Lorentz Center - Model order reduction, coupled problems and optimization from 19 Sep 2005 through 23 Sep 2005
Current Workshop  |   Overview   Back  |   Home   |   Search   |

Abstracts

# Passivity-Preserving Model Reduction for Large-Scale Systems

We will discuss model reduction methods for passive linear systems.   Preserving passivity in reduced-order models is an important task in circuit simulation and microsystems technology.  We introduce several algorithms  related to  positive-real balancing that can be proven to preserve passivity. The applicability and computational properties of this methods will be compared using several examples from various application areas.

## C. Tebaldi

POD Analysis of Periodic and Quasi-periodic Behaviour in Two-dimensional Navier-Stokes Equations

For systems who show turbulent behaviour like fluids, the concept of coherent structures (strongly persistent spatio-temporal structures) has provided an efficient descriptive tool as well as the possibility of low-dimensional reductions. The introduction of Proper Othogonal Decomposition (POD) has been found  a successful technique to find coherent structures. The methodology has been used to characterize the transitions to periodic and quasi-periodic behaviour in two-dimensional Navier-Stokes equations, obtaining good reduction results.

# Structure preserving methods for pole-zero and stability analysis

We discuss existing and novel eigenvalue techniques that are useful for pole-zero and stability analysis. We present structure preserving pole and zero finders for both first-order and second-order systems, and will pay special attention to the stability analysis: the efficient  computation of rightmost eigenvalues.

# Rational Krylov for Eigenvalue Computation and Model Reduction

The Rational Krylov algorithm is a generalization of the shifted and inverted Arnoldi eigenvalue algorithm, where several shifts are used in one run. In a linear model reduction context, it

corresponds to moment matching. The reduced model is given by the computed Hessenberg matrix, its eigenvalues are approximations to the poles of the system. Moment matching corresponds to interpolation of the response function at the shifts.

Moment matching (MM) is one of the major approaches considered for model reduction.  A judicious choice of interpolation points, can give us a good matching over a wide frequency range. We are currently developing an adaptive strategy where, in each step, we choose between increasing the order of matching in the current point, and finding a new shift which demands factorization of a new shifted matrix.

We have previously reported some experiences in linear model reduction on systems coming from VLSI design and CFD.

Domenico Lahaye
Space-Mapping Applied to Linear Actuator Design

Many electromechanical devices are nowadays designed using computationally expensive finite element models. The space-mapping technique aims at speeding up the design process using auxiliary models that are less accurate but cheaper to compute. In this talk we will illustrate the application of the space-mapping technique in the design of linear actuators. Coarse models employed are simplified finite element discretizations and magnetic equivalent circuits. Numerical results show that carefully chosen coarse models lead to very efficient design schemes.

# Groundwater models coupling flow, transport, heat transfer and geochemistry

Author : J. Erhel, INRIA, Rennes, France

Many environmental studies rely on modelling geochemical reactions combined with hydrodynamic processes such as groundwater flow, transport of solutes by advection and diffusion, heat transfer in porous media. Some issues concern aquifer contamination, underground waste disposal, etc. Density-driven flows and reactive transport aim at coupling these processes in a unique model, in order to understand the system as a whole. Reactive transport models are complex non-linear PDEs, coupling the transport engine with the reaction operator. Density-driven flow and transport models are also complex non-linear PDEs, coupling the flow operator with the transport engine. We discuss efficient and robust numerical methods, based on DAE solvers, combined with either a Gauss-Seidel or on a modified Newton method with a powerful linear solver.

# POD as a Tool for Frequency-Averaging in Radiative Heat Transfer

The simulation of radiative heat transfer (RHT) poses several challenging problems due to its high numerical complexity. During the last decade many people worked on the derivation of a whole hierarchy of  macroscopic models ranging from moment approximations to diffusive approximations. The main goal was to have no dependence on the angular variable of the intensity anymore. Further, also frequency averaging techniques were developed to reduce the number of frequency bands.

These give either crude approoxations, like grey models, or they rely on the knowledge of the engineer, who chooses a good averaging.  We use instead POD to compute automatically a  well suited averaging. The efficiency of our approach will be underlined by numercial examples and by comparisons with other averaging techniques.

