"Neue Funktionenräume auf Gebieten und ihre diskrete Charakterisierung"

• Antragsteller: Professor Dr. Stephan Dahlke
• Fachliche Zuordnung: Mathematik
• Förderung: Förderung von 2017 bis 2022
• Projektkennung: Deutsche Forschungsgemeinschaft (DFG) - Projektnummer 373295677

• Erstellungsjahr: 2024

Zusammenfassung der Projektergebnisse

The aim of the research project was to develop new numerical discretizations for operator equations that are capable of handling more general operator and singularity structures than classical finite elements. The original motivation for this came from signal processing, where in the early 2000’s a host of new function decompositions have been proposed, each one being capable of optimally approximating functions with certain types of singularities. These systems were originally conceived for approximation and compression purposes. The goal of this project was to develop them further into a tool for solving operator equations. For this purpose, several practical problems need to be addressed: most such systems are defined on all of Rd whereas many operator equations are posed on finite domains with suitable boundary conditions imposed. Furthermore, efficient numerical schemes require fast computatbility of the operator Gram matrix, which amounts to the development of efficient quadrature schemes. In a general form, our goal can be phrased as follows: Develop novel numerical discretization methods into useful tools for solving PDEs, and thus widen the scope of PDEs that can be efficiently addressed by numerical methods. The aforementioned goal was definitely achieved. During the course of this project, new numerical methods were developed and analyzed that significantly outperform the previous state of the art in the numerical solution of high-dimensional Kolmogorov PDEs and electronic Schrödinger equations – two notoriously challenging problems. In addition several new results could be obtained on the approximation behaviour of modern non-standard approximation methods, such as, for example, anisotropic wavelets, or neural networks.

Projektbezogene Publikationen (Auswahl)

• Numerically solving parametric families of high-dimensional kolmogorov partial differential equations via deep learning. Advances in Neural Information Processing Systems, 33:16615–16627.
  J. Berner, M. Dablander & P. Grohs
  
• Deep Neural Network Approximation Theory. IEEE Transactions on Information Theory, 67(5), 2581-2623.
  Elbrachter, Dennis; Perekrestenko, Dmytro; Grohs, Philipp & Bolcskei, Helmut
  
• Anisotropic Triebel-Lizorkin spaces and wavelet coefficient decay over one-parameter dilation groups, II. Monatshefte für Mathematik, 201(2), 431-464.
  Koppensteiner, Sarah; van Velthoven, Jordy Timo & Voigtlaender, Felix
  
• Anisotropic Triebel–Lizorkin spaces and wavelet coefficient decay over one-parameter dilation groups, I. Monatshefte für Mathematik, 201(2), 375-429.
  Koppensteiner, Sarah; van Velthoven, Jordy Timo & Voigtlaender, Felix
  
• Towards a transferable fermionic neural wavefunction for molecules. Nature Communications, 15(1).
  Scherbela, Michael; Gerard, Leon & Grohs, Philipp