Vorträge

Lower bounds for variances are often needed to derive central limit theorems. In this talk, we establish a specific lower bound for the variance of a Poisson functional that uses the difference operator of Malliavin calculus.
Poisson functionals, i.e. random variables that depend on a Poisson process, are widely used in stochastic geometry. In this talk, we show how to apply our lower variance bound to statistics of spatial random graphs, the $L^p$ surface area of random polytopes and the total edge length of hyperbolic radial spanning trees. This talk is based on joint work with M. Schulte.

Spectral Deferred Correction (SDC) methods are iterative solvers for collocation discretization of ordinary differential equations, but each iterate can also be interpreted as particular Runge-Kutta (RK) scheme. In contrast to fixed RK schemes, viewing SDC as a fixed point iteration allows combining them with various kinds of deliberate perturbations resulting from mesh adaptivity or algebraic adaptivity in PDEs, lossy compression in parallel-in-time solvers, or inexact computations in scale-separated long time integrations, for improved performance. It also fosters a deeper understanding of SDC approximation error behavior, and the construction of more efficient preconditioners. In the talk, we will touch several of these aspects, and provide a - necessarily incomplete - overview of the astonishing flexibility of SDC methods.

The talk discusses the benefits of pressure-robust discretizations in the scope of optimal control of incompressible flows.
Here, gradient forces appearing in the data can have a negative impact on the accuracy of state and control and can only be correctly balanced if their
$L^2$-orthogonality onto discretely divergence-free test functions is restored. Perfectly orthogonal divergence-free discretizations or divergence-free reconstructions
of these test functions do the trick and lead to much better analytic a priori estimates that are also validated in numerical examples.

This is joint work with Christian Merdon (WIAS)

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