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Chair of Applied Mathematics / Numerical Analysis
Bergische Universität Wuppertal
Faculty of Mathematics and Natural Sciences
Gaußstraße 20
D42119 Wuppertal
Germany
Phone: +49 202 439 5296
Fax: (Fax currently unavailable)
EMail: sekamna{at}math.uniwuppertal.de
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Model Order Reduction
Model Order Reduction (MOR) is the art of reducing a system's complexity while preserving its inputoutput 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 inputoutput 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 inputterminals translated to outputpins?
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 EUMarie Curie Initial Training Network COMSON, attention was concentrated on MOR for Differential Algebraic Equations. Members that have been working on MOR in the EUMarie Curie Transfer of Knowledge project OMOORENICE! 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
3. 
Multiscale Modeling and Multirate TimeIntegration of Field/Circuit Coupled Problems
of Elektrotechnik
Publisher: VDI Verlag, Düsseldorf
2011
ISBN: 9783183398218
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. 
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. 
Model order reduction for semiexplicit systems of differential algebraic equations
In Troch, I. and Breitenecker, F., editor,
Proceedings MATHMOD 09 Vienna
2009
Note: ARGESIM Report 