Vorträge 1 bis 10 von 525 | Gesamtansicht
Observability for the (anisotropic) Hermite semigroup from finite volume or decaying sensor sets*
Ivan Veselic, TU Dortmund, Fakultät für Mathematik, Lehrstuhl LSIX
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.
|04.02.22||13:30||Zoom (same as coffee chat)||
Discontinuous Galerkin Spectral Element Methods - Space-Time Formulations and Efficient Solvers
Lea Miko Versbach
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.
Schleifen und Mehrfachkanten im Konfigurationsmodell [Bachelorarbeit]
Happy Khairunnisa Sariyanto
A new approach to the hot spots conjecture
Dr. Jonathan Rohleder, Stockholm University, Sweden
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.
Solution of the vibrational Schrödinger equation using neural networks [Masterarbeit]
Behavior of Nonlinear Water Waves in the Presence of Random Wind Forcing
Specific solutions of the nonlinear Schrödinger equation, such as the Peregrine breather, are considered to be prototypes of extreme or freak waves in the oceans. An important question is whether these solutions also exist in the presence of gusty wind. Using the method of multiple scales, a nonlinear Schrödinger equation is obtained for the case of wind-forced weakly nonlinear deep water waves. Thereby, the wind forcing is modeled as a stochastic process. This leads to a stochastic nonlinear Schrödinger equation, which is calculated for different wind regimes. For the case of wind forcing which is either random in time or random in space, it is shown that breather-type solutions such as the Peregrine breather occur even in strong gusty wind conditions.
|06.01.22||13:00||Am Schwarzenberg-Campus 3 (E), Raum 3.074||
Bachelorarbeit: Task-basierte Implementierung von Parareal mittels torcpy
Florentine Meerjanssen, Institut für Mathematik
Low-Rank Updates for Schur Complement Preconditioners
Atmospheric dynamics can be described by the Boussinesq approximation which models bouyancy-driven fluid flows. Its simulation involves the repeated solution of the Navier-Stokes equations. This requires numerical solution methods for the dense Schur complement. In this talk, we will be concerned with Schur complement preconditioners. Furthermore, we will discuss a low-rank update for the Schur complement preconditioners. The update method is based on the error between the preconditioned Schur complement and the identity. It will be illustrated with some numerical results.
A Block Householder Based Algorithm for the QR Decomposition of Hierarchical Matrices
Hierarchical Matrices are dense but data-sparse matrices that use low-rank factorisations of suitable submatrices to allow for storage with linear-polylogarithmic complexity. Furthermore, efficient approximations of matrix operations like matrix-vector and matrix-matrix multiplication, matrix inversion and LU decomposition are available. There are several approaches for the computation of QR factorisations in the hierarchical matrix format, however, they suffer from numerical drawbacks that limit their use in many applications. In this talk, I will present a new approach based on block Householder transformations that improves upon some of those problems. To prevent unnecessary high ranks in the resulting factors and increase speed as well as accuracy the algorithm meticulously tracks for which intermediate results low-rank factorisations are available.
I will try to keep things as simple as possible and give a short introduction to hierarchical matrices as well. Previous knowledge of them is not necessary to understand the basic ideas and main obstacles of the new algorithm. I will focus on aspects, that I haven't talked about yet in similar talks in the past, mainly on how a cost estimate is possible although the hierarchical structure of the resulting QR decomposition is step-wise created during the algorithm and not defined beforehand.
* Vortrag im Rahmen des Kolloquiums für Angewandte Mathematik