# Mike Botchev

Krylov-subspace matrix functions evaluations for time integration of large scale Maxwell's equations models

Time-stepping schemes involving matrix functions (so-called "exponential" or "cosine" schemes) are rapidly becoming a popular tool for time integration of large scale problems. The key issue here is evaluation of the matrix-function-vector products which is normally done with a Krylov subspace technique (Arnoldi or Lanczos processes).  In this talk we show how this can be done efficiently within the framework of finite element discretizations of 3D Maxwell's equations. This talk is based on joint work with Davit Harutyunyan and Jaap van der Vegt.

# Kees Vuik

Deflation acceleration for Computational Fluid Dynamics problems

Accurate solution of the discretized incompressible Navier-Stokes equations is important for many applications. The resulting (non)-linear system cannot be solved by direct solution methods. For this reason iterative methods are used to solve the large systems. In many packages this is combined with domain decomposition or another method to parallelize the methods. Due to the zero block in the continuity equation, the system is alway indefinite, which leads to slow convergence. Several methods are known to enhance the convergence properties of the iterative methods. In our applications we use the GCR-SIMPLE method or some operator splitting method (pressure correction). This implies that smaller linear systems have to be solved in the preconditioner. This is also done by an iterative method. It appears that the pressure equation can cost up to 80 percent of the used CPU time. Therefor a fast parallel solver for this system is very important. Using a domain decomposition algorithm combined with approximate solution of the subdomain problems, we observe that the convergence of the iterative method deteriorates if the number of subdomains increases. Various methods are known to solve these problems as there are: (additive) Coarse Grid Correction (CGC) and the Balancing Neumann-Neumann (BNN) method. Recently, it appears that Deflation can also be used to accelerate the convergence. For all these methods a good choice of the projection vectors is important. In this paper we present the following choices for the projection vectors: eigenvectors, physical vectors and coarse grid vectors. Furthermore, a comparison of Deflation with CGC and BNN is given. For the theoretical comparison we restrict ourselves to eigenvectors as projection vectors. Some numerical experiments illustrate the theoretical results.

# Kenui Fujimoto

Singular value analysis of nonlinear operators and its application to model reduction of nonlinear control systems

This research is devoted to characterize singular values of nonlinear operators. Although eigenvalue and spectrum analysis for nonlinear operators has been studied by many researchers in mathematics literature, singular value analysis has not been investigated so much. In this paper, a novel framework of singular value analysis is proposed which is closely related to the operator gain. The proposed singular value analysis is based on the eigenvalue analysis of a special class of nonlinear operators called differentially self-adjoint. Some properties of those operators are clarified which are natural generalization of the linear case results. Furthermore, thus proposed singular value analysis framework is applied to model reduction problem for nonlinear control systems. Also some numerical examples demonstrate its effectiveness.

# Olivier Bruls

Reduced-Order Modeling of Flexible Mechanisms

In flexible multibody dynamics, the nonlinear Finite Element method leads to high-order equations of motion with kinematic constraints. A reduced-order model is thus desirable for control, simulation or optimization purpose. For instance, the component-mode synthesis is a standard reduction technique, which can be applied to any flexible body of the mechanism. Instead of working at the body level, we propose to define a reduced parameterization at the mechanism level. Hence, the overall motion is described in terms of rigid and flexible modes, which have a global physical interpretation in the configuration space. The reduction operators are fundamentally nonlinear and the reduction procedure combines the component-mode technique with an approximation strategy in the configuration space. Several examples illustrate the efficiency of the approach, such as a four-bar mechanism, a parallel kinematic machine-tool, and a long-reach manipulator.

Daniel Ioan, Gabriela Ciuprina

Politehnica University Bucharest, Electrical Engineering Department, Numerical Methods Laboratory

313 Spl. Independentei, Bucharest, Romania, lmn@lmn.pub.ro, www.lmn.pub.ro.

The RomWorkbench – a Matlab Based Tool for the Testing of a-posteriori Model Order Reduction Methods

The need for evaluation of simulation tools and reduced order-models (ROM) for passive on-chip components has led us to the formulation and execution of the European project: CODESTAR (IST-2001-34058) [1] that was dedicated to study the high-frequency effects in a design environment of on-chip integrated passives and interconnects.

