Nonlinear Optimization and Inverse Problems

The group contributes to the following mathematical research topics of WIAS:

The work is focussed on models for inductive heating of steel and for light scattering by periodic surface structures. For this the quasi-stationary Maxwell equation is coupled with nonlinear partial differential equations and the timeharmonic Maxwell equation is combined with special radiation conditions, respectively. The convergence of numerical methods and several inverse promblems are analyzed. [>> more]

Thermomechanical models are the basis for the description of numerous technological processes. The consideration of phase transitions and of inelastic constitutive laws raise exciting new questions regarding the analysis of direct problems as well as the identification of materials parameters. [>> more]

Models of real-world phenomena inevitably include uncertainties which influence the solutions in a nonlinear way. Numerical methods for PDE with stochastic data enable to quantify such uncertainties of the solution in dependence of the stochastic input data. Due to the high problem complexity, modern compression techniques are mandatory. [>> more]

The theory of Optimal Transport has been immensely influential in connecting partial differential equations, geometry, and probability. On the one hand, research at WIAS is focused on applying methods and tools from Optimal Transportation Theory to problems in statistics, such as semi-supervised and unsupervised learning, clustering, text classification, as well as in image retrieval, clustering, segmentation, and classification by developing and analyzing new numerical algorithms and schemes. On the other hand, the theory of optimal transport is extended, e.g., towards unbalanced optimal transport and connections to evolutionary partial differential equations via gradient systems. [>> more]

Many processes in nature and technics can only be prescribed by partial differential equations,e.g. heating- or cooling processes, the propagation of acoustic or electromagnetic waves, fluid mechanics. Additionally to challenges in modeling, in various applications the manipulation or controlling of the modeled system is also of interest in order to obtain a certain purpose... [>> more]

In many applications the quantities of interest can be observed only indirectly, or they must be derived from other measurements. Often the measurements are noisy and the reconstruction of the quantities of interest from noisy measurements is unstable. [>> more]

Stochastic Optimization in the widest sense is concerned with optimization problems influenced by random parameters in the objective or constraints. [>> more]