154 research outputs found
Geometric partial differential equations: Theory, numerics and applications
This workshop concentrated on partial differential equations involving stationary and evolving surfaces in which geometric quantities play a major role. Mutual interest in this emerging field stimulated the interaction between analysis, numerical solution, and applications
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Mathematical Imaging and Surface Processing
Within the last decade image and geometry processing have become increasingly rigorous with solid foundations in mathematics. Both areas are research fields at the intersection of different mathematical disciplines, ranging from geometry and calculus of variations to PDE analysis and numerical analysis. The workshop brought together scientists from all these areas and a fruitful interplay took place. There was a lively exchange of ideas between geometry and image processing applications areas, characterized in a number of ways in this workshop. For example, optimal transport, first applied in computer vision is now used to define a distance measure between 3d shapes, spectral analysis as a tool in image processing can be applied in surface classification and matching, and so on. We have also seen the use of Riemannian geometry as a powerful tool to improve the analysis of multivalued images.
This volume collects the abstracts for all the presentations covering this wide spectrum of tools and application domains
Numerical approximation of boundary value problems for curvature flow and elastic flow in Riemannian manifolds
We present variational approximations of boundary value problems for
curvature flow (curve shortening flow) and elastic flow (curve straightening
flow) in two-dimensional Riemannian manifolds that are conformally flat. For
the evolving open curves we propose natural boundary conditions that respect
the appropriate gradient flow structure. Based on suitable weak formulations we
introduce finite element approximations using piecewise linear elements. For
some of the schemes a stability result can be shown. The derived schemes can be
employed in very different contexts. For example, we apply the schemes to the
Angenent metric in order to numerically compute rotationally symmetric
self-shrinkers for the mean curvature flow. Furthermore, we utilise the schemes
to compute geodesics that are relevant for optimal interface profiles in
multi-component phase field models.Comment: 42 pages, 21 figure
Variational Convergence and Discrete Minimal Surfaces
This work is concerned with the convergence behavior of the solutions to parametric variational problems. An emphasis is put on sequences of variational problems that arise as discretizations of either infinite-dimensional optimization problems or infinite-dimensional operator problems. Finally, the results are applied to discretizations of the Douglas-Plateau problem and of a boundary value problem in nonlinear elasticity
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Nonlinear Data: Theory and Algorithms
Techniques and concepts from differential geometry are used in many parts of applied mathematics today. However, there is no joint community for users of such techniques. The workshop on Nonlinear Data assembled researchers from fields like numerical linear algebra, partial differential equations, and data analysis to explore differential geometry techniques, share knowledge, and learn about new ideas and applications
Hydrodynamic interactions in polar liquid crystals on evolving surfaces
We consider the derivation and numerical solution of the flow of passive and
active polar liquid crystals, whose molecular orientation is subjected to a
tangential anchoring on an evolving curved surface. The underlying passive
model is a simplified surface Ericksen-Leslie model, which is derived as a
thin-film limit of the corresponding three-dimensional equations with
appropriate boundary conditions. A finite element discretization is considered
and the effect of hydrodynamics on the interplay of topology, geometric
properties and defect dynamics is studied for this model on various stationary
and evolving surfaces. Additionally, we consider an active model. We propose a
surface formulation for an active polar viscous gel and exemplarily demonstrate
the effect of the underlying curvature on the location of topological defects
on a torus
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