In order to get a deep insight into the underlying process, dynamics, structure or devices, modeling and simulation are unavoidable in many research and application fields. The resulting mathematical models are usually in the form of partial differential equations. To simulate such models, spatial (-time) discretization is necessary, which results in large-scale, complex systems with enormous number of equations. The simulation becomes time-consuming because of the large scale and complexity of the systems.
Developed from well-established mathematical theories and robust numerical algorithms, model order reduction (MOR) has been recognized as an efficient tool for fast simulation. Using model order reduction, small systems of far less number of equations (reduced models) are derived, and can substitute the original large-scale systems for simulation. As a result, the simulation can be sped up by several orders of magnitude.
The technique of model order reduction (MOR) has been successfully applied in the communities of mechanical engineering, fluid dynamics, control, circuit simulation, microelectromechanical systems (MEMS) simulation, uncertainty quantification, etc. for decades. The robustness of MOR has been demonstrated in all application areas above. In the CSC group, with wide and successful cooperation with international scientist, novel MOR methods, algorithms and software have been developed and are still being investigated for the above applications. More results for MOR especially focusing on nonlinear and parametrized systems are being expected in the near future.
MOR in Process Engineering, Molecular Simulations
(Parametric) model order reduction ((P)MOR) techniques are widely used in many areas such as electronics, micro-electro-mechanical systems, acoustics, structures and vibrations. However, their applications in process engineering and molecular simulations are still rare, partly due to the complexity of the systems and the lack of cooperative research. MPI-Magdeburg, a research institute emphasizing the cooperation between engineering and mathematics, serves as an ideal place to conduct these researches. Up to now, we have four joint projects with the groups PCF, PSE, PSD and MSD.
Model Order Reduction Algorithms and Techniques
Model order reduction is required in many fields of application to replace large complex systems by significantly smaller ones. These make simulations more efficiently. It is important to consider how large this efficiency gain is, and how (hopefully) small the error is.
Within the CSC Group new methods for model order reduction are developed, their properties and applicability to systems from different fields of application are investigated. A major effort is to handle the many parameters (geometric, material,...) coupled in the linear or nonlinear systems, which remains an active research topic in the MOR community. Moreover, the systems may be nonlinear or stochastic. Model order reduction combined with stochastic methods dealing with uncertainty quantifications (UQ) is in the early stage of research internationally. Model reduction methods applicable to stochastic systems are of high interest and of major significance for solving complex UQ problems in applications. A good selection of complex systems for testing newly developed methods is available in a benchmark collection that has been initiated by the CSC group and has been made available to the public in the so called MOR-Wiki.