TY - GEN A1 - Schneider, Christopher A1 - Wachsmuth, Gerd T1 - Regularization and discretization error estimates for optimal control of ODEs with group sparsity T2 - ESAIM: Control, Optimisation and Calculus of Variations (COCV) N2 - It is well known that optimal control problems with L¹-control costs produce sparse solutions, i.e., the optimal control is zero on whole intervals. In this paper, we study a general class of convex linear-quadratic optimal control problems with a sparsity functional that promotes a so-called group sparsity structure of the optimal controls. In this case, the components of the control function take the value of zero on parts of the time interval, simultaneously. These problems are both theoretically interesting and practically relevant. After obtaining results about the structure of the optimal controls, we derive stability estimates for the solution of the problem w.r.t. perturbations and L²-regularization. These results are consequently applied to prove convergence of the Euler discretization. Finally, the usefulness of our approach is demonstrated by solving an illustrative example using a semismooth Newton method. KW - Optimal control KW - group sparsity KW - directional sparsity KW - bang-bang principle KW - stability analysis KW - discretization error estimates Y1 - 2018 U6 - https://doi.org/10.1051/cocv/2017049 SN - 1292-8119 SN - 1262-3377 VL - 24 IS - 2 SP - 811 EP - 834 ER - TY - GEN A1 - Schneider, René A1 - Wachsmuth, Gerd T1 - Achieving optimal convergence order for FEM in control constrained optimal control problems T2 - Proceedings in Applied Mathematics and Mechanics (PAMM) N2 - We propose a new heuristic approach to overcome convergence order restrictions implied by the low regularity of the optimal control due to the activity interface of control constraints. Aligning the mesh with the interface yields an improved approximation of the control. Utility of the approach is demonstrated by numerical experiments. Y1 - 2015 U6 - https://doi.org/10.1002/pamm.201510350 SN - 1617-7061 VL - 15 IS - 1 SP - 731 EP - 734 ER - TY - GEN A1 - Zschech, Ehrenfried A1 - Geisler, Holm A1 - Rinderknecht, Jochen A1 - Schneider, Gerd A1 - Spolenak, Ralph A1 - Schmeißer, Dieter T1 - Nano-scale analysis using synchrotron-radiation: Applications in the semiconductor industry T2 - Current nanoscience KW - synchrotron-radiation KW - semiconductor Y1 - 2008 SN - 1573-4137 VL - 4 IS - 3 SP - 256 EP - 266 ER - TY - GEN A1 - Schneider, René A1 - Wachsmuth, Gerd T1 - A-posteriori error estimation for control-constrained, linear-quadratic optimal control problems T2 - SIAM Journal on Numerical Analysis N2 - We derive a posteriori error estimates for control-constrained, linear-quadratic optimal control problems. The error is measured in a norm which is motivated by the objective. Our abstract error estimator is separated into three contributions: the error in the variational inequality (i.e., in the optimality condition for the control) and the errors in the state and adjoint equation. Hence, one can use well-established estimators for the differential equations. We show that the abstract error estimator is reliable and efficient if the utilized estimators for the differential equations have these properties. We apply the error estimator to two distributed optimal control problems with distributed and boundary observation, respectively. Numerical examples exhibit a good error reduction if we use the local error contributions for an adaptive mesh refinement. KW - adaptive finite elements KW - a posteriori error analysis KW - optimal control KW - control constraints Y1 - 2016 U6 - https://doi.org/10.1137/15M1020460 SN - 1095-7170 SN - 0036-1429 VL - 54 IS - 2 SP - 1169 EP - 1192 ER -