Keywords. shape analysis and matching, inverse problems, optimization, (nonlinear) elasticity theory, (nonlinear) finite elements, computer vision and graphics, nonlinear dispersive PDEs, functional analysis, monotone operators, unbounded operators
Elasticity Based Models for Shape Matching
During my PhD studies at the Weizmann Institute of Science, Israel, I was part of the computer vision and graphics group. My PhD thesis was focused on the development of physics based methods (PDE models) for the shape alignment problem which has applications in medical imaging, computer graphics and vision.
Outcome. A set of three methods for (pairwise) shape matching that account for unknown (observed) large deformations of shapes, e.g., anatomical shapes, in 2D and 3D. An improved understanding of the origin of deformations can be provided by finding a possible physical cause (external forces) in a suitable optimization framework.
Nonlinear Dispersive PDEs
We investigated the applicability of novel so-called time averaging methods to systems of dispersive PDEs. These methods are simpler than those coming from harmonic analysis and by-pass the use of complicated dispersive Sobolev spaces. Furthermore, we believe that these methods reveal the regularizing character of dispersion (e.g., fast rotations) much clearer than powerful but complicated methods form harmonic analysis.
Outcome. We can show well-posedness of a dispersive system that models the interaction of baroclinic and barotropic waves in the mid-latitudes with relatively simple means. Furthermore, we show that it is not necessary to invert a relatively complicated ODE analytically which was done in the original paper. Instead, we employ a mode splitting into high and low modes and achieve well-posedness in spaces of relatively rough functions.
Monotone Operators for Nonlinear Integro-Differential
Equations in Aeronautical Engineering (Diploma thesis)
My diploma thesis (German equiv. to master thesis) is about well-posedness of a class of integro-differential equations that arise in aeronautical engineering of aircraft wings. These equations involve singular integrals and often fairly complicated nonlinearities.
Outcome. My thesis summarizes results in the literature in a comprehensive and detailed way in order to make the results and methods accessible and clearer. Furthermore, we point out the essential difficulties and necessities to obtain well-posedness of the relevant models and suggest generalizations.