At the start of the project, a lot of reducing order methods (ROM) were already available in the literature. Since the final goal was to obtain a reduced order model of CODESTAR type problems, one important issue was to decide which ROM technique is the most appropriate for the proposed benchmarks. For this reason, a new tool called CODESTAR ROM Workbench (fig. 1) was conceived. Its aim is to allow the user to reduce models by means of as many ROM techniques as possible, and to compare the results. In this way the behaviour of every reduction method applied to a CODESTAR model (output of the field solvers described above) is investigated and a specific reduction strategy to be applied for every type of CODESTAR benchmark and solver could be recommended. Basically, the ROM workbench consists of:

·         A series of benchmark problems;

·         A set of model order reduction methods;

·         Criteria for results evaluation and comparison.

Figure 1 shows the main blocks of the ROM workbench. Thick lines illustrate its main goal.

The benchmark problems are either linear time invariant systems described by means of state space matrices, frequency characteristics described by the variation of impedance, admittance or S-parameter matrices with respect to the frequency, or net lists described in the SPICE language.

The reduction can be carried out by means of various methods. These methods include: explicit moment matching, Krylov subspace techniques [2], Laguerre techniques [3], a two step Lanczos strategy, also a new two step reduction strategy, based on a PRIMA technique followed by a truncated balanced reduction, and truncated balanced realization procedures [4]. A very robust technique included in the ROM Workbench is the vector fitting method proposed in [5].

The workbench is able to compare responses obtained for different systems. The comparison can be carried out either on the time responses (step, impulse, etc.) or on the frequency responses (Bode, Nyquist, Smith, etc). Lumped parameters, quality factors or line parameters can also be compared. Since the available measurements are for S parameters, the main criteria used for comparison is the computation of the an error estimator based on the Frobenius norm ||.||F:

rms ||Sref-San||F / maxf ||Sref||F,,

where Sref are S parameters for the reference system (output of field solver or measurements), and San are S parameters for the analysed system (reduced one).

The vector fitting procedure proved to be the best one for all the CODESTAR benchmarks, allowing the reduction to very low orders (less than 10) with an extremely low computational effort (less than 1 sec), the relative error between a simulation result and its reduced order model being less than 1 %. . Therefore, the challenging task of the project was not the reduction, but the modelling and simulation of passive on-chip components and interconnects.

Considering the final aim of the Codestar project, i.e. the generation of reduced model synthesized by a SPICE circuit, the ROM workbench includes techniques for circuit synthesis as well. There are two methods implemented. One of them is the Direct Stamping method, more appropriate for reduced models that generated reduced state space matrices (such as Krylov type methods). The other method is the Differential Equation Macromodel, more suitable for reduced order methods that generated transfer functions (such as vector fitting) [6]. In this way, the ROM workbench allowed not only the testing of ROM techniques but also of SPICE synthesis algorithms. Comparison between measurements and SPICE simulation results of the synthesised reduced order models are given in section 7.

The implementation of the ROM workbench is under Matlab, with GUI. As approaching the end of the Codestar project, various tools of the ROM Workbench became independent objects that were linked to the Codestar software.

 Figure 1:  Main blocks of the ROM Workbench.

The RomWorkbench proved to be a very helpful tool in reaching CODESTAR goals. Our presentation will illustrate its main capabilities.

#### References

 1 CODESTAR website www.imec.be/codestar 2 Mustafa Celik, Lawrence Pileggi, Altan Odabasioglu, IC Interconnect Analysis, Kluwer Academic Publishers,2002 3 L. Knockaert and D. De Zutter, Laguerre-SVD reduced order modeling,Electrical Performance of Electronic Packaging,1999, pp.249-252 4 The Control and Systems Library SLICOT, available at 5 B. Gustavsen and A. Semlyen, Rational Approximation of Frequency Domain Responses by  Vector Fitting, IEEE Trans. Power Delivery, 1999, 14,(3), pp 1052-1061 6 Timo Palenius, Time-domain simulation of reduced-order interconnect models, Helsinki University of Technology, available at http://www.aplac.hut.fi/publications/dt-timppa/thesis.pdf, 2002.

