 SN - 0022-247X VL - 382 IS - 2 SP - 802 EP - 813 ER - TY - GEN A1 - Etling, Tommy A1 - Herzog, Roland A1 - Loayza, Estefania A1 - Wachsmuth, Gerd T1 - First and Second Order Shape Optimization Based on Restricted Mesh Deformations T2 - SIAM Journal on Scientific Computing N2 - 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. KW - shape optimization KW - shape gradient descent KW - shape Newton method KW - restricted mesh deformations Y1 - 2020 U6 - https://doi.org/10.1137/19M1241465 SN - 1095-7197 SN - 1064-8275 VL - 42 IS - 2 SP - A1200 EP - A1225 ER - TY - CHAP A1 - Herzog, Roland A1 - Meyer, Christian A1 - Wachsmuth, Gerd ED - Leugering, Günter T1 - Optimal Control of Elastoplastic Processes: Analysis, Algorithms, Numerical Analysis and Applications T2 - Trends in PDE Constrained Optimization N2 - 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. KW - Mathematical programs with complementarity constraints in function space KW - Variational inequalities KW - Elastoplasticity KW - Regularization KW - Optimality conditions Y1 - 2014 SN - 978-3-319-05082-9 SN - 978-3-319-38114-5 SN - 978-3-319-05083-6 U6 - https://doi.org/10.1007/978-3-319-05083-6_4 SP - 27 EP - 41 PB - Birkhäuser CY - Cham ET - 1. Auflage ER -