The Challenge

Improved computational techniques and advanced computer architectures will help cope with the important industrial, technological, scientific and societal issues that our world is facing, and contribute to innovations and new insights. This depends crucially on the availability of accurate and realistic mathematical models for the multitude of phenomena we are observing, in many different fields. Fortunately, many well-established models exist, and by using combinations of several models the necessary high level of accuracy can be achieved. However, whereas the simulation of a single model within reasonable time has been mastered in most cases (turbulent flow simulation remains rather demanding), the co-simulation of several models to obtain realistic simulations is often still out of reach, even on today’s computers and those of the near future. In many situations, this is due to the fact that the individual models lead to much superfluous detail for the combined simulation. Often, what matters are the dominant features of a model, interacting with the models describing another aspect of the problem at hand. An example is the simulation of interconnect structures in a chip, where the Maxwell equations need to be coupled to the system of electronic circuit equations. If the full system of 3-D Maxwell equations is solved, this introduces millions of extra degrees of freedom on top of the large system of circuit equations. However, it turns out that the electromagnetic problem can be greatly reduced, often to only a few hundreds of degrees of freedom, thereby still describing accurately the performance of the circuit under the influence of the metallic interconnect structure and its parasitic electromagnetic effects.

Model reduction is a field that has been rapidly growing since the beginning of this century, and is able to provide solutions to this important challenge. Using model reduction techniques, dominant features can be captured, thereby reducing computation times enormously and enabling realistic simulations to be carried out in acceptable time. However, classified in a mathematical way, the variety of problems is very large, ranging from simple linear time-varying to nonlinear and autonomous systems. Simulations are combined with experimental data, parameters may be present, problem information (such as coefficients in the models) can be uncertain. Furthermore, often specific model properties need to be preserved by the reduced order models. Last but not least, engineers are interested in optimizing their designs in such a way that prescribed specifications are met, leading to a sequence of many simulations. While for (non- parameterized) linear problems the theories are relatively well-developed, and a wealth of methods is available, this is certainly not the
case for problems that contain one or more of the aforementioned aspects. Many researchers have turned their attention towards the field of model reduction, and are addressing the many aspects as well as working on many different applications. Conferences like the successful SIAM (Society for Industrial and Applied Mathematics) series on Computational Science and Engineering or the series of ICIAM (International Congress for Industrial and Applied Mathematics) conferences nowadays contain many sessions and mini-symposia devoted to model reduction. The field of ‘model reduction’ attracts researchers from tensor analysis, reduced basis methods, and other disciplines.

FORMULATION OF THE CHALLENGE

This COST Action will address the challenge of significantly bringing down computation times for realistic simulations and co-simulations of industrial, scientific, economic and societal models by developing appropriate ‘model reduction’ methods for linear and nonlinear, autonomous and nonautonomous, parameterized and non-parameterized models, with or without variability and/or additional experimental data, and enabling optimization of designs. The Action will also lead to a unified framework of ‘model reduction’ techniques to achieve this challenge.