Talks

Approximating functions by a linear span of truncated basis sets is a standard procedure for the numerical solution of differential equations. Commonly used concepts of approximation methods are well-posed and convergent, by provable approximation orders. On the down side, however, these methods often suffer from the curse of dimensionality, which limits their approximation behavior. Nonlinear approximation methods, such as neural networks, were shown to be very efficient approximating high-dimensional functions. We investigate nonlinear approximation methods that are constructed by composing standard basis sets with normalizing flows. Such models yield richer approximation spaces while maintaining the density properties of the initial basis set, as we show. We investigate such approximation schemes for solving molecular Schrödinger equations and provide linear and nonlinear convergence analysis.

Zoomlink:
https://tuhh.zoom.us/j/84729171896?pwd=ODArbForaUxMM3Q3VTJsNG1kaVNYQT09

Stochastic optimization techniques have become an essential tool for training of
neural networks. One prominent algorithm is stochastic gradient descent
(SGD). Under smoothness and convexity assumptions one can show
convergence of SGD to a minimizer. However, the analyses of variants of
SGD require different techniques. In this talk, we look at recent
advances in modelling SGD by a continuous-time process defined by a
stochastic differential equation to obtain a unified framework. In
particular, we motivate the connection between the discrete and
continuous process and investigate in what sense they convergence to one
another. Further, we present examples of how the continuous-time model
behaves in practice.

Zoomlink:
https://tuhh.zoom.us/j/84729171896?pwd=ODArbForaUxMM3Q3VTJsNG1kaVNYQT09

Describing aspects of physical phenomena by forming abstract mathematical models is a common practice in scientific work: the mathematical formalism allows for permeation of the mathematical model as a means of creating insights and knowledge over the described real-world phenomenon.

In this talk, I will present how the topics of my dissertation contribute to the theory of popular mathematical models ranging from quantum physics to mathematical fluid mechanics.

In particular, you will find out

(I) how to classify periodic potentials of discrete Schrödinger operators with respect to the applicability of the finite section method,
(II) how to prove final-state observability for time-dependent diffusion problems, and
(III) how to improve the regularity of weak solutions to the Navier-Stokes equations on rough domains.

Link to slides:
https://math.fabian-gabel.de/talks/fabian_gabel_dissertation_pres.pdf

Link to video recording:
https://youtu.be/_2W-b-vXeZE