Talks

In this talk I will show how transfer matrix techniques can be used to compute the asymptotic spectra of non-Hermitian tridiagonal finite-block Toeplitz matrices. In this way one can recover Widoms results on the asymptotic spectra of such matrices. Special attention will be given to topological eigenvalues arising from matrices with a chiral symmetry and the associated bulk-boundary correspondence. Going beyond Widoms theory, I will also show how the transfer matrix approach can be used to compute the asymptotic spectra of Toeplitz matrices with a perturbation on a finite number of sites. One can then see how the different parts of the spectra depend on the perturbations. Varying the ranks of the perturbations one can now also interpolate between open and closed boundary conditions. The results will be illustrated by numerics.

This is joint work with Hermann Schulz-Baldes.

We explore examples of Dirac operators on bounded domains exhibiting an interval of essential spectrum. In particular, we consider three-dimensional Dirac operators on Lipschitz domains with critical electrostatic and Lorentz scalar shell interactions supported on a compact smooth surface. Unlike typical bounded-domain settings where the spectrum is purely discrete, we show that the criticality of these interactions can generate a nontrivial essential spectrum interval, whose position and length are explicitly controlled by the coupling constants and surface curvatures.

Based on joint work with J. Behrndt (TU Graz), M. Holzmann (TU Graz), and K. Pankrashkin (Univ. Oldenburg).

With today’s exascale computers requiring 20 to 40 MW and some cloud centers exceeding 100MW, with no slacking of demand in sight, computing is a nonnegligible factor in climate change. For the past three years, we have been finalists in the Gordon Bell Prize with computations that do more with less – that scale up while squeezing out operations and data transfers that do not ultimately impact application accuracy requirements. Scientific and engineering computing has a history of “oversolving” inherited from a period when its cost was small enough to neglect. Today’s market for computing hardware is driven by machine learning applications that are able to exploit lower precision arithmetic. Traditional computational science and engineering are therefore being reinvented to employ lower precision arithmetic and replacement of blocks of operator and field data by low-rank substitutes, where possible without impacting accuracy. We provide examples from various applications, including Gordon Bell Prize finalist research in 2022 in environmental statistics, in 2023 in seismic processing, and in 2024 in genomics and again in climate emulation. The last was awarded the 2024 Gordon Bell Prize in Climate Modeling. In this talk, we will elucidate the algorithmic “secret sauce” shared by these diverse applications for which the (Gordon) Bell tolls.

Iterative parallel-in-time algorithms like Parareal can extend scaling beyond the saturation of purely spatial parallelization when solving initial value problems.
However, they require the user to build coarse models to handle the inevitable serial transport of information in time.
This is a time-consuming and difficult process since there is still limited theoretical insight into what constitutes a good and efficient coarse model.
Novel approaches from machine learning to solve differential equations could provide a more generic way to find coarse-level models for parallel-in-time algorithms.
This talk demonstrates that a physics-informed Fourier Neural Operator (PINO) is an effective coarse model for the parallelization in time of the two-asset Black-Scholes equation using Parareal.
We demonstrate that PINO-Parareal converges as fast as a bespoke numerical coarse model and that, in combination with spatial parallelization by domain decomposition, it provides better overall speedup than both purely spatial parallelization and space-time parallelization with a numerical coarse propagator.

Zoomlink:
https://tuhh.zoom.us/j/81920578609?pwd=TjBmYldRdXVDT1VkamZmc1BOajREZz09

Conlon and Gowers in 2016 described a general approach to proving sparse random analogues of extremal results in combinatorics, such as bounding the minimum and maximum number of triangles in any subgraph of G(n,p) with a given number of edges. The general part of this approach is a functional-analytic statement which, given a sparse setting, constructs a dense model. However, there is a condition which must be shown to hold with high probability to apply the dense model theorem. In Conlon and Gowers' work, there is a technical difficulty with the probabilistic part which leads to a rather involved proof, which applies only in a restricted setting (for example, they can handle triangles but not triangles with a pendant edge), and with quite poor bounds on 'high probability'.

We revisit Conlon and Gowers' approach, and show how to avoid their technical problem, giving a simpler proof of their counting result which applies in a general setting and with optimal probability bounds. As a corollary, we prove the 'Counting KLR' theorem of Conlon, Gowers, Samotij and Schacht, but for general hypergraphs and with optimal probability bounds. This is joint work with Julia Boettcher, Joanna Lada and Domenico Mergoni.

Ungefähre Themen des Vortrags:
1. Probleme der Intervallberechnung.
2. Die Beziehung zwischen Intervallanalyse und Wahrscheinlichkeitstheorie.
3. Bayesianisches Modell der kollektiven Entscheidungsfindung und seine Intervallverallgemeinerung.