Optimal Control of PDE-constrained Systems

This thesis proposes new computational methodologies that contribute to ensure fast, memory-efficient and robust solution of large-scale optimization problems that are constrained by partial differential equations (PDEs). As a step towards real-time optimization of large-scale processes, the proposed tools are intended to serve as building blocks in closed-loop controllers based on e.g. nonlinear model predictive control (NMPC). Overall, the thesis comprises eight research papers that include two published journal papers, four published conference papers, one submitted paper that is currently under review, and one arXiv preprint. The associated contributions fall into two distinct parts: Part I New methods for computational efficient solution of multi-objective optimization problems that arise in oil reservoir management. Part II Customized iterative solvers for stationary and time-dependent PDE-constrained optimization problems that are governed by systems of non-linear diffusion-reaction (DR) equations.
Part I. The new methodologies for oil production optimization address the issues of risk that arise with uncertain and errant model data. Conventional risk mitigation strategies typically rely on multi-query solution of large-scale PDE-constrained optimization problems. As a consequence, conventional methods often become computational intractable in practice. To meet this challenge, the new risk mitigation strategies seek to address uncertainty using approaches based on PDE-constrained multi-objective optimization. To establish proof-of concept, a selection of numerical case studies demonstrate that the new risk mitigation strategies pose a computational attractive and scalable alternative to conventional methods of the oil literature.
Part II. The iterative solvers are based on a new high-order approach to PDE-constrained optimization that combines customized spectral discretization schemes with appropriate Krylov subspace (KSP) methods. The solvers specifically target distributed control problems in separable domains; examples include control of the Schlögl model, FitzHugh-Nagumo problems and coupled systems of diffusion-reaction equations that govern chemical reactions. The solvers allow for both subdomain control and additional point-wise bound constraints. As a key feature, the high-order approach introduces a new type of Poisson like (PL) preconditioners that are tailored for efficient solution of the large-scale saddlepoint problems that constitute the computational bottleneck of Newton-like optimization algorithms such a Sequential Quadratic Programming (SQP) methods. By construction, the PL preconditioners are matrix-free, they are scalable with respect to the spatial problem dimension, and they are prone to parallelization. For simple stationary linear-quadratic model problems, spectral analysis shows that the PL preconditioners are ideal in the sense that the spectra of the preconditioned systems are bounded independently of the problem dimensionality. Numerical studies indicate that the PL preconditioners remain ideal for more complex cases of non-linear and time-dependent DR problems. Further, numerical results show that the new iterative solvers outperform state-of-the art direct methods and may pose viable alternatives to the widely-used constellation of low-order schemes and Schur-complement block preconditioners.