T9: Multiscale processes and effective behaviour

Many processes in physics, engineering and life sciences involve multiple spatial and temporal scales, where the underlying geometry and dynamics on the smaller scales typically influence the emerging structures on the coarser ones. A unifying theme running through this research topic is to identify the relevant spatial and temporal scales governing the processes under examination. This is achieved, e.g., by establishing sharp scaling laws, by rigorously deriving effective scale-free theories and by developing novel approximation algorithms which balance various parameters arising in multiscale methods.

In optimisation and design problems, multiple scales can either emerge as optimal structures, or they can be a priori present in the system that one wants to control. In the first case, the main challenge is to characterise emergence of multiple scales and to develop numerical methods that can identify the highly complex optimisers. In the second case, the challenge is to analytically or numerically approximate the macroscopic system behaviour with sufficiently high efficiency such that it can form part of each step in an iterative optimisation.

Determination and optimisation of structures

Investigators: Jentzen, Simon, Wirth

In this unit, we illustrate how our strong track record in the area of optimisation, numerical complexity reduction and multiscale modelling are used to tackle these problems. We also propose approximation algorithms that solve optimal control problems for Hamilton–Jacobi–Bellman equations without the so-called curse of dimensionality, thus allowing a very large number of control variables.

Homogenisation and multiscale methods

Investigators: Mukherjee, Ohlberger, Rave, Wirth, Zeppieri

In this unit, we consider multiscale problems which can be recast into the theory of non-linear homogenisation. Concretely, we study the convergence rate to the homogenised limit of systems of non-linear elliptic PDEs defined in randomly perforated media where perforations overlap with high probability. Moreover, motivated by applications to combustion and propagation of fronts in random environments, we study homogenisation of viscous Hamilton-Jacobi-Bellman (HJB) equations on continuum percolation clusters, which bring fundamental challenges due to the inherent non-stationarity, non-ellipticity and lack of global Lipschitz properties. Model order reduction (MOR) methods will also be developed and analysed for efficient approximation of dynamic multiscale problems and related PDE constrained optimisation and inverse problems.

Mixing and equilibration

Investigators: Engwer, Ohlberger, Pirner, Seis, Stevens

Here we study multiscale problems through the lens of mixing and equilibration. Our incentive is guided by pivotal questions in the realm of fluid dynamics such as identifying the maximal rates of mixing and studying enhanced dissipation within certain fluid flows. Our investigation also extends to treating kinetic equations and developing numerical methods to minimise numerical dissipation and address transport and conservation laws.