Construction of New Smoothness Spaces on Domains

This project is concerned with a set of problems in signal analysis which are currently of great interest. They are connected with the overall goal of data analysis: extraction of relevant information for a concrete application. Big Data is one of the keywords in this context, and applied mathematics is certainly able to make substantial contributions to this area. An important asset is the multitude of representation systems which have been developed in recent years. In the ideal case sparse representation of signals allows to extract specific information, such as directional information from images. A natural goal for the current research is therefore to investigate potential role of such representation systems for the numerical treatment of operator equations. Since operator equations form the basis for the modeling of most problems in science and engineering, the constructive approximation of their solutions is a field of enormous importance. This requires to treat the following fundamental problem: While traditional expansions work for functions defined over the full Euclidean space, the numerics of operator equations requires to have similar expansions over bounded domains or on manifolds. It is one of the declared goals of this project to contribute to this topic. Hence we plan to develop shearlet or Gabor frames on general domains to the extent possible. Besides exploiting the potential of suitable extension operators and domain decomposition approaches, we also want to bring in the power of coorbit theory. We expect that efficient numerical algorithms can be established by investigating the compression properties of the stiffness matrices associated with such expansion systems as well as corresponding approximation properties. Finally, we will describe the regularity properties of important classes of operator equations with respect to optimally adapted discretizations through frame expansions.

Partial differential equations (PDEs) are a fundamental tool for modeling our physical world. Since these equations are generally not solvable exactly by formulas, numerical methods are required to approximate their solutions. The most commonly used method employs "finite elements" as trial functions. It can be mathematically proven that this approach is optimal for certain classes of PDEs. However, there are many important problems-such as high-dimensional problems or those with complex singularities-for which finite elements are unsuitable. For these problems, traditional methods become so complex that even the most advanced supercomputers reach their limits. This project developed new systems of trial functions that enable the efficient numerical solution of high-dimensional and singular problems. On one hand, the mathematical foundations were established to make these systems usable for problems on complex geometries. On the other hand, modern deep learning methods were developed to solve various high-dimensional problems. Examples include the Black-Scholes equation from financial mathematics and the electronic Schrödinger equation from quantum chemistry. Furthermore, theoretical results concerning the optimal convergence order of adaptive methods have been proven. These serve as benchmarks for the development of new discretization methods. Additionally, new approach systems have been developed using Coorbit theory, which will be used in the near future for the numerical solution of PDEs on manifolds, such as on spheres.