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Quite often it is necessary to preserve some parameters within system matrices. Provided that parameters enter system matrices linearly, one can generalize moment matching to multivariate moment matching in order to generate the projection subspace that does not depend on parameter values.

There are results from several groups showing that the approach is working. Yet it is still unclear how to choose an optimal reduced model because there is no theory for error estimates. However, in some applications it can be effectively done with some heuristic approach.

Tutorial in Mathematica on how to use parametric model reduction to preserve several film coefficients in the symbolic form during model reduction is available: Download, Browse, Manual.

*L. H. Feng, E. B. Rudnyi, J. G. Korvink.*

**Preserving the film coefficient as a parameter in the compact thermal
model for fast electro-thermal simulation.**

IEEE Transactions on Computer-Aided Design of Integrated Circuits and
Systems, December, 2005, v. 24, N 12, p. 1838-1847.

Final paper at IEEE.

Compact thermal models are often used during joint electro-thermal simulation of MEMS and circuits. Formal model reduction allows us to generate compact thermal models automatically from high-dimensional finite element models. Unfortunately, it requires us to fix a film coefficient employed to describe the convection boundary conditions. As a result, compact models produced by model reduction do not comply with the requirements of being boundary condition independent. In the present paper, we suggest an approach of successive series expansion with respect to the film coefficient as well as to the frequency during model reduction, which allows us to overcome the problem and keep the film coefficient as a symbolic parameter in the reduced model. The approach is justified with a numerical example of electro-thermal simulation of a microthruster unit.

*L. Feng, D. Koziol, E. B. Rudnyi, and J. G. Korvink.*

**Parametric Model Reduction for Fast Simulation of Cyclic Voltammograms
**.

Sensors Letters, v. 4, N 2, p. 165-173, 2006.

Final Paper at IngentaConnect.

Model order reduction is a well-established technique for fast simulation of large-scale models based on ordinary differential equations, especially those in the field of integrated circuits and micro-electro-mechanical systems. In this paper, we propose the use of parametric model reduction for fast simulation of a cyclic voltammogram. Instead of being considered as a time varying system, the model for a cyclic voltammogram is treated as a system with a parameter (applied voltage) which is to be preserved during model reduction. Because voltage is preserved in the symbolic form during model reduction, we can simulate the cyclic voltammogram with a reduced system and therefore invest much less time and memory as compared with direct simulation based on the original large-scale model. We present our approach for a case study based on scanning electrochemical microscopy.

*E. B. Rudnyi, C. Moosmann, A. Greiner, T. Bechtold, J. G. Korvink.*

Parameter Preserving Model Reduction for MEMS System-level Simulation and Design.

5th MATHMOD, Proceedings. Volume 1: Abstract Volume, p. 147, Volume 2: Full Papers CD, 8 pp, February 8 - 10, 2006, Vienna University of Technology, Vienna, Austria. ISBN 3-901608-30-3.

Model reduction is a very helpful tool to generate compact models for system-level simulation. Quite often however, system matrices depend on design parameters and the new goal is not only to reduce the original system but also to preserve system parameters in the symbolic form during model reduction. We introduce multivariate moment matching as a possible solution to this problem. We consider several examples from MEMS to demonstrate the feasibility of the approach: a device cooled by airflow, a microhotplate, a flow meter (anemometer) and a microelectrode. Finally, we discuss problems that should be overcome in order to use this technique in software for engineering design applications.

*D. S. Weile, E. Michielssen, E. Grimme, and K. Gallivan.*

**A method for generating rational interpolant reduced order models of two-parameter linear systems.**

Applied Mathematics Letters, vol. 12, pp. 93-102, 1999.

Paper at ScienceDirect.

*P. Gunupudi, R. Khazaka, and M. Nakhla.*

**Analysis of transmission line circuits using multidimensional model reduction techniques.**

IEEE Transactions on Advanced Packaging, vol. 25, pp. 174-180, 2002.

Paper at IEEEXplore.

*L. Daniel, O. C. Siong, L. S. Chay, K. H. Lee, and J. White.*

**A Multiparameter Moment-Matching Model-Reduction Approach for Generating Geometrically Parameterized Interconnect Performance Models.**

IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, vol. 23, pp. 678-693, 2004.

Paper at IEEEXplore.

*Lihong Feng.*

**Parameter independent model order reduction.**

Mathematics and Computers in Simulation 2005, v. 68, N 3, p. 221-234.

Paper at ScienceDirect.

Gianluigi Rozza,

Shape design by optimal flow control and reduced basis techniques: applications to bypass configurations in haemodynamics

Evgenii B. Rudnyi

Designed by

Masha Rudnaya