Data Compression, Process Optimization, Aerodynamics: A Tour Through the Scales

Increasing computational power facilitates a more and more precise analysis of complex systems in science and engineering through computer simulation. Mathematics as the interface between the real and the digital world provides on the one hand the foundations for the formulation of necessarily simplified models of reality. On the other hand it also presents the basic methodological principles for designing efficient algorithms, which can be used to gain quantitative information from such models. The fact, that real systems are mostly driven by phenomena covering a large range of time and length scales, constitutes the major challenge. Therefore, the development of mathematical methods explicitly dealing with the multiscale nature is of great importance. In this article, this issue will be explained and illustrated on the basis of most recent developments in different fields of application to demonstrate how many “birds” can be caught with a single “mathematical stone”. In particular, fundamental algorithmic concepts relying on wavelet decomposition will at first be explained in a tutorial manner in the context of image compression and coding. Then it will be demonstrated that these concepts carry over to different applications such as analysis of measurement data in process plants, large-scale optimization in the process industries as well as complex fluid dynamics problems in aerodynamics. Stable decomposition in different time and length scales facilitates the implementation of adaptive concepts, which are capable of placing computational resources automatically where needed to realize the desired quality of the solution, e.g. in terms of error tolerances, at the lowest computational effort.