Roland Griesse

A semismooth Newton method for Tikhonov functionals with sparsity constraints

Minimization problems in l2 for Tikhonov functionals with sparsity constraints are considered. Sparsity of the solution is ensured by a weighted l1 penalty term. The necessary and sufficient condition for optimality is shown to be slantly differentiable (Newton differentiable), hence a semismooth Newton method is applicable. Local superlinear convergence of this method is proved. Numerical examples are provided which show that our method compares favorably with existing approaches.

A Primal-Dual Active Set Strategy for Optimal Boundary Control of a Nonlinear Reaction-Diffusion System

This paper is concerned with optimal boundary control of an instationary reaction-diffusion system in three spatial dimensions. This problem involves a coupled nonlinear system of parabolic differential equations with bilateral as well as integral control constraints. We include the integral constraint in the cost by a penalty term whereas the bilateral control constraints are handled explicitly. First- and second-order conditions for the optimization problem are analyzed. A primal-dual active set strategy is utilized to compute optimal solutions numerically. The algorithm is compared to a semismooth Newton method.

Distributed optimal control of lambda–omega systems

The formulation, analysis, and numerical solution of distributed optimal control problems governed by lambda–omega systems is presented. These systems provide a universal model for reaction-diffusion phenomena with turbulent behavior. Existence and regularity properties of solutions to the free and controlled lambda–omega models are investigated. To validate the ability of distributed control to drive lambda–omega systems from a chaotic to an ordered state, a space-time multigrid method is developed based on a new smoothing scheme. Convergence properties of the multigrid scheme are discussed and results of numerical experiments are reported.

Optimal Control for a Stationary MHD System in Velocity-Current Formulation

An optimal control problem for the equations governing the stationary problem of magnetohydrodynamics (MHD) is considered. Control mechanisms by external and injected currents and magnetic fields are treated. An optimal control problem is formulated. First order necessary and second order sufficient conditions are developed. An operator splitting scheme for the numerical solution of the MHD state equations is analyzed.

Numerical Sensitivity Analysis for the Quantity of Interest in PDE-Constrained Optimization

In this paper, we consider the efficient computation of derivatives of a functional (the quantity of interest) which depends on the solution of a PDE-constrained optimization problem with inequality constraints and which may be different from the cost functional. The optimization problem is subject to perturbations in the data. We derive conditions under with the quantity of interest possesses first and second order derivatives with respect to the perturbation parameters. An algorithm for the efficient evaluation of these derivatives is developed, with considerable savings over a direct approach, especially in the case of high-dimensional parameter spaces. The computational cost is shown to be small compared to that of the overall optimization algorithm. Numerical experiments involving a parameter identification problem for Navier-Stokes flow and an optimal control problem for a reaction-diffusion system are presented which demonstrate the efficiency of the method.

Lipschitz Stability of Solutions to Some State-Constrained Elliptic Optimal Control Problems

In this paper, optimal control problems with pointwise state constraints for linear and semilinear elliptic partial differential equations are studied. The problems are subject to perturbations in the problem data. Lipschitz stability with respect to perturbations of the optimal control and the state and adjoint variables is established initially for linear–quadratic problems. Both the distributed and Neumann boundary control cases are treated. Based on these results, and using an implicit function theorem for generalized equations, Lipschitz stability is also shown for an optimal control problem involving a semilinear elliptic equation.

Lipschitz stability for elliptic optimal control problems with mixed control-state constraints

A family of linear-quadratic optimal control problems with pointwise mixed state-control constraints governed by linear elliptic partial differential equations is considered. All data depend on a vector parameter of perturbations. Lipschitz stability with respect to perturbations of the optimal control, the state and adjoint variables, and the Lagrange multipliers is established.

Parametric sensitivity analysis in optimal control of a reaction-diffusion system – part II: practical methods and examples

In this article, we consider a control-constrained optimal control problem governed by a system of semilinear parabolic reaction-diffusion equations. The optimal solutions are subject to perturbations of the dynamics and of the objective. In Part I of the article, local optimal solutions, as functions of the perturbation parameter, have been proved to be Lipschitz continuous and directionally differentiable. The directional derivatives, also known as parametric sensitivities, have been characterized as the solutions of auxiliary quadratic programing problems, i.e., linear-quadratic optimal control problems. In this article, we devise a practical algorithm that is capable of solving both the unperturbed optimal control problem and the parametric sensitivity problem. A numerical example with complete data is given in order to demonstrate the applicability of our method. To verify our results, we provide a second-order expansion of the minimum value function and compare it to the objective values at true perturbed solutions.

Parametric Sensitivity Analysis for Optimal Boundary Control of a 3D Reaction-Diffusion System

A boundary optimal control problem for an instationary nonlinear reaction-diffusion equation system in three spatial dimensions is presented. The control is subject to pointwise control constraints and a penalized integral constraint. Under a coercivity condition on the Hessian of the Lagrange function, an optimal solution is shown to be a directionally differentiable function of perturbation parameters such as the reaction and diffusion constants or desired and initial states. The solution’s derivative, termed parametric sensitivity, is characterized as the solution of an auxiliary linear-quadratic optimal control problem. A numerical example illustrates the utility of parametric sensitivities which allow a quantitative and qualitative perturbation analysis of optimal solutions.

Update strategies for perturbed nonsmooth equations

Nonsmooth operator equations in function spaces are considered, which depend on perturbation parameters. The nonsmoothness arises from a projection onto an admissible interval. Lipschitz stability in L ∞ and Bouligand differentiability in L p of the parameter-to-solution map are derived. An adjoint problem is introduced for which Lipschitz stability and Bouligand differentiability in L ∞ are obtained. Three different update strategies, which recover a perturbed from an unperturbed solution, are analysed. They are based on Taylor expansions of the primal and adjoint variables, where the latter admits error estimates in L ∞. Numerical results are provided.