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In this paper we discuss the stability and model order reduction of coupled linear
time-invariant systems. Sufficient conditions for a closed-loop system to be asymptotically stable are
given. We present a model reduction approach for coupled systems based on reducing the order of the
subsystems and coupling the reduced-order subsystems by the same interconnection matrices as for
the original model. Such an approach allows to obtain error bounds for the reduced-order closed-loop
system in terms of the errors in the reduced-order subsystems. Model reduction of coupled systems
with unstable subsystems is also considered. Numerical examples are given.
Classical results about the local existence and uniqueness of
DAE solutions are based on the derivative array [2] or on a geometrical
approach [13]. Thus these results can't be applied to equations with nonsmooth
coefficients. Also, sufficient conditions that guarantee solvability
are hard to check in general [6, 13]. In this paper a new approach to proving
local existence and uniqueness of DAE solutions is presented. The
main tool is a decoupling procedure that makes it possible to split DAE
solutions into their characteristic parts. Thus it is possible to weaken
the smoothness requirements considerably. In order for the decoupling
procedure to work we require a certain structural condition to hold. In
contrast to results already known, this condition can be easily verified.
In this work we deal with the numerical solution of some problems of air pollution.
Since the problems are posed on unbounded domains we have to introduce
artificial boundaries to confine the computational region.
We construct and analyse (discrete) transparent boundary conditions
for an implicit difference scheme.
We discuss the concepts of positivity and monotonicity of
difference schemes and briefly consider these
properties of difference schemes for advection-diffusion equations
arising in problems of air (and water) pollution.
The efficiency and accuracy of our method is illustrated by an example.
In this paper a method for solving large-scale Sylvester equations is presented. The method is based on the sign function iteration and is particularly
effective for Sylvester equations with factorized right-hand side. In this case, the solution will be computed in factored form as it is for instance required in model reduction.
The hierarchical matrix format and the corresponding formatted arithmetic is integrated in the iteration scheme to make the method feasible for large-scale computations.
In this work we construct and analyse transparent boundary conditions (TBCs)
for general systems of parabolic equations. These TBCs are constructed for the
fully discrete scheme (-method, finite differences), in order to maintain unconditional
stability of the scheme and to avoid numerical re
ections. The discrete
transparent boundary conditions (DTBCs) are discrete convolutions in time and
are constructed using the solution of the Z{transformed exterior problem. We will
analyse the numerical error of these convolution coefficients caused by the inverse
Z{transformation. Since the DTBCs are non{local in time and thus very costly to
evaluate, we present approximate DTBCs of a sum{of{exponentials form that allow
for a fast calculation of the boundary terms. Finally, we will use our approximate
DTBCs for an example of a
uid stochastic Petri net and present numerical results.
This paper deals with the efficient numerical solution of the two-dimensional one
way Helmholtz equation posed on an unbounded domain. In this case one has to
introduce artificial boundary conditions to confine the computational domain. The
main topic of this work is the construction of so{called discrete transparent boundary
conditions for state-of-the-art parabolic equations methods, namely a split-step
discretization of the high{order parabolic approximation and the split-step Padle
algorithm of Collins. Finally, several numerical examples arising in optics and underwater
acoustics illustrate the efficiency and accuracy of our approach.
Transparent boundary conditions (TBCs) are an important tool for the truncation of the compu-
tational domain in order to compute solutions on an unbounded domain. In this work we want
to show how the standard assumption of `compactly supported data' could be relaxed and derive
TBCs for the wide angle parabolic equation directly for the numerical scheme on the discrete level.
With this inhomogeneous TBCs it is not necessary that the starting field lies completely inside the
computational region. However, an increased computational effort must be accepted.
We discuss the nonstandard problem of using the finite difference
method to solve numerically a partial differential equation posed on
an unbounded domain. We propose different strategies to construct
so-called discrete articial boundary conditions (ABCs) and present
an efficient implementation by the sum-of-exponential ansatz. The
derivation of the ABCs is based on the knowledge of the exact solution,
the construction of asymptotic solutions or the usage of a continued
fraction expansion to a second-order difference equation. Our approach
is explained by means of three different types of partial differential
equations arising in option pricing, in quantum mechanics and
in (underwater) acoustics. Finally, we conclude with an illustrating
numerical example from underwater acoustics showing the superiority
of our new approach.
In this work we are interested in the numerical solution of a coupled model of
differential algebraic equations (DAEs) and partial differential equations (PDEs).
The DAEs describe the behavior of an electrical circuit that contains semiconductor
devices and the partial differential equations constitute drift-diffusion equations
modelling the semiconductor devices in the circuit.
After space discretization using a finite element method, the coupled system results
in a differential-algebraic system with a properly stated leading term. We
investigate the structure and the properties of this DAE system. In particular, we
develop structural criteria for the DAE index. This is of basic interest since DAE
properties like stability, existence and uniqueness of solutions depend strongly on
its index.