Pieter Heres (Eindhoven University of Technology)

Treatment of the redundancy in Krylov subspace methods for Model Order Reduction

Krylov subspace methods are often the method of choice for practical and industrial applications. Reasons for this chioce is that they are relatively cheap, while still accurate. Examples of such methods are PRIMA and Laguerre-SVD.

It is well-known however that the system approximation generated by Krylov subspace methods is non-optimal. As a consequence, the reduced systems still contain information that is not necessary for a good approximation. The approximation can be made smaller, without harming the accuracy of the model. In our research we investigated a method that deflates columns from the Krylov space, that are converged. In this way one can stop iterating for one port of the system, while proceeding for others. This deflation does hardly add computational cost to the method and leads to models that are smaller than the original models, while the accuracy is maintained. This approach can be used to cure part of the redundancy of Krylov subspace. Further, explicit removal of redundant poles from the system can be applied, after a full eigenvalue decomposition is calculated. This eigenvalue decomposition serves a double purpose, it can be used twice; it can also be used to formulate the reduced system in terms of a passive RLC-circuits as well. This offers great perspective for frequency and time domain simulations of the reduced system.

Michele Benzi
Numerical solution of saddle point problems
Large linear systems of saddle point type arise frequently in the numerical solution of PDEs, in constrained optimization, in the analysis of electric networks, and elsewhere. Such linear systems can be rather challenging to solve, and many solution methods have appeared in the literature. In this talk I will give an overview of recent developments in this very active research area, with a focus on a few selected applications and current challenges.

Erik I. Verriest, Georgia Institute of Technology

Nonlinear Balanced Realizations
Balancing for linear systems and its application to model reduction via the balanced truncation method are briefly reviewed.  Its generalization for nonlinear systems may be expected to be based upon three sound principles:
1) Balancing should be defined with respect to a nominal flow;
2) Only Gramians defined over small time intervals should be used in order to preserve the accuracy of the linear perturbation model and;
3) Linearization should commute with balancing, in the sense that the linearization of a globally balanced model should correspond to the balanced linearized model in the original coordinates.
It will be shown that an integrability condition generically provides an obstruction towards such a notion of a globally balanced realization in the strict sense.  By relaxing the conditions of "strict balancing" in various ways we shall obtain useful system  approximations.  In particular, an interpolation method (Mayer-Lie interpolation) is proposed.
As with linear systems, the metric provided by a local canonical gramian is shown to provide useful information about the dynamics of the system and the topology of the state space.

Andy.Wathen
Implicit factorization preconditioners for saddle-point systems
Saddle-point systems arise ubiquitously from problems with constraints. For problems where one can assume little additional structure, constraint preconditioners (in which constraint blocks are reproduced in the preconditioner) have attractive theoretical properties and allow for the use of the projected Conjugate Gradient method as an effective iterative solution method.
Implicit factorization preconditioners are a practical way of constructing effective constraint preconditioners. We will discuss the background, describe the simplest classes of implicit factorization preconditioner (based on the Schilders factorization) as well as introduce more general classes for regularized saddle-point problems.
This work is joint with Sue Dollar, Nick Gould and Wil Schilders

Joost Rommes

Computing specific poles of transfer functions

Given a transfer function, one is often interested in poles with  a specific property. If one wants to compute a reduced representation of  the transfer function, usually only a small number of poles is needed. In  stability analysis, typically the rightmost poles

are wanted. In this talk  approaches to compute such poles with specific properties will be  discussed and illustrated by practical examples.

