Generalised Euler equations: theory, numerics, and computational anatomy

Research Project, 2012 – 2015

The mathematical field of generalised Euler equations concerns a class of differential equations with a rich geometric structure. There is a vast range of applications in which such equations are central. A new and upcoming application is computational anatomy, which provides a framework for comparison of medical images by computing geodesic curves on some infinite dimensional manifold of mappings. The field of Euler equations also connects to the field of optimal transport, which is of importance in probability theory, information theory, statistical mechanics, and quantum mechanics. When solving generalised Euler equations numerically, it is essential, in order to capture the correct dynamical behaviour, to preserve various geometric structures, such as the associated Lie?Poisson structure. The field of geometric numerical integration is concerned with development and analysis of numerical integration algorithms that preserve geometric features in phase space. The purpose of the proposed project is to contribute, both in theory and in application, to the field of generalised Euler equations and computational anatomy. The roles of the participants in the project are as follows. The international host contributes with expertise on generalised Euler equations, optimal transport, and medical imaging. The host in Sweden contributes with expertise on numerical analysis of PDEs. The main applicant contributes with expertise in geometric numerical integration.

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