This work is concerned with transparent boundary conditions (TBCs) for systems of Schrödinger-type equations, namely
the time-dependent kp-Schrödinger equations. These TBCs are constructed for the fully discrete scheme (Crank-Nicolson,
finite differences), in order to maintain unconditional stability of the scheme and to avoid numerical reflections. The discrete
transparent boundary conditions (DTBCs) are discrete convolutions in time and are constructed using the Z-transformed
solution of the exterior problem. We will analyse the numerical error of these convolution coeffficients caused by the inverse
Z-transformation. Since the DTBCs are non-local in time and thus very costly to evaluate, we present approximate DTBCs
of a sum-of-exponentials form that allow for a fast calculation of the boundary terms.
Traveling wave equations are used to model the dynamics of multisection semiconductor lasers. To perform a bifurcation analysis of this system of 1-D partial differential equations its low dimensional approximations are constructed and considered. Along this paper this analysis is used for the extensive study of the pulsations in a three section distributed feedback laser. Namely, stability of pulsations, different bifurcation scenaria, tunability of the pulsation frequency and its locking by the frequency of electrical modulation are considered. All these pulsation qualities are highly important when applying lasers in optical communication systems.
We prove the existence, uniqueness, regularity and smooth dependence
of the weak solution on the initial data for a certain class of semilinear first order
dissipative hyperbolic systems with spacially discontinuous coefficients. Such
kind of hyperbolic problems have succesfully been used to describe the dynamics
of distributed feedback multisection semiconductor lasers in recent years. We
show that in a suitable function space of continuous functions the weak solutions
generate a smooth semiflow.
The Modified Nodal Analysis leads to differential algebraic equations
with properly stated leading terms. In this article a special structure of the DAEs
modelling electrical circuits is exploited in order to derive a new decoupling for
nonlinear index-2 DAEs. This decoupling procedure leads to a solvability result and
is also used to study general linear methods, a class of numerical schemes that covers
both Runge-Kutta and linear multistep methods. Convergence for index-2 DAEs is
proved.
We simulate and analyse a 1D-PDE model describing the dynamics of multisection semiconductor lasers. We demonstrate how a semi-analytical computation of the spectrum and the corresponding eigenfunction expansion of the computed solutions provides a useful information allowing to achieve a better understanding of the laser dynamics. Basic algorithms implemented into a corresponding software tool are described.
An in-depth theoretical as well as experimental analysis of the nonlinear dynamics in semiconductor lasers
with active optical feedback is presented. Use of a monolithically integrated multisection device of submillimeter
total length provides access to the short-cavity regime. By introducing an amplifier section as a special
feature, phase and strength of the feedback can be separately tuned. In this way, the number of modes involved
in the laser action can be adjusted. We predict and observe specific dynamical scenarios. Bifurcations mediate
various transitions in the device output, from single-mode steadystate to self-pulsation and between different
kinds of self-pulsations, reaching eventually chaotic behavior in the multimode limit.
Im Zentrum der Arbeiten soll die Chaos- und Kohärenzkontrolle von Halbleiterlasern mit gegenseitiger optischer Kopplung stehen. Diese Fragestellung ist von erheblicher praktischer Relevanz, da Rauschen und chaotisches Verhalten generelle Probleme in der optischen Hochgeschwindigkeitskommunikation sind. Die Kontrolle von optischen Systemen mit komplexer Selbstorganisation stellt aber auch aus grundsätzlicher Sicht Neuland dar. Hierfür geeignete Konzepte sind bisher weder überzeugend theoretisch beschrieben noch experimentell umgesetzt.
We describe the basic ideas behind the concept of distributed
feedback (DFB) lasers with short optical feedback for the
generation of high-frequency self-pulsations and show the theoretical
background describing realized devices. It is predicted by
theory that the self-pulsation frequency increases with increasing
feedback strength. To provide evidence for this, we propose a novel
device design which employs an amplifier section in the integrated
feedback cavity of a DFB laser.We present results from numerical
simulations and experiments. It has been shown experimentally
that a continuous tuning of the self-pulsation frequency from 12
to 45 GHz can be adjusted via the control of the feedback strength.
The numerical simulations, which are in good accordance with experimental
investigations, give an explanation for a self-stabilizing
effect of the self-pulsations due to the additional carrier dynamic
in the integrated feedback cavity.
Abstract. We consider a mathematical model (the so-called traveling-wave system) which describes longitudinal
dynamical effects in semiconductor lasers. This model consists of a linear hyperbolic system
of PDEs, which is nonlinearly coupled with a slow subsystem of ODEs. We prove that a corresponding
initial-boundary value problem is well posed and that it generates a smooth infinite-dimensional dynamical
system. Exploiting the particular slow–fast structure, we derive conditions under which there exists a lowdimensional
attracting invariant manifold. The flow on this invariant manifold is described by a system
of ODEs. Mode approximations of that system are studied by means of bifurcation theory and numerical
tools.
We generalize an alternating direction implicit method and the Smith method for
large-scale projected generalized Lyapunov equations. Such equations arise in model reduction for
descriptor systems. Low rank versions of these methods are also presented, that can be used to
compute low rank approximations to the solution of projected generalized Lyapunov equations with
low rank symmetric, positive semidefinite right-hand side. Numerical examples are presented.