D.C. Sorensen, Department of Computational and Applied Mathematics
Rice University
Gramian Based Model Reduction with Symmetry Constraints
Model reduction seeks to replace a large-scale system of differential or difference equations by a system of substantially lower dimension, that ideally, has the same response characteristics as the original system, yet requires far less computational resources for realization.
This talk will give an overview of Gramian based methods for model reduction. For linear time invariant systems this leads to balanced reduction which has ideal approximation properties including a global a-priori error bound.   The relationship between this approach and proper orthogonal decomposition (POD) and also principal component analysis (PCA) for dimension reduction in nonlinear problems will be discussed.
Principle Component Analysis based upon the singular value decomposition has many applications in various aspects of computational biology. Dimension reduction in molecular dynamics simulations is one very important application.  In many biological molecules, such as HIV1 protease, reflective  or rotational symmetry should be present in the molecular configuration.  Determining this symmetry allows one to create SVD modes of motion that best describe the symmetric movements of the protein. We present a method to compute the plane of reflective symmetry or the axis of rotational symmetry of a large set of points.  Moreover, we develop a symmetry preserving SVD that best approximates the given set while respecting the symmetry.  Our method is suitable for very large scale problems and is also robust in the presence of noise.

V. Simoncini, Dipartimento di Matematica, Universita' di Bologna

Analysis of projection-type methods for approximating the matrix exponential operator

Krylov subspace methods are often successfully employed to approximate the action of the matrix exponential exp(A) on a vector v in the numerical solution of differential

equations stemming from various time-dependent application problems.

In this talk we review some key properties of this approximation when A is Hermitian negative semidefinite or skew-Hermitian. In particular, we emphasize the role of the projection (reducing) space in acceleration procedures and in preserving some structural properties of the original exact problem.

This contribution is a joint work  with Luciano Lopez, Universita' di Bari.

Karen Willcox
Variable-fidelity methods provide a computationally effective strategy to achieve design and optimization of complex, large-scale, multidisciplinary systems. In this research, we combine concepts from model reduction and space mapping with a variable-fidelity optimization framework. A new method has been developed to combine space mapping with model correction within a trust-region model management framework.  This method is provably convergent to a local minimizer of the high-fidelity problem, and is  experimentally shown to give improved results over space mapping. A further significant challenge is to develop new methods in which the low-fidelity and high-fidelity objective functions can have differing numbers of design variables. We will discuss a new method that addresses this issue by using the Proper Orthogonal Decomposition to establish a mapping between the sets of design variables, which is subsequently incorporated within a trust-region model management framework. This method is also provably convergent and is shown to have significant computational  cost savings, relative to optimization in the high-fidelity space, on an airfoil design problem.

Patrick Dewilde

Generalized (non-stationary) model reduction
We present a systematic overview of non-stationary model-reduction theory and its application to modeling and simulation problems. Topics that will be covered are:
- the notion of semi- and quasi-separability, multipole approximation and the origin of quasi-separability, the inversion of full quasi-separable matrices
- maximum entropy interpolation for band and multiple band matrices
- model reduction theory for quasi-separable systems
- extensions to hierarchical and pseudo-hierarchical quasi-separability
- applications (e.g. preconditioning) and open problems.

Zhaojun Bai, University of California, Davis

Recent Advances in Structure-Preserving Model Order Reduction of Dynamical Systems and Applications
In this talk, we will report our recent work on computational techniques of structure-preserving model order reduction for large-scale RCS circuits, second-order systems, systems of coupling substructures and bilinear systems. Applications of these computational techniques to circuit and MEMS simulations and structure dynamics analysis will be presented.
This is a joint work with Ren-Cang Li, Ben-Shan Liao, Karl Meerbergen,
Daniel Skoogh, Yangfeng Su, and Xuan Zeng.

