2009-2012 FriNat project
The aim of this project is the design, analysis, and implementation of leading edge algorithms for the numerical integration of differential equations arising in mechanics and control applications.
In the GeNuIn project we consider simulation and modelling of multi-body dynamics, nonholonomic systems and problems arising in biomedical imaging. Our main scientific goal is to achieve more accurate and reliable numerical simulations of the considered physical phenomena by exploiting the qualitative properties of the underlying models. Our starting point are models with an underlying geometric structure, like for instance differential equations whose solution evolves on a manifold or admits a symmetry. This branch of numerical analysis of differential equations is known under the name Geometric Numerical Integration.
For example the computer simulation of complicated real life mechanical structures is achieved by first describing the physical problem with models of differential equations using geometry and analysis, then performing a numerical discretization. Testing and validation of the control strategies applied to operate the mechanical structures are often essential goals of the simulations.
Control operations first come into play in the modeling phase, where they are designed based on physical principles and analyzed with mathematical tools. Their faithful reproduction under discretization is depending on qualitatively correct numerical tools able to transfer the most important physical and geometrical features from the continuous model to its discrete numerical approximation.
In modern medicine, the physician extensively bases the diagnosis and choice of therapy on interpretation of images. Image-guided therapy has a rapid growing impact on clinical decision making. However, the potential benefits of all these imaging methods have not been fully exploited. In this project we will consider problems of image registration and denoising for medical images, that are fundamental to any other process of data extraction from the images.
Both rod dynamics and imaging methods can be described by mathematical models that admit a Lagrangian variational formulation and share a common feature: they possess an underlying geometric structure that has signicant importance on the dynamical behaviour of the system, as it generally corresponds to a concrete physical feature of the problem.
We are convinced that the exploitation of Geometric Numerical Integration in a more applied problem setting will lead to significant advance in the considered applications and give new insight for the design of superior numerical strategies.