Talks

In this talk, we review the underappreciated theorem by Lotz that tells us that every strongly continuous operator semigroup on a Grothendieck space with the Dunford-Pettis property is automatically uniformly continuous. A large class of spaces that carry these geometric properties are L^\infty for non-negative measure spaces. This shows once again that $L^\infty$-spaces have to be treated differently.

Atmospheric motion covers a broad range of temporal and spatial scales. Low- and high-pressure systems can influence us for days or even weeks, and they extend up to hundreds of kilometers. In contrast, sound waves pass by in seconds with wavelengths of centimetres to meters. Implicit-explicit (IMEX) time stepping methods can help avoid drastic limitations on the time step induced by this variety of scales without requiring computationally expensive fully nonlinear implicit solves.
We discuss (IMEX-) Spectral Deferred Correction (SDC) methods in the context of Runge-Kutta methods (RKM) and apply SDC to test cases, which are relevant to numerical weather prediction. First, we use RKM theory to:
1. Construct new SDC methods that increase convergence order by 2 per iteration, in contrast to the increase of 1 in general SDC methods;
2. Construct SDC methods that conserve quadratic invariants;
3. Show that SDC can be symmetric but not symplectic for finite iterations.
Second, we apply implicit-explicit (IMEX) SDC to fluid dynamical problems that are relevant to numerical weather prediction. In particular, we compare IMEX-SDC, multistep and RKM time integrators for the Galewsky test case using the Python spectral method framework Dedalus. We demonstrate that SDC methods have superior stability properties and can provide a shorter time to solution for comparable errors. In addition, we outline strategies that could further reduce simulation times by using the SDC residual to minimise the computational effort. Finally, we apply SDC to the compressible Euler equations using compatible finite element methods, demonstrating their applicability to more complex atmospheric models.

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

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.