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Chair of Applied Mathematics / Numerical Analysis
Bergische Universität Wuppertal
Faculty of Mathematics and Natural Sciences
Gaußstraße 20
D-42119 Wuppertal
Germany

Phone: +49 202 439 5296
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E-Mail: sek-amna{at}math.uni-wuppertal.de

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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

References
 3 Sebastian Schöps Multiscale Modeling and Multirate Time-Integration of Field/Circuit Coupled Problems of Elektrotechnik Publisher: VDI Verlag, Düsseldorf 2011 ISBN: 978-3-18-339821-8 http://elpub.bib.uni-wupp...thematik/diss2011/schoeps Abstract: This treatise is intended for mathematicians and computational engineers that work on modeling, coupling and simulation of electromagnetic problems. This includes lumped electric networks, magnetoquasistatic field and semiconductor devices. Their coupling yields a multiscale system of partial differential algebraic equations containing device models of any dimension interconnected by the electric network. It is solved in time domain by multirate techniques that efficiently exploit the structure. The central idea is the usage of lumped surrogate models that describe latent model parts sufficiently accurate (e.g. the field model by an inductance) even if other model parts (e.g. the circuit) exhibit highly dynamic behavior. We propose dynamic iteration and a bypassing technique using surrogate Schur complements. A mathematical convergence analysis is given and numerical examples are discussed. They show a clear reduction in the computational costs compared to single rate approaches. 2 Kasra Mohaghegh; Michael Striebel; E. Jan W. ter Maten; Roland Pulch Nonlinear model order reduction based on trajectory piecewise linear approach: comparing different linear cores In Roos, Janne and Costa, L., editor, Scientific Computing in Electrical Engineering. Mathematics in Industry of 14 , page 563 -- 570. Publisher: Springer, Berlin 2010 1 Kasra Mohaghegh; Roland Pulch; Michael Striebel; E. Jan W. ter Maten Model order reduction for semi-explicit systems of differential algebraic equations In Troch, I. and Breitenecker, F., editor, Proceedings MATHMOD 09 Vienna 2009 Note: ARGESIM Report
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