• analysis of stochastic systems
• regularization methods
• qualitative and numerical analysis of dynamical systems
• nonlinear and combinatorial optimization
• optimization and analysis of discrete structures
• Bayesian spatio-temporal prediction and design
• parameter identification
The interaction of and distinction between discrete and continuous modeling concepts has a long standing history in dynamical systems. Here it is common knowledge that similar concepts can be used to assess the asymptotic behavior of discrete and continuous dynamical systems but, e.g., replacement of a continuous time with discrete time steps has a crucial influence on finite time convergence. The situation is somewhat different in optimization, where mutual exchange of methodology between the combinatorial and the continuous optimization community is developed to a much lesser extent, a fact which is also reflected in the status of knowledge of PhD students in these fields. Quite recently crucial breakthroughs for combinatorial problems have been achieved by appropriate continuous relaxations. While this research direction has so far been focusing on general finite dimensional mixed integer nonlinear programming, still much room is left for enhancing methodology for structured infinite dimensional problems such as control and identification in the context of ordinary and partial differential equation (ODE, PDE) systems.
An additional aspect that increasingly gains practical importance and theoretical interest is uncertainty. Stochastic programming and stochastic differential equations are highly active fields of research. However, even on an advanced education level, the deterministic and the stochastic world are still largely separated. To advance the field of stochastic control and estimation towards theory based methods for real world problems modeled by differential equations, there is a strong need for enhancing the stochastics training of young researchers interested in PDEs and ODEs.
The research in this DP will be driven by two fields of engineering applications, represented by associated faculty members, where the mentioned requirement of intimate cooperation between experts on differential equations, optimization and stochastics becomes most evident. Robustness with respect to perturbations is vital both in smart sensor and actuator systems and in signal processing for communication systems. Here, the further development of tools for quantifying and possibly reducing measurement uncertainty leads to challenges in the analysis and numerics of parameter and state estimation methodologies in dynamical systems as well as in continuous and discrete optimization (e.g., sensor placement). Additionally we intend to closely cooperate with the doctoral program “Modeling, Simulation and Optimization in Business and Economics (MSOBE)” on methods and applications in the fields of operations and supply chain management, agent-based computational economics, analyses of macroeconomic policy problems, and mechanism design in microeconomics.
It is the aim of this DP to first of all provide PhD students with the necessary mathematical and interdisciplinary knowledge required to attack challenging applications of this kind. Once the young researchers have gained the combined knowledge on differential equations, optimization, discrete mathematics, stochastics, and inverse problems, they will be able to develop completely new ideas and find approaches leading to groundbreaking methodological innovations applicable in a much wider range of problems.