Abstract. Optimization problems with convex but non-smooth cost functional subject to an elliptic partial differential equation are considered. The non-smoothness arises from a L 1 -norm in the objective functional. The problem is regularized to permit the use of the semi-smooth Newton method. Error estimates with respect to the regularization parameter are provided. Moreover, finite element approximations are studied. A-priori as well as a-posteriori error estimates are developed and confirmed by numerical experiments.Mathematics Subject Classification. 49M05, 65N15, 65N30, 49N45.

Abstract. Semilinear elliptic optimal control problems involving the L 1 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 [13] are also obtained. Numerical experiments confirm the convergence rates.Key words. optimal control of partial differential equations, non-differentiable objective, sparse controls, finite element discretization, a priori error estimates

Abstract. We study optimal control problems in which controls with certain sparsity patterns are preferred. For time-dependent problems the approach can be used to find locations for control devices that allow controlling the system in an optimal way over the entire time interval. The approach uses a nondifferentiable cost functional to implement the sparsity requirements; additionally, bound constraints for the optimal controls can be included. We study the resulting problem in appropriate function spaces and present two solution methods of Newton type, based on different formulations of the optimality system. Using elliptic and parabolic test problems we research the sparsity properties of the optimal controls and analyze the behavior of the proposed solution algorithms.

In this paper we consider Newton's problem of finding a convex body of least resistance. This problem could equivalently be written as a variational problem over concave functions in R 2 . We propose two different methods for solving it numerically. First, we discretize this problem by writing the concave solution function as a infimum over a finite number of affine functions. The discretized problem could be solved by standard optimization software efficiently.Second, we conjecture that the optimal body has a certain structure. We exploit this structure and obtain a variational problem in R 1 . Deriving its Euler-Lagrange equation yields a program with two unknowns, which can be solved quickly.

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