Kinetic theory is of key importance in mathematical physics and provides fundamental ap- proaches to modern computational fluid dynamics. New challenges in accurate numerical sim- ulation methods require the application and development of novel mathematical techniques for efficiently treating kinetic models with typically high dimensional phase spaces. Models with high dimensional parameter spaces arise frequently also in other mathematical areas, including uncertainty quantification and optimization. The projects in group B on kinetic and parametric models aim at exploiting suitable sparsity structures for the development of efficient algorithms to tackle these high dimensional problems.
From kinetic theory to numerical methods.
Kinetic theory has been a cornerstone of mathematical physics over the past decades leading to seminal results in applied analysis including two Fields medals (C. Villani and P.-L. Lions). Furthermore, it provides the basis of continued progress in computational fluid dynamics applications. Rigorous mathematical insights and novel theoretical concepts on, e.g., entropy, compensated compactness, hypocoercivity and more recently on convex integration, lead to advances also in mathematical fields outside classical kinetic theory. In parallel, the methods and tools of kinetic theory have been applied to understand emerging challenges in various problems in natural sciences as, e.g., collective behavior in bio- and sociological models, advances in semi-conductor theory, the behavior of mean field games, or uncertainty quantification in physical problems like initial confinement fusion or radiative transfer, to name but a few.
A key ingredient in kinetic theory and the corresponding solution methods is the possibility to extract different active scales for dynamics due to an inherent mathematical model hierarchy that allows for the development and application of different mathematical methods. Besides the analytical treatment, the multiscale aspect is of immanent importance in numer- ical simulation. New applications and challenges in the development of numerical methods require the application of novel mathematical methods to treat the typically high dimensional phase space of kinetic models encoding information leading, e.g., to the observed collective behavior. In order to extract the relevant information and to understand the basic driving mechanism, usually lower dimensional but nonlinear submanifolds are considered. This is a very close link to many other mathematical areas where high dimensional and/or para- metric problems arise, including uncertainty quantification, and model order reduction using nonlinear sparse or low rank approximations.
Numerical methods for parametric and high-dimensional problems.
In this CRC we aim to exploit techniques from compressive sensing, algebraic systems theory, or adaptive low rank approximations, leading to novel methodologies and reduced order modeling strategies. This should be understood as a pathway to address pressing issues in the current understanding of, e.g., the rise of emerging phenomena in kinetic models outside classical fluid dynamics. The same methods that need to be developed for high-dimensional phase spaces are expected to be applicable to a variety of parametric and stochastic models. Vice versa, advances in the latter areas can provide advances in the understanding and efficient treatment of kinetic models. Consequently, complementary developments on kinetic and parametric models will create beneficial synergy effects. For example, in subsurface flow or heat transfer models the corresponding model parameters are typically stochastic quantities that need to be traced through the complex physical system in order to quantify their effects across multiple scales appropriately.
Partial differential equations can arise directly in higher dimension (as explained above) but often the PDE itself is posed in moderate “physical” dimensions and the so-called curse of dimensionality enters via a high-dimensional parameter dependency. Such parameters may be stochastic and require a subsequent quantization of the uncertainty, giving rise even to infinite dimensions. We propose a variety of approaches in order to tackle high-dimensional and multiscale problems, both for forward and inverse problems with parametric and stochastic dependencies.
The following projects constitute group B:
B01 Nonlinear reduced modeling for state and parameter estimation (Markus Bachmayr, Wolfgang Dahmen)
B02 Robust sparse low rank approximation of multiparametric partial differential equations (Markus Bachmayr, Lars Grasedyck)
B03 Robust data-driven coarse-graining for surrogate modeling (Sebastian Krumscheid)
B04 Sparsity promoting patterns in kinetic hierarchies (Michael Herty, Manuel Torrilhon)
B05 Sparsification of time-dependent network flow problems by discrete optimization (Christina Büsing, Michael Herty, Arie Koster)
B06 Kinetic theory meets algebraic systems theory (Michael Herty, Eva Zerz)