Multilevel methods for heterogeneous spatial networks

• Applicant: Dr. Moritz Hauck
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 549356871

In this project we develop multilevel methods for the efficient simulation of heterogeneous spatial network models. Spatial networks are a popular tool for the simplified discrete representation of complex geometric structures and applications include the modelling of porous media or fibre-based materials. They typically exhibit strong heterogeneities due to their complex geometry and possibly highly varying fibre lengths and thicknesses. This makes them difficult to handle by standard discretisation methods that resolve all scales globally. A remedy is provided by methods such as domain decomposition or multiscale methods, which localise the global computations and incorporate the resulting local information into a two- or multilevel method. In the context of today's massively parallel computational resources, the limiting factor of two-level methods typically is the solution of the resulting coarse system of equations. Therefore, this project aims at the construction and analysis of multilevel methods for spatial network models that overcome this limitation. A cornerstone for this will be a new theoretical framework which, at sufficiently coarse scales, allows to transfer important results known from the continuous setting, such as Poincaré's inequality, also to spatial networks. Within this project, we pursue two different multilevel approaches for heterogeneous spatial network models. The first approach is a multigrid-type solver which is specifically tailored to the spatial network setting. It carefully blends between strategies from geometric and algebraic multigrid methods and uses a smoother based on domain decomposition techniques to bridge scale differences. In this way, we aim to construct an efficient method with a low setup cost. The second multilevel approach addresses particularly strong heterogeneities and employs techniques known from multiscale methods. It is based on problem-adapted coarse spaces constructed by solving local spectral problems in the space of locally operator-harmonic functions. By using such problem-adapted coarse spaces, we aim to construct a multilevel solver with strong robustness properties, but at a higher setup cost. Employing the aforementioned theoretical framework for spatial networks, we aim to provide a rigorous convergence analysis of both multilevel methods. Particular emphasis will be placed on the dependence of the constants on the number of levels and the choice of coarse spaces. The target application for evaluating the practical performance of the two multilevel methods is a biological application concerned with the simulation of stress fibres in cells, which are crucial for cell mechanics. This application is motivated by an ongoing interdisciplinary research collaboration at Heidelberg University, the host institution for this project.

• DFG Programme: WBP Position