Seminar: Nonlinear and high-dimensional approximation (WS 2025/2026)

Prof. Dr. Markus Bachmayr


Topics

In this seminar, we will discuss recent literature and ongoing research on numerical approximation methods for high-dimensional functions, in particular for partial differential equations on high-dimensional domains, and on related subjects. In particular, the topics will include:

  • Low-rank tensor approximations for time-dependent problems.
  • Approximations by linear combinations of Gaussian functions and their connections to optimal transport problems.
  • High-dimensional eigenvalue problems and recent developments in numerical methods in quantum chemistry.
  • Stability properties of approximations by compositions of functions such as neural networks.

Registration, organization and credits

The seminar will accommodate talks both by MSc students and by PhD students and postdoctoral researchers. The talks by MSc students will take place at the end of the semester.

If you are interested in participating in the seminar as an MSc student, please contact Prof. Bachmayr directly. Registration is via RWTHonline. If you register for the seminar you will automatically get access to the course room in RWTHmoodle. Further information (including on the first meeting in the first week of the semester) will be provided there.



Literature

Some selected papers:

  • A. Cohen, R. DeVore, G. Petrova and P. Wojtaszczyk. Optimal Stable Nonlinear Approximation. Foundations of Computational Mathematics, 2021.
  • F. Cucker. Probabilistic analyses of condition numbers. Acta Numerica, 2016.
  • G. Ceruti, J. Kusch and Ch. Lubich. A rank-adaptive robust integrator for dynamical low-rank approximation. BIT, 2022.
  • M. Bachmayr. Low-rank tensor methods for partial differential equations, Acta Numerica, 2023.
  • Eric Cancès et al. On basis set optimisation in quantum chemistry. ESAIM: Proceedings and Surveys, 2023.