Research
interests
Homogenization of parabolic geometric partial differential equations
in the group of
Prof. Desbrun,
Computing + Mathematical Sciences,
Applied Geometry Lap, joined work with
Prof. Owhadi.
Two step time discretization of the
anisotropic Willmore flow
in the group of
Prof. Rumpf, Institute for Numerical Simulation,
University of Bonn, cf.
homepage.
Using the isotropic Willmore functional does not lead to satisfying results if
an edge or a corner of the surface is destroyed. The anisotropic
Willmore energy is a natural generalization of the Willmore energy
which has crystal-shaped surfaces like cubes or octahedra as
minimizers. We extend the two step time discretization for discrete
isotropic Willmore flow to the anisotropic case. The approach is applied to
polygonal curves, where the anisotropy could be chosen almost
crystalline. Various numerical examples underline again the
stability of the new scheme, which enables time steps of the order
of the spatial grid size.
Surface restoration based on the two step
time discrete isotropic Willmore flow
in the group of
Prof. Rumpf, Institute for Numerical Simulation,
University of Bonn, cf.
homepage.
Extending the two step time discretization of the isotropic Willmore
flow to boundary conditions, we are able to restore surfaces with
smooth boundary conditions. E.g. we apply the new scheme to a real
world restoration problem, where we reconstruct damaged regions of
an Egea sculpture.
in the group of
Prof. Rumpf, Institute for Numerical Simulation,
University of Bonn, cf.
homepage.
Based on a natural approach for the time discretization of gradient
flows we develop a new time discretization for discrete Willmore
flow of polygonal curves and triangulated surfaces. The approach is
variational and takes into account an approximation of the
L2-distance between the surface at the current time step and the
unknown surface at the new time step as well as a fully implicity
approximation of the Willmore functional at the new time step. The approach is applied to polygonal curves and
triangular surfaces and is independent of the co-dimension. The new
scheme is stable and enables time steps of the order of the spatial
grid size.
Flow Visualization via Segmentation
in E-Dur
in the group of
Prof. Rumpf, Institute for Numerical Simulation,
University of Bonn, cf.
homepage.
The visualization of
flow is an important and challenging topic in scientific
visualization. In the third project
Weiterentwicklung der Rechenprogramme
d3f und r3t (E-DuR) funded by the German Federal Ministry of
Education and Research, we develop a Mumford-Shah model for
visualizing flow fields via segmentation.
Visualization in GRAPE
in the group of
Prof. Rumpf, Institute for Numerical Simulation,
University of Bonn, cf.
homepage.
The software package
GRAPE
has been developed at the Collaborative Research Center 256 at the University of Bonn
and at the
Institute for Applied Mathematics at the University of
Freiburg. My main interests in scientific visualization using GRAPE are
adaptively hierachical postprocessing and visualization methods for
large data sets accessed via a procedural interface.
A number of these methods are included in the postprocessing and
visualization tool that has been developed in cooperation with the
Gesellschaft für Anlagen-
und Reaktorsicherheit in three projects funded by the German
Federal Ministry of Education and Research, Entwicklung eines
schnellen Programms zur Modellierung von Grundwasserströmungen
mit variabler Dichte (d3f), Entwicklung eines Programmes zur
dreidimensionalen Modellierung des Schadstofftransportes (r3t)
and Weiterentwicklung der
Rechenprogramme d3f und r3t (E-DuR).
in the group of
Prof. Rumpf, Institute for Numerical Simulation,
University of Bonn, cf.
homepage.
Establishing a correspondence between two surfaces is a basic
ingredient in many geometry processing applications. Existing
approaches, which attempt to match two embedded meshes directly, can
be cumbersome to implement and it is often hard to produce accurate
results in reasonable time. A new variational method for matching
surfaces that addresses these issues is presented.
in the group of Prof. Rumpf, University of Duisburg-Essen, cf.
homepage.
For a two-dimensional surface in IR3 that has gender zero,
low-distortion conformal parameterizations are described in terms of
minimizers of suitable energy functionals. Appropriate distortion
measures are derived from principles of rational mechanics, closely
related to the theory of non-linear elasticity.
in the group of Prof. Rumpf, University of Duisburg-Essen, cf.
homepage.
The
visualization of time-dependent flow is an important and challenging
topic in scientific visualization. Its aim is to represent transport
phenomena governed by time-dependent vector fields in an intuitively
understandable way, using images and animations. Here we pick up the
recently presented anisotropic diffusion method, expand and
generalize it to allow a multiscale visualization of long-time,
complex transport problems.
