In this article we propose a novel nonstationary iterated Tikhonov (NIT)-type method for obtaining stable approximate solutions to ill-posed operator equations modeled by linear operators acting between Hilbert spaces. Geometrical properties of the problem are used to derive a new strategy for choosing the sequence of regularization parameters (Lagrange multipliers) for the NIT iteration. Convergence analysis for this new method is provided. Numerical experiments are presented for two distinct applications: (I) a two-dimensional elliptic parameter identification problem (inverse potential problem); and (II) an image-deblurring problem. The results obtained validate the efficiency of our method compared with standard implementations of the NIT method (where a geometrical choice is typically used for the sequence of Lagrange multipliers).

Partial differential equation (PDE) models are widely used in engineering and natural sciences to describe spatio-temporal processes. The parameters of the considered processes are often unknown and have to be estimated from experimental data. Due to partial observations and measurement noise, these parameter estimates are subject to uncertainty. This uncertainty can be assessed using profile likelihoods, a reliable but computationally intensive approach. In this paper, we introduce an integration based approach for the profile likelihood calculation for inverse problems with PDE constraints. While existing approaches rely on repeated optimization, the proposed approach exploits a dynamical system evolving along the likelihood profile. We derive the dynamical system for the reduced and the full estimation problem and study its properties. To evaluate the proposed method, we compare it with stateof-the-art algorithms for a simple reaction-diffusion model for a cellular patterning process. We observe a good accuracy of the method as well as a significant speed up as compared to established methods. Integration based profile calculation facilitates rigorous uncertainty analysis for computationally demanding parameter estimation problems with PDE constraints.

Abstract. Online parameter identification is of importance, e.g., for model predictive control. Since the parameters have to be identified simultaneously to the process of the modeled system, dynamical update laws are used for state and parameter estimates. Most of the existing methods for infinite dimensional systems either impose strong assumptions on the model or cannot handle partial observations. Therefore we propose and analyze an online parameter identification method that is less restrictive concerning the underlying model and allows for partial observations and noisy data. The performance of our approach is illustrated by some numerical experiments.

Particle accelerators are enabling tools for scientific exploration and discovery in various disciplines. However, finding optimised operation points for these complex machines is a challenging task due to the large number of parameters involved and the underlying non-linear dynamics. Here, we introduce two families of data-driven surrogate models, based on deep and invertible neural networks, that can replace the expensive physics computer models. These models are employed in multi-objective optimisations to find Pareto optimal operation points for two fundamentally different types of particle accelerators. Our approach reduces the time-to-solution for a multi-objective accelerator optimisation up to a factor of 640 and the computational cost up to 98%. The framework established here should pave the way for future online and real-time multi-objective optimisation of particle accelerators.