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.