Roland W. Freund, Department of Mathematics, University of California

Structure-preserving model reduction --- What we have and what we still need
In recent years, there has been a lot of interest in dimension reduction of large-scale linear dynamical systems, and a large number of reduction methods have been proposed.  Most of these methods are formulated within the framework of first-order descriptor systems.  However, many linear dynamical systems that arise in actual applications, such as RCL networks in VLSI circuit simulation, are not given in first-order form.  The standard approach to dimension reduction of such systems is to first re-write them in equivalent first-order form, and then employ reduction techniques for first-order systems.  However, the resulting reduced systems are in first-order form, and in general, the special structure of the original dynamical system is not preserved.  To remedy this shortcoming, much of the recent research in model reduction of large-scale dynamical systems has focused on structure-preserving techniques that by-pass any equivalent first-order formulation and directly generate reduced model of the same form as the
original large-scale system.
In this talk, we first survey structure-preserving model reduction techniques, in particular reduction methods based on Krylov subspaces. We present theoretical results that clarify the connection between specialized reduction algorithms for second-order and higher-order linear dynamical systems, and the standard approach of applying Krylov subspace-based reduction methods to equivalent first-order formulations.  These theoretical results are illustrated with numerical examples from VLSI circuit simulation.
We then present an assessment of the state-of-the-art of structure-preserving
model reduction, and we describe a number of open problems in this area.

Michel Nakhla, Carleton University, Ottawa, Canada

Model Reduction of high-speed interconnectsusing integrated congruence transform

Passive model-order reduction of distributed interconnects via usage of the Hilbert-space moments is discussed. An implicit orthogonalization procedure is presented that can be used to compute an orthogonal basis for any set of elements that are related through a differential operator in a generalized Hilbert space. This procedure is utilized to construct an orthogonal basis that spans the Hilbert subspace spanning the first few moments of the voltages and currents across the distributed element. This basis is then used in the integrated congruence transform to cast the Telegrapher’s partial differential equations into a set of ordinary differential equations that can be linked easily with time-domain simulator without any form of spatial discretization. Application to uniform and nonuniform interconnects will be demonstrated.  This talk is based on joint work with Emad Gad (University of Ottawa).

P.W. Hemker

Space Mapping and Defect Correction

Moody T. Chu, North Carolina State University

## Large Quadratic Inverse Eigenvalue Problem

When modeling complex physical systems, the resulting mathematical models are sometimes of a very high order too expensive for simulation. One remedy is the notion of model reduction that assists in approximating very high order mathematical models with lower order models. Model reduction has been under extensive study and rapid development over the past few years with applications to many physical and engineering areas. On the other hand, due to inaccurate modeling and unknown disturbances, precise mathematical models of physical systems are rarely available in practice. It becomes necessary, when compared with realistic data, to {\em update} a primitive model to attain consistency with empirical results. This procedure of updating or revising an existing model is another essential ingredient for establishing an effective model.

This work concerns the quadratic model:

M\ddot{\mathbf{x}} + C\dot{\mathbf{x}} + K\mathbf{x} = f(t),

\label{qep_ode}

where $\mathbf{x} \in {\mathbb R}^{n}$ and $M$, $C$, $K \in {\mathbb R}^{n \times n}$ usually are structured. In most applications involving (\ref{qep_ode}), specifications of the underlying physical system are embedded in the matrix coefficients $M$, $C$ and $K$ while the resulting bearing of the system usually can be interpreted via its eigenvalues and eigenvectors. The process of analyzing and deriving the spectral information and, hence, inducing the dynamical behavior of a system from {\em a priori} known physical parameters such as mass, length, elasticity, inductance, capacitance, and so on is referred to as a {\em direct} problem. The {\em inverse} problem, in contrast, is to validate, determine, or estimate the parameters of the system according to its observed or expected behavior. The concern in the direct problem is to express the behavior in terms of the parameters whereas in the inverse problem the concern is to

express the parameters in term of the behavior. The inverse problem is just as important as the direct problem in applications.

The inverse eigenvalue problem is a diverse area full of research interests and activities.  Among current developments, the quadratic inverse eigenvalue problem  is particularly more important and challenging with many unanswered questions. The main emphasis in this work is to take into account one critical constraint arisen in practice --- in a large or complicated physical system, it is often impossible to obtain the entire spectral information. Furthermore, quantities related to high frequency terms in a \emph{finite model} generally are susceptible to measurement errors due to the finite bandwidth of measuring devices.  Spectral information, therefore, should not be used at its full extent. For these reasons, it might be more sensible to consider an inverse eigenvalue problem where only a {\em portion} of eigenvalues and eigenvectors is prescribed.

In this presentation, the speaker plans to outline current understanding about the solvability, computability, sensitivity, and feasibility  concerning the construction and updating of the quadratic pencil $(M,C,K)$ from partially prescribed spectral information.

