PDE-constrained optimization: Preconditioners and diffuse domain methods

This thesis is mainly concerned with the efficient numerical solution of optimization problems subject to linear PDE-constraints, with particular focus on robust preconditioners and diffuse domain methods. Associated with such constrained optimization problems are the famous first-order KarushKuhn-Tucker (KKT) conditions. For certain minimization problems, the functions satisfying the KKT conditions are also optimal solutions of the original optimization problem, implying that we can solve the KKT system to obtain the optimum; the so-called “all-at-once” approach. We propose and analyze preconditioners for the different KKT systems we derive in this thesis. In papers I and II we study PDE-constrained optimization problems with inequality constraints and problems subject to total variation regularization, respectively. These are both non-linear problems, so we apply iterative methods; the Primal Dual Active Set algorithm and the split Bregman method, resulting in iterative schemes where we must solve a sequence of linear KKT systems. Using Riesz maps to form preconditioners, we get iteration numbers independent of the mesh parameter h, and we are able to prove a maximum growth in MINRES iteration numbers of order O([log(α−1)]2) as the regularization parameter α→ 0. Furthermore, we present numerical simulations with the improved rate of order O(log(α−1)). To derive a solver which is completely robust with respect to both the mesh parameter h and the regularization parameter α is, from a functional analysis perspective, a matter of finding weighted Sobolev spaces in which all the stability estimates are independent of h and α. If such topologies are obtained, the Riesz maps associated with the underlying normed spaces will form a natural preconditioner for the KKT system, resulting in solvers with hand α-independent iteration numbers. The third paper concerns the derivation of such a robust preconditioner for a specific PDE-constrained optimization problem. More specifically, we i study an elliptic control problem with boundary observations only and locally defined control functions. A careful analysis reveals that there exists an isomorphism between the control space and the space of Lagrange multipliers, leading to stability estimates of the associated KKT system independent of the mesh parameter h and regularization parameter α. Consequently, we obtain a completely hand α-robust Krylov subspace solver. The problem studied in Paper III was motivated by the inverse problem of electrocardiography (ECG). Finally, in papers IV and V, we are concerned with the computational representation of the involved domains. In applications, the domains are often complex or not exactly known. We apply the diffuse domain method, an embedding technique, to solve PDE-constrained optimization problems posed on such domains. A full theoretical investigation is undertaken, and strict convergence rates, with respect to the diffuse domain parameter , is obtained. We must also handle topologies depending on the parameter , which increases the complexity of deriving robust KKT solvers. A completely -robust iterative solver is, nevertheless, achieved from a careful construction of topologies. All the theoretical investigations, presented in this thesis, are supported by numerical simulations, and we obtain very good agreement between the theoretical and numerical results.