Data-driven Optimization and Control

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

Stochastic numerical algorithms for optimal stopping and control problems are required for the evaluation of usually high-dimensional callable or cancelable products, or the determination of optimal decision strategies in systems involving high-dimensional underlying quantities. In this respect, primal methods provide suboptimal exercise strategies, hence lower estimations of the target value (e.g. price), while dual methods provide upper estimations. Naturally, the gap between lower and upper bounds due to these approaches should be as small as possible. [>> 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]

The term statistical inference summerizes methods to extract information from observed data in order to characterize properties in populations. This involves statistical modeling, estimation and uncertainty assessment of parameters, testing of hypothesis. [>> more]

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

Many physical phenomena can be described by suitable functionals, whose critical points play the role of equilibrium solutions. Of particular interest are local and global minimizers: a soap bubble minimizes the surface area subject to a given volume and an elastic body minimizes the stored elastic energy subject to given boundary conditions. At WIAS, methids from the calcuus of variations are applied and further developed to solve problems in physics and technology such as continuum mechanics, quantum mechanics, and optimal control. [>> more]