Qiang Ye, Department of Mathematics, University of Kentucky
Krylov Subspace Methods for Model Reduction of Quadratic Matrix Polynomial
We consider model reduction problems for a quadratic matrix polynomial $I-A\lambda-B\lambda^2$ or more generally a two parameter matrix function $I-A\lambda - B\mu$, where $A$ and $B$ are large and sparse matrices. We shall present new Arnoldi and
Lanczos type processes which operate on the same space as $A$ and $B$ live and construct  projections of $A$ and $B$ to produce a quadratic matrix polynomial with the coefficient matrices of much smaller size, which is used to approximate the original problem.
We shall apply the new processes to the model reduction problem and discuss their convergence properties. Some special cases where the new algorithms are particularly effective will be identified. Our new processes are also extendable to cover a general matrix polynomial of any degree.

Tatjana Stykel
Model reduction of coupled systems
We discuss model order reduction of coupled linear systems. The behavior of such systems is determined by different interconnected subsystems that are usually governed by entirely different physical laws and often act in different scales. Instead of reducing the order of the entire system it seems to be more efficient to reduce the order of the subsystems and to couple the reduced-order subsystems through the same interconnection matrix. An important property of this approach is that every subsystem can be reduced by a most suitable model reduction method that takes into consideration the structure of the subsystem. Moreover, it is possible to obtain error bounds for the approximate closed-loop system in terms of the errors in the subsystems.

Paul Van Dooren, CESAME, UCL, Belgium
In this talk we describe general techniques of model reduction for large scale models of mechanical systems, with emphasis on vibration control in large buildings and mechanical structures. We first survey the technology that is available in this area. We then spend some time to explain the constraints that are often imposed in models for such structures. We focus in particular on systems of second order equations and on interconnected systems. We also discuss a particular example that is being studied in collaboration with research teams of Florida State University, Rice University and Purdue University.

Dave Bekers

Order Reduction for Large Antenna Arrays: The Eigencurrent Approach

On April 30, 1904, Christian Hülsmeyer patented his ‘Telemobiloskop’, which became the first operational radar system for detecting ships through the transmission and reception of electromagnetic waves. Nowadays, radar systems are widely used, e.g., to control air traffic, to measure vehicle speeds, and to detect and track objects like ships and airplanes. Often, the transmit/receive unit of such a system is an array composed of separate antennas, their number varying from a dozen to many hundreds. Since the development of antenna arrays is complex and costly, designs from simulations are made prior to the development.

At present, brute-force numerical approaches applied to a large array are still far too computationally expensive. Therefore, an array is often considered as an infinite periodic structure, where symmetry is used to restrict the analysis to a single antenna of the array. This approach, however, cannot completely describe the characteristic electromagnetic behavior of antenna arrays. In particular, it cannot predict the occurrence of standing-wave phenomena that limits their bandwidth severely. In this presentation, we propose an approach that describes the characteristic behavior of finite arrays accurately. Besides the prediction of standing-wave phenomena, the approach can indicate how these phenomena can be reduced for the entire scan range of the array.

The main aspect of the approach is the description of the behavior of an array by its ‘eigenvibrations’ or eigencurrents. These eigencurrents are the eigenfunctions of the impedance operator that relates the currents on the separate antennas to their excitation fields. From a physical point of view, the eigencurrents are standing waves of the array. The concept of eigencurrent appears extremely useful for the design, because eigencurrents are one-to-one related to properties of the array, like sum patterns, difference patterns, grating lobes, modulated impedance oscillations, and impedance variations attributed to surface waves. Besides a physical interpretation, the approach with eigencurrents leads to rapidly executable simulations; for, although the performance parameters of an array vary as a function of the geometry parameters, the eigencurrents vary hardly. Moreover, eigencurrents of (large) arrays are approximated as compositions of a small number of eigencurrents of the individual antennas in an array

Michael Guenther

Generalized network models for coupled systems: model order reduction and partial differential-algebraic equations

# J. Scherpen

Axis singular value functions and normalized coprime factorizations.

[Back]