Vorträge

Vortrag (PDF, 228KB)

Scattered data interpolation refers to an interpolation problem where the data sites are distributed irregularly within some domain. An interpolant may be constructed as a linear combination of radial basis functions centered at the data sites. Finding the coefficients in this representation leads to linear equations where the system matrices are large, dense, indefinite and ill-conditioned. These matrices can be approximated using the framework of hierarchical matrices. We will compare different approximation methods and discuss how to construct algebraic preconditioners.

Es werden große dünnbesetzte Sattelpunktprobleme betrachtet, wie sie z.B. in der Strömungsmechanik auftreten. Diese i.A. unsymmetrischen und indefiniten Systeme können mittels iterativer Krylovraum-Verfahren, inkl. geeigneter Präkonditionierer, gelöst werden. Insbesondere werden sogenannte induzierte Dimensions-Reduktions-Methoden (IDR), im Speziellen QMRIDR(s), verwendet, welche zusätzlich mit einem Deflationsansatz gepaart werden. Dabei werden Informationen aus früheren Durchläufen derart recycelt, sodass es möglich ist, Sequenzen von linearen Systemen effektiv zu lösen. Als Beispiel werden die diskretisierten Oseen-Gleichungen betrachtet; weitere Anwendung kann dies darüber hinaus z.B. bei inneren Punkte-Verfahren in der linearen Optimierung finden.

The goal of an a posteriori error analysis for an approximate PDE
solution is to establish the equivalence of error and a posteriori
estimator. Unfortunately, this equivalence is often only up to so-
called oscillation terms.

In this talk we shall clarify the reasons for the presence of
oscillation. Moreover, we propose a new approach to a posteriori error
estimation, where oscillation can be bounded by the error and so does
not longer spoil the aforementioned equivalence.

This is joint work with Christian Kreuzer (Bochum).

We present a space-time discontinuous Galerkin (DG) method for linear
wave propagation problems.
The special feature of the scheme is that it is a Trefftz method,
namely that trial and test functions are solution of the partial
differential equation to be discretised in each element of the
(space-time) mesh.
The DG scheme is defined for unstructured meshes whose internal faces
need not be aligned to the space-time axes.
The Trefftz approach can be used to improve and ease the
implementation of explicit schemes based on ``tent-pitched'' meshes.
We show that the scheme is well-posed, quasi-optimal and dissipative,
and prove a priori error bounds for general Trefftz discrete spaces.
A concrete discretisation can be obtained using piecewise polynomials
that satisfy the wave equation elementwise, for which we show high
orders of convergence.
If time allows, we will describe a similar Trefftz-DG method for the
Helmholtz equation, i.e. wave equation in time-harmonic regime, for
which non-polynomial basis functions are used and quite a complete
theory has been established.

Projektarbeit

The evolution of a plasma in external and self-consistent fields is modelled by the Vlasov equation for the distribution function in six dimensional phase space. Due to the high dimensionality and the development of small structures the numerical solution is very challenging. Grid-based methods
for the Vlasov equation have been shown to give accurate results but their use has mostly been limited to simulations in two or four dimensional phase space due to extensive memory requirements in higher dimensions. Compression of the solution via high-order singular value decomposition can help in reducing the storage requirements and the hierarchical Tucker format provides efficient basic linear algebra routines for low-rank representations of tensors.

In this talk, I will present a semi-Lagrangian scheme for the solution of the Vlasov equation represented as a low-parametric tensor. Interpolation formulas for the low-parametric tensor format as well as efficient implementations will be discussed. Numerical simulations for the Vlasov-Poisson equation are shown for the Landau damping test case in two, four, and six dimensional phase space as well as simulations with a constant magnetic field. Depending on the test case, the memory
requirements reduce by a factor $10^2$-$10^3$ in four and a factor $10^5$-$10^6$ in six dimensions compared to the full-grid method.