Department of Mathematics and Computer Science

Complex processes involving cascades of scales are ubiquitous in nature. Such processes have more than two characteristic scales, their smallest and largest scales are widely separated, and many of their scales are important in determining the characteristics of the process. Experiments and observations often provide only limited insights into such processes, but with increasing computing power there is hope for progress via simulations. Such simulations remain very challenging, however, as the wide range of scales is associated with many degrees of freedom that usually make brute-force full-detail models unfeasible. Moreover, the coupling between the smallest, largest, and intermediate scales often renders established coarse-graining or model reduction theories and methods ineffective as most of them are well founded only for problems with two well-separated scales.

To exemplify the main challenges for this Collaborative Research Center (CRC), let us consider a prototypical complex process. An efficient simulation would require a controlled distribution of the computational resources over its cascade of scales such that each scale and subprocess is represented just adequately with respect to its impact on the quantities of interest provided this impact cannot be represented explicitly by mathematical means. Yet, there are no systematic means of meeting this requirement today, even if a complete mathematical “root model” is available that describes the process in all detail. Moreover, in many practical situations the best available model for such a process is given only in the form of a complex computer code that is not accessible to mathematical analysis. In the worst case scenario, scientists are still searching for a satisfactory root model, and its development is to be pursued together with that of methods for bridging the scales involved in the quantities of interest.

CRC 1114 has focused on this problem field since 2014 with a strongly interdisciplinary approach in which Mathematicians join forces with scientists from Biochemistry, Physics, and the Geosciences who contribute hard application problems involving scaling cascades. Our primary aim is to achieve:

Methodological developments for the modelling and simulation of complex processes involving cascades of scales derived from prototypical challenges in the natural sciences.

We believe that we have made considerable progress on several prototypical challenges already. For example, new methods developed in the CRC allowed us, for the first time, to simulate the dynamical mechanism of a molecular process at the timescale of seconds with all-atom resolution.

Our secondary aim is to indeed tackle these prototypical application challenges and demonstrate that theory and methods developed in the CRC 1114 can go all the way to making relevant impact in application fields such as molecular and cellular processes, moist atmospheric dynamics, and geophysics.

The added value of CRC 1114 arises from communication and cooperation across the boundaries of projects and disciplines, aiming at a common understanding of cascade-of-scales-problems and at generalisations of the developed methods to broader problem classes. Mathematical abstraction takes a leading role in the identification of overarching structures and the formulation of general concepts in a common language. Over the 12-year funding perspective, the CRC will place the methods development for cascade-of-scales-processes on the international research agenda and advance the field of multiscale modelling and simulation by mastering a number of open challenges in the natural sciences.