AEI-DFG: Function spaces and high-dimensional approximation

• Applicants: Professorin Dr. Dorothee Haroske; Professor Dr. Thomas Kühn
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 541400746

Function spaces of Besov or Sobolev type have been studied for many years. The history and interest in these spaces is closely linked with the study of PDEs, this remains a mainspring for the systematic study of such function spaces up to now. Their investigation involves a variety of classical and modern methods of functional analysis and harmonic analysis; conversely, this also has an impact on the further development of these areas. It concerns questions of interpolation theory, representation of distributions via wavelets or atoms, various problems of approximation. This extends from rather abstract settings like the approximation of compact linear operators by special, one-dimensional operators (the famous concept of nuclearity of an operator), to essential numerical questions of so-called high-dimensional approximation: While it might be satisfactory from a functional-analytic point of view to determine the asymptotic behaviour of certain approximation quantities (like approximation numbers), except for general constants which may depend on the setting of the problem, but not on the degree of approximation, this turns out to be unsatisfactory, perhaps even useless, for numerical purposes, as these universal constants may spoil the approximation rate for small numbers of approximations. Here "small" means with respect to the dimension of the underlying problem. So in case of high-dimensional problems, the theoretical findings for infinitely many approximation steps mentioned above, are less important than the preasymptotic estimates. Such phenomena appear, for instance, in finance, stochastic PDEs or quantum chemistry. As a result, these theoretical foundations and recent results received attention in the scientific community, also from people working in IBC (Information-Based Complexity), Machine Learning, neural networks, and further areas. In our application we study Besov spaces as multiplication algebras (this property turns out to be important when solving Navier-Stokes equations or nonlinear heat equations in the context of Besov spaces). We investigate the real interpolation of so-called Besov-Lorentz spaces (here exist a number of open questions which could be partly answered recently by the applicants, using rather new methods). We concentrate on the topic of nuclearity of an operator acting in Besov spaces of dominating mixed smoothness (such generalisation of Besov spaces are important for applications). And we finally focus on high-dimensional approximation, both in the context of Gevrey spaces, and in the sense of the preasymptotic behaviour described above. We are convinced that our joint expertise and various preliminary results in functional analysis and the theory of function spaces will lead to a number of essential new results which in turn will also be important for applications in PDEs, Numerics.

• DFG Programme: Research Grants
• International Connection: Spain
• Partner Organisation: Agencia Estatal de Investigación
• Cooperation Partners: Professor Dr. Fernando Cobos; Professorin Dr. Luz M. Fernández-Cabrera