C4. Geometry-based modelling, approximation, and reduction.
In mathematical modelling and its application to the sciences, the notion of geometry enters in multiple related but different flavours: the geometry of the underlying space (in which e.g. data may be given), the geometry of patterns (as observed in experiments or solutions of corresponding mathematical models), or the geometry of domains (on which PDEs and their approximations act). We will develop analytical and numerical tools to understand, utilise and control geometry, also touching upon dynamically changing geometries and structural connections between different mathematical concepts, such as PDE solution manifolds, analysis of pattern formation, and geometry.
We will interpret data from different contexts (in particular measurements from the life sciences and physics, shapes from computer graphics applications, and solutions to parameterised PDEs) as elements of an underlying non-linear (infinite-dimensional) geometric space, e.g. a Riemannian manifold. This geometric structure will be exploited for the development of data processing tools. A focus will lie on variational and numerical methods for data fitting and regression via submanifolds, on singular perturbation methods, and on asymptotics and model reduction for parameterised PDEs by decomposing each solution into an element of a linear space and a Lie group element acting on it.
The geometry of spatial patterns often determines the average, effective properties of such structures, e.g. in a material, its effective material properties. Motivated by particular patterns and their defects, as observed in biological organisms or materials, we will examine their macroscopic, homogenised properties and their stability with respect to pattern perturbations. The effective, homogenised structure will again be described in geometric terms: For instance, the homogenised free energy of carbon nanotubes may depend on their bending curvature, the transport efficiency of molecules in strongly layered biological membranes of cell organelles may depend on an effective distance metric, and defects in atomic crystals can be related to specific singularities of two-dimensional surfaces.
Applications like shape optimisation or shape reconstruction problems are concerned with the identification of a geometry. Typically, there is an additional PDE constraint for which the sought geometry serves as the PDE domain. It is a challenge to efficiently approximate this geometry. We will develop concepts that quantify the efficiency of the geometry approximation in terms of the involved computational effort per desired accuracy, and we will investigate numerical schemes that can efficiently deal with complex (time-)varying PDE domains without the need for remeshing.
This unit consists of the following projects:
- Modelling of submanifolds in nonlinear spaces. (Böhm, Ohlberger, Rave, Wilking, Wirth)
- Patterns and effective geometries. (Böhm, Friedrich, Lohkamp, Stevens, Wilking, Wirth)
- Numerical approximation concepts for geometries. (Engwer, Ohlberger, F. Schindler, Schürmann, Wirth)