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Model order reduction, coupled problems and optimization |
Peter
Benner
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. Hochstenbach
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. Axel Ruhe
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 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. J. Erhel
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. Rene
Pinnau
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_{ }||S_{ref}-S_{an}||_{F} /
max_{f} ||S_{ref}||_{F,}, where S_{ref}
are S parameters for the reference system (output of field solver or
measurements), and S_{an }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.
The RomWorkbench proved to be a very
helpful tool in reaching CODESTAR goals. Our presentation will illustrate its main
capabilities. References
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 Erik I. Verriest, Georgia Institute of Technology Nonlinear Balanced Realizations Andy.Wathen 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 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
Patrick Dewilde Generalized (non-stationary) model
reduction Zhaojun Bai, University of California, Davis Recent
Advances in Structure-Preserving Model Order Reduction of Dynamical Systems and
Applications Roland W. Freund, Department of Mathematics, University
of California Structure-preserving model reduction
--- What we have and what we still need
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 See http://www.lc.leidenuniv.nl/lc/web/2005/20050919/Hemker.pdf 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: \begin{equation} M\ddot{\mathbf{x}}
+ C\dot{\mathbf{x}} + K\mathbf{x} = f(t), \label{qep_ode} \end{equation} 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 Tatjana Stykel Paul Van Dooren, CESAME, UCL, Belgium 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
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] |