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Arbeitsgruppe Angewandte Mathematik / Numerische Analysis
Bergische Universität Wuppertal
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Model Order Reduction

Model Order Reduction (MOR) is the art of reducing a system's complexity while preserving its input-output behavior as much as possible.

Processes in all fields of todays technological world, like physics, chemistry and electronics, but also in finance, are very often described by dynamical systems. With the help of these dynamical systems, computer simulations, i.e. virtual experiments, are carried out. In this way, new products can be designed without having to build costly prototyps.

Due to the demand of more and more realistic simulations, the dynamical systems, i.e., the mathematical models, have to reflect more and more details of the real world problem. By this, the models' dimensions are increasing and simulations can often be carried out at high computational cost only.

In the design process, however, results are needed quickly. In circuit design, e.g., structures may need to be changed or parameters may need to be altered, in order to satisfy design rules or meet the prescribed performance. One cannot afford idle time, waiting for long simulation runs to be ready.

Model Order Reduction allows to speed up simulations in cases where one is not interested in all details of a system but merely in its input-output behavior. That means, considering a system, one may ask:

• How do varying parameters influence certain performances ?
Using the example of circuit design: How do widths and lengths of transistor channels, e.g., influence the voltage gain of a circuit.
• Is a system stable?
Using the example of circuit design: In which frequency range, e.g., of voltage sources, does the circuit perform as expected
• How do coupled subproblems interact?
Using the example of circuit design: How are signals applied at input-terminals translated to output-pins?

Classical situations in circuit design, where one does not need to know internals of blocks are optimization of design parameters (widths, lengths, ...) and post layout simulations and full system verifications. In the latter two cases, systems of coupled models are considered. In post layout simulations one has to deal with artificial, parasitic circuits, describing wiring effects.

Model Order Reduction automatically captures the essential features of a structure, omitting information which are not decisive for the answer to the above questions. Model Order reduction replaces in this way a dynamical system with another dynamical system producing (almost) the same output, given the same input with less internal states.

MOR replaces high dimensional (e.g. millions of degrees of freedom) with low dimensional (e.g. a hundred of degrees of freedom ) problems, that are then used instead in the numerical simulation.

The working group "Applied Mathematics/Numerical Analysis" has gathered expertise in MOR, especially in circuit design. Within the EU-Marie Curie Initial Training Network COMSON, attention was concentrated on MOR for Differential Algebraic Equations. Members that have been working on MOR in the EU-Marie Curie Transfer of Knowledge project O-MOORE-NICE! gathered knowledge especially in the still immature field of MOR for nonlinear problems.

Current research topics include:

• MOR for nonlinear, parameterized problems
• structure preserving MOR
• MOR for Differential Algebraic Equations
• MOR in financial applications, i.e., option prizing

Group members working on that field

Publications

Referenzen
 19 M. Striebel; R. Pulch; E. J. W. ter Maten; Z. Ilievski; W. H. A. Schilders Model order reduction and sensitivity analysis Günther, M., Autoren, Coupled Multiscale Simulation and Optimization in Nanoelectronics Band 21 aus Mathematics in Industry Kapitel 5.3, Seite 319--341 and 355--358. Herausgeber: Springer, 2015 18 D. Harutyunyan; R. Ionutiu; E. J. W. ter Maten; J. Rommes; W. H. A. Schilders; M. Striebel Advanced Topics in Model Order Reduction Günther, M., Autoren, Coupled Multiscale Simulation and Optimization in Nanoelectronics Band 21 aus Mathematics in Industry Kapitel 6, Seite 361--432. Herausgeber: Springer Berlin Heidelberg, 2015 17 M. Striebel; E. J. W. ter Maten Model order reduction for nonlinear network problems Günther, M., Autoren, Coupled Multiscale Simulation and Optimization in Nanoelectronics Band 21 aus Mathematics in Industry Kapitel 6.1, Seite 362-380 and 426-428. Herausgeber: Springer, 2015 16 D. Harutyunyan; J. Rommes; E. J. W. ter Maten; W. H. A. Schilders Simulation of mutually coupled oscillators using nonlinear phase macromodels and model order reduction techniques Günther, M., Autoren, Coupled Multiscale Simulation and Optimization in Nanoelectronics Band 21 aus Mathematics in Industry Kapitel 6.3, Seite 398--425 and 430--432. Herausgeber: Springer, 2015 15 E. J. W. ter Maten; M. Striebel Model order reduction for chip design Günther, M. and Bartel, A. and Brunk, M. and Schöps, S. and Striebel, M., Autoren, Progress in Industrial Mathematics at ECMI 2010 Band 17 aus Mathematics in Industry , Seite 125-129. Herausgeber: Springer, 2012 14 Model Reduction for Circuit Simulation Benner, P. and Hinze, M. and ter Maten, E. J. W., Autoren, Band 74 aus Lecture Notes in Electrical Engineering Herausgeber: Springer Verlag, 2011 ISBN: 9400700881 https://www.ebook.de/de/p...r_circuit_simulation.html 13 T. Bechtold; D. Hohlfeld; E. B. Rudnyi; M. Günther Efficient extraction of thin film thermal parameters from numerical models via parametric model order reduction J. Micromech. Microeng., 20 2010 ISSN: 045030 12 A. Verhoeven; M. Striebel; J. Rommes; E. J. W. ter Maten; T. Bechtold Proper Orthogonal Decomposition Model Order Reduction of Nonlinear IC Models Fitt, A. D. and Norbury, J. and Ockendon, H. and Wilson, E., Autoren, Progress in Industrial Mathematics at ECMI 2008 Band 15 aus Mathematics in Industry Seite 441--446. Herausgeber: Springer Berlin Heidelberg, 2010 11 A. Verhoeven; M. Striebel; E. J. W. ter Maten Model Order Reduction for Nonlinear IC Models with POD Roos, J. and Costa, L. R. J., Autoren, Scientific Computing in Electrical Engineering at SCEE 2008 Band 14 aus Mathematics in Industry Seite 571--578. Herausgeber: Springer Berlin Heidelberg, 2010 10 K. Mohaghegh; M. Striebel; E. J. W. ter Maten; R. Pulch Nonlinear Model Order Reduction Based on Trajectory Piecewise Linear Approach: Comparing Different Linear Cores Roos, J. and Costa, L. R. J., Autoren, Scientific Computing in Electrical Engineering at SCEE 2008 Band 14 aus Mathematics in Industry Seite 563--570. Herausgeber: Springer Berlin Heidelberg, 2010 9 E. J. W. ter Maten Introduction to Part V (Model Order Reduction) Roos, J. and Costa, L. R. J., Autoren, Scientific Computing in Electrical Engineering at SCEE 2008 Band 14 aus Mathematics in Industry Seite 463--467. Herausgeber: Springer Berlin Heidelberg, 2010 8 A. Verhoeven; J. ter Maten; M. Striebel; R. Mattheij Model Order Reduction for Nonlinear IC Models Korytowski, A. and Malanowski, K. and Mitkowski, W. and Szymkat, M., Autoren, System Modeling and Optimization, IFIP AICT 312 Band 312 aus IFIP Advances in Information and Communication Technology Seite 476--491. Herausgeber: Springer Berlin Heidelberg, 2009
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