Certified reduced basis methods for parametrized PDE-constrained optimization problems

We present a new reduced basis approach for the efficient and reliable solution of parametrized PDE-constrained optimization problems. This kind of problems arise when a system modeled by PDEs is optimized in order to achieve a desired behavior. Typically, the numerical solution of these optimization problems is challenging in terms of required computational cost, in particular if one is interested in multiple solutions for different parameter values. The reduced basis method is a technique for parametric model order reduction, which provides both efficient and computationally stable approximations. It further provides rigorous error bounds and is well established for a large class of parametrized PDEs of elliptic and parabolic nature. In this thesis, we extend the reduced basis methodology to parametrized PDE-constrained optimization problems. These optimization problems may contain parameters in the PDE constraint itself, as well as in the objective function (such as the desired state or regularization parameter). We consider both elliptic and parabolic problems, as well as finite-dimensional and distributed controls. Our approach is completely online-efficient: The online computational cost for the reduced basis approximation and the associated error bounds depends only on the dimension of the reduced basis space. It is independent of the original high-dimensional finite element approximation. Our approximation is based on a reduced basis space for the state and the adjoint variables. By integrating state and adjoint snapshots into a single reduced basis space, we ensure both a consistent approximation and the stability of the reduced model. In addition, we propose a reduction of the control space for distributed controls. The focus of this thesis is to derive efficiently computable and rigorous a posteriori error bounds for various quantities of interest. To this end, we present two approaches: The first approach is based on a perturbation argument, whereas the second bound is directly derived from the error residual equations of the optimality system. Both approaches yield bounds for the errors in the optimal control, in the optimal value of the cost functional, and in the optimal state and adjoint variables. More generally, by introducing an additional dual problem, it is possible to bound the error of arbitrary linear output functionals depending on the state, adjoint, and control variables. All of the proposed bounds are completely offline-online separable and hence can be evaluated efficiently. This makes our approach relevant in the many-query or real-time context. In particular, the bounds involve only constants (or their upper/lower bounds), which can be computed with low effort. We present numerical results for various parametrized PDE-constrained optimization problems to demonstrate the effectiveness of the proposed method

A Posteriori Error Estimation for Reduced Order Solutions of Parametrized Parabolic Optimal Control Problems

We consider the efficient and reliable solution of linear-quadratic optimal control problems governed by parametrized parabolic partial differential equations. To this end, we employ the reduced basis method as a low-dimensional surrogate model to solve the optimal control problem and develop a posteriori error estimation procedures that provide rigorous bounds for the error in the optimal control and the associated cost functional. We show that our approach can be applied to problems involving control constraints and that, even in the presence of control constraints, the reduced order optimal control problem and the proposed bounds can be efficiently evaluated in an offline-online computational procedure. We also propose two greedy sampling procedures to construct the reduced basis space. Numerical results are presented to confirm the validity of our approach