Vorträge

In the early 1980s, Erd\H{o}s and S\'os initiated the study of the classical Tur\'an problem with a uniformity condition: the uniform Tur\'an density of a hypergraph $H$ is the infimum over all $d$ for which any sufficiently large hypergraph with the property that all its linear-size subhyperghraphs have density at least $d$ contains $H$. In particular, they raise the questions of determining the uniform Tur\'an densities of $K_4^{(3)-}$ and $K_4^{(3)}$. The former question was solved only recently in [Israel J. Math. 211 (2016), 349--366] and [J. Eur. Math. Soc. 20 (2018), 1139--1159], while the latter still remains open for almost 40 years.
In addition to $K_4^{(3)-}$, the only $3$-uniform hypergraphs whose uniform Tur\'an density is known are those with zero uniform Tur\'an density classified by Reiher, R\"odl and Schacht~[J. London Math. Soc. 97 (2018), 77--97] and a specific family with uniform Tur\'an density equal to $1/27$.

In this talk, we give an introduction to the concept of uniform Tur\'an densities, present a way to obtain lower bounds using color schemes, and give a glimpse of the proof for determining the uniform Turán density of the tight $3$-uniform cycle $C_\ell^{(3)}$, $\ell\ge 5$.

We study the observability and null control problem for
the semigroup generated by the harmonic oscillator
and the partial harmonic oscillator.
We identify sensor sets which ensure null controlabillity
improving and unifying previous results for such problems.
In particular, it is possible to observe the Hermite semigroup
from finite volume sensor sets.
This is joint work with A.Dicke and A. Seelmann.

Since I am new to our institute, I will start by introducing myself and presenting the results of my master thesis on second order information in
neural network training.

Traditionally, neural networks are trained using gradient-based optimization methods like Adagrad or Adam. Using second order methods might result in faster convergence (e.g. locally quadratic convergence in Newton's method). Furthermore, curvature information can provide some insight into the optimization process and help to characterize the cost function of a neural network.

For large problems, applying Newton's method and Quasi-Newton-methods to the cost function of a neural net is only possible through implicit Hessian-vector-products. For this reason, Krvlov subspace methods are particularly well suited for solving the linear system with the Hessian that appears in Newton's method. Krylov subspace methods use matrix-vector-products instead of operating on the full matrix.

Two data sets are used: The first one is constructed to allow the exact calculation (except for rounding errors) of the Hessian and its eigenvalues. Here, we observe that the largest eigenvalue can be approximated with a small number of steps of a Krylov subspace method and with high accuracy. The second data set is the famous MNIST data set for handwritten digit classification. For MNIST and the given computational resources, we cannot calculate the full Hessian of the cost function. Instead, the Krylov subspace method is used to approximate eigenvalues from implicitly calculated Hessian-vector-products. On both data sets, the largest eigenvalue can be observed to be coupled to the value of the cost function.

An inexact Quasi-Newton-method and the L-BFGS method are used to train a neural network on both data sets.

Furthermore, I will talk about first ideas for my dissertation on nonlinear finite element methods with applications in ship structural design.

We are interested in constructing cheap and efficient implicit high order
solvers for compressible turbulent
flow problems. These problems arise for
example in the design of next generation jet engines, air frames, wind tur-
bines or star formation. A suitable high order discretization for these prob-
lems are discontinuous Galerkin spectral element methods (DG-SEM). In
this talk we discuss challenges of solvers for DG-SEM discretizations in space
combined with implicit time-stepping methods.
One option to yield implicit DG-SEM solvers is to apply a space-time
DG-SEM discretization, i.e. discretizing space and time simultaneously with
DG-SEM. We present two approaches for the formulation and implementa-
tion of space-time DG-SEM: Either time is treated as an additional coor-
dinate direction and the Galerkin procedure is applied to the entire prob-
lem. Alternatively, the method of lines is used with DG-SEM in space and
the fully implicit Runge-Kutta method Lobatto IIIC in time. The two ap-
proaches are mathematically equivalent in the sense that they lead to the
same discrete solution. However, in practice they differ in several important
respects, including the terminology used to the describe them, the struc-
ture of the resulting software, and the interaction with nonlinear solvers.
We present challenges and merits of the two approaches and show their im-
pact on numerical tests using implementations based on the Distributed and
Unified Numerics Environment (DUNE).
Another option to construct implicit DG-SEM solvers is the classical
method of lines approach. The spatial directions are discretized with DG-
SEM and any implicit time-stepping method can be applied to the resulting
ODE. This yields large nonlinear systems and a solver has to be chosen
carefully. We suggest to use a preconditioned Jacobian-free Newton-Krylov
method. The challenge here is to construct a preconditioner without con-
structing the Jacobian of the spatial discretization. Our idea is to make use
of a simplified replacement operator for the DG operator and a multigrid
method. We discuss the idea of our suggested preconditioner and present
numerical results to show the potential of this preconditioning technique.

Vortrag (PDF, 73KB)

https://tuhh.zoom.us/j/82516486683?pwd=RnV4ZEcvREhXeDYyZXdiUE1kUmh1QT09

Meeting-ID: 825 1648 6683
Kenncode: 329040

It is a conjecture going back to J. Rauch (1974) that the hottest and coldest spots in an insulated homogeneous medium such as an insulated plate of metal should converge to the boundary, for "most" initial heat distributions, as time tends to infinity. This so-called hot spots conjecture can be phrased alternatively as follows: the eigenfunction(s) corresponding to the first non-zero eigenvalue of the Neumann Laplacian on a Euclidean domain should take its maximum and minimum on the boundary only. This has been proven to be false for certain domains with holes, but it was shown to hold for several classes of simply connected or convex planar domains. One of the most recent advances is the proof for all triangles given by Judge and Mondal (Annals of Math. 2020). The conjecture remains open in general for simply connected or at least convex domains. In this talk we provide a new approach to the conjecture. It is based on a non-standard variational principle for the eigenvalues of the Neumann and Dirichlet Laplacians.

Jonathan Rohleder is an associate professor at Stockholm University, Sweden. His work focusses on spectral theory.