Nonlinear Optimization and Inverse Problems

The group contributes to the following application oriented research topics of WIAS:

Diffractive optical elements employ the controlled use of light propagation on microstructered interfaces and can be applied in diffractive measurement technique, spectroscopy, astronomy, optical communication techniques, etc. These elements are manufactured with photolitography method, relying, in turn on precise knowledge of the optic properties of the materials. Our joint effort with the cooperation partners from PTB is in mathematical modelling and numeric simulations of the measurement process. A statistical description of the measurement result can be given as a solution of a Bayesian inverse problem. We focus on developing and applying novel methods of Bayesian inversion, stemming from optimal transport ang gradient flow theory. [>> more]

Optimization problems in energy management are concerned with the planning of production and distribution of different energy sources (power, gas), in order to cover a given customer's demand. In this context, the consideration of uncertainties (e.g., loads, meteorological parmeters, prices) in transportation networks represents a major challenge. The aim is to find cost optimal decisions which are robust at the same time with respect to uncertainties. The additional consideration of markets and the physica of energy transport then lead to risk-averse optimal control problems with equilibrium constraints. [>> more]

Industrial processes are currently experiencing their fourth revolution. The entire production process is connected and equipped with sensor technology, which makes huge amounts of data available. Workers are not only supported by technical visualization and information processing, but parts of the decision-making processes are even carried out independently with the help of AI systems. The large amount of data and the fully automated process pose new challenges for mathematics, but offer unprecedented opportunities for optimization algorithms, not only in the optimization of individual production steps, but across the entire value chain. [>> more]