Image registration is crucial in many imaging applications such as medical imaging or computer vision. The goal of finding a suitable transformation between two images poses similar restrictions and regularity requirements on the set of admissible transformations as shape optimization problems. In the scope of this talk, we build on an approach that models image registration as an optimization problem that is constrained by a linear hyperbolic transport equation. We use a higher-order discontinuous Galerkin finite element method for discretization and motivate the numerical upwind scheme and its limitations from the continuous weak space-time formulation of the transport equation. Moreover, we build on recent theoretical results to model the optimization problem. To discuss the potential of the proposed algorithm, we apply it to patient specific brain mesh generation from magnetic resonance images (MRI). This can be a time consuming task and require manual corrections, e.g., for meshing the ventricular system or defining subdomains. The idea is to use the registration of an input MRI to a respective target in order to obtain a new mesh from a high-quality template mesh.

Robust optimal control problems, when expressed in discrete time, generally take a min-max form over continuous variable spaces. In this talk, we focus on the problem of choosing decision variables that optimize the performance of a system in the worst-case realization of uncertainties drawn from bounded continuous spaces. We also consider systems with non-unique state trajectories that arise from the introduction of modelling variables through smoothing approaches, as well as existence constraints, which are useful in describing various logical conditions. Under certain assumptions on the continuity of the functions involved, we demonstrate how this problem can be cast as a semi-infinite program (SIP) and solved using a local reduction technique, which replaces the infinite-dimensional constraints by an optimally chosen finite scenario set. The solution procedure is therefore split into a number of lower level problems that find the uncertainty scenarios that maximize the constraint violations given the current best guess for the decision variables, and a high level problem that minimizes the original objective under the obtained scenario set. The existence constraints, meanwhile, can be reformulated as separate SIPs, and the worst-case violations for these constraints can be obtained by solving these nested SIPs using the same local reduction approach. Our approach solves a wide range of problems and exhibits better scaling than scenario trees or traditional global optimization techniques. We illustrate this through a number of examples drawn from discrete-time robust optimal control.

Column liquid chromatography plays an important role in the downstream processing of biopharmaceuticals, where the goal is to capture and purify a target protein from a mixture. Our goal is to employ a model-based approach for process optimization in order to improve the quality of the product, while also achieving further economical and ecological benefits.

Rate models in combination with suitable reaction schemes that model the specific adsorption process are often employed to describe chromatographic processes. Regarding the optimization of such processes, due to the presence of discrete control decisions, we are eventually facing mixed-integer optimal control problems governed by advection-diffusion-reaction-type partial differential equations with high nonlinearities. Furthermore, since at least one flow reversal is typically performed in practical applications to obtain sharper elution profiles, we additionally have to deal with switching dynamics. Lastly, it is important to determine robust solutions in order to safeguard against, e.g., uncertain model parameters, such as reaction rates and feed composition.

In this talk we present developments towards robustly optimal switching control applied to chromatographic separation processes and discuss the obtained results.

In this presentation, the stability of popular first-order optimization algorithms is examined through the lens of passivity. The passivity theorem ensures input-output stability of a passive plant connected in a negative feedback loop with a very strictly passive (VSP) system. Combining existing work on control interpretation of first-order optimization algorithms with loop transformation techniques, it is shown that gradient descent (GD) can be rendered passive, while the more recent triple momentum (TM) method is input strictly passive (ISP). It is shown that the sector boundedness of the gradient of an L-smooth, m-strongly convex function renders it being VSP. Therefore, the passivity theorem can ensure input-output stability of GD when the gradient is L-smooth, m-strongly convex.