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In this paper we consider the rational interpolation problem consisting in finding a rational matrix-valued function that
interpolates a given set of parameters. We briefly describe two different numerical methods for solving this problem. These are
the vector fitting and the frequency domain subspace identification method. Several numerical examples are given that compare
the properties of these methods. Furthermore, we discuss the computation of a (minimal) state space realization of a rational
function. Model order reduction methods such as modal approximation and balanced truncation are also presented. These
methods can be used to compute a reduced-order approximation of the realized dynamical system.
We review some known results for POD model reduction applied to ODEs. Then, these
results are generalized to several types of DAEs. We provide algorithms for the
model reduction and error bounds for the reduced order models. Some limits of
the approach are pointed out and alternative methods for reduced order subspace approximation
are presented. The POD approach is tested and evaluated for a medium sized DAE example
from multibody dynamics.
The purpose of this paper is the analysis of relaxation methods for the numerical integration
of coupled systems of ODEs and DAEs. We will investigate convergence of relaxation methods and put
special emphasis on the Jacobi- and Gauss-Seidel methods. Furthermore, the fundamental difference
in the convergence behaviour of coupled ODEs and DAEs is pointed out. This difference is used to explain
why certain relaxation methods for coupled DAEs may fail. Finally, a remedy to this undesirable effect
is proposed that makes use of a so-called preconditioned dynamic iteration strategy. This regularization
also allows significant reduction of relaxation steps.
The numerical simulation of very large scale integrated
circuits is an important tool in the development of new
industrial circuits. This topic has received increasing attention
within the last years. The main problem in circuit simulation is
that the model equations lead to differential algebraic equations
(DAEs). One known property of circuit DAEs is that they
may have an index larger than one, i.e., they may contain socalled
hidden constraints. The increased index has numerous
disadvantages on the numerical treatment of circuit DAEs.
The determination of these hidden constraints can be done
investigating the circuit topology. Until now, this information
has only been used for the consistent initialization of the circuit
equations. A recent approach has been to reduce the index of the
circuit DAE in order to improve their numerical behaviour. This
paper will give graph theoretical methods that lead to constraints
in a favorable formulation. Furthermore, the index reduction via
minimal extension will be performed for circuit DAEs, using these
constraints.
Element-based Topological Index Reduction for Differential-Algebraic Equations in Circuit Simulation
(2005)
The numerical simulation of very large scale integrated
circuit is an important tool in the development of new
industrial circuits. In the course of the last years, this topic has
received increasing attention. Common modeling approaches for
circuits lead to differential-algebraic systems (DAEs). In circuit
simulation, these DAEs are known to have index 2, given some
topological properties of the network. This higher index leads
to several undesirable effects in the numerical solution of the
DAEs. Recent approaches try to lower the index to improve
the numerical behaviour. These methods usually involve costly
algebraic transformations of the differential-agebraic equations.
Especially, for large scale circuit equations, these transformations
become too costly to be efficient.
We will present methods that change the topology of the network
itself, while replacing certain elements in oder to obtain a
network that leads to a DAE of index 1. This procedure can
be performed prior to the actual numerical simulation. The
decreasing of the index usually leads to significantly improved
numerical behaviour.
The purpose of this paper is the analysis of dynamic iteration methods for
the numerical integration of coupled systems of ODEs and DAEs.
We will investigate convergence of these methods and put special emphasis
on the {\sc Jacobi}- and {\sc Gauss-Seidel} methods. Furthermore, the
fundamental difference in the convergence behaviour of coupled ODEs and DAEs
is pointed out. This difference is used to explain why certain relaxation methods
for coupled DAEs may fail. Finally, a remedy to this undesirable
effect is proposed that makes use of a so-called {\em preconditioned dynamic
iteration} strategy. This regularization also allows significant reduction of
dynamic iteration steps.