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We study the class of linear differential-algebraic m-input m-output systems
which have a transfer function with proper inverse.
A sufficient condition for the transfer function to have proper inverse
it that the system has 'strict and non-positive relative degree'.
We present two main results:
First, a so called 'zero dynamics form' is derived: this form is - within the class of system equivalence - a simple "almost normal" form of the DAE; it is a counterpart to the well-known Byrnes-Isidori form for
ODE systems with strictly proper transfer function.
The 'zero dynamics form' is exploited to characterize structural properties such as
asymptotically stable zero dynamics,
minimum phase, and high-gain stabilizability.
The zero dynamics are characterized by (A,E,B)-invariant subspaces.
Secondly, it is shown that the 'funnel controller' (that is a static nonlinear output error feedback) achieves, for all DAE systems with asymptotically stable zero dynamics and transfer function with proper inverse, tracking of a reference signal by the output signal within a pre-specified funnel. This funnel determines the transient behaviour.
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.
We consider discretizations for reaction-diffusion systems with nonlinear
diffusion in two space dimensions. The applied model allows to handle heterogeneous
materials and uses the chemical potentials of the involved species as primary variables.
We propose an implicit Voronoi finite volume discretization on regular Delaunay
meshes that allows to prove uniform, mesh-independent global upper and lower L1
bounds for the chemical potentials. These bounds provide the main step for a convergence
analysis for the full discretized nonlinear evolution problem. The fundamental
ideas are energy estimates, a discrete Moser iteration and the use of discrete
Gagliardo-Nirenberg inequalities. For the proof of the Gagliardo-Nirenberg inequalities
we exploit that the discrete Voronoi finite volume gradient norm in 2d coincides
with the gradient norm of continuous piecewise linear finite elements.
For a system of globally pulse-coupled phase-oscillators,
we derive conditions for stability of the completely synchronous
state and all possible two-cluster states and explain
how the different states are naturally connected via
bifurcations. The coupling is modeled using the phaseresponse-
curve (PRC), which measures the sensitivity of
each oscillator’s phase to perturbations. For large systems
with a PRC, which turns to zero at the spiking threshold,
we are able to find the parameter regions where multiple
stable two-cluster states coexist and illustrate this by an
example. In addition, we explain how a locally unstable
one-cluster state may form an attractor together will its
homoclinic connections. This leads to the phenomenon
of intermittent, asymptotic synchronization with abating
beats away from the perfect synchrony.
We use the traveling wave model for simulating and analyzing
nonlinear dynamics of complex semiconductor ring laser devices.
This modeling allows to consider temporal-spatial distributions
of the coun\-ter-pro\-pa\-ga\-ting slowly varying optical fields
and the carriers, what can be important when studying
non-homogeneous ring cavities, propagation of short pulses or fast switching.
By performing numerical integration of the model
equations we observe several dynamic regimes as well as transitions
between them. The computation of ring cavity modes explains some
peculiarities of these regimes.
A new approach to derive transparent boundary conditions (TBCs) for wave, Schrödinger, heat and drift-diffusion equations is presented. It relies on the pole condition and distinguishes between physical reasonable and unreasonable solutions by the location of the singularities of the spatial Laplace transform of the exterior solution. To obtain a numerical algorithm, a Möbius transform is applied to map the Laplace transform onto the unit disc. In the transformed coordinate the solution is expanded into a power series. Finally, equations for the coefficients of the power series are derived. These are coupled to the equation in the interior, and yield transparent boundary conditions.
Numerical results are presented in the last section, showing that the error introduced by the new approximate TBCs decays exponentially in the number of coefficients.
The pole condition approach for deriving transparent boundary conditions is extended to the time-dependent, two-dimensional case. Non-physical modes of the solution are identified by the position of poles of the solution's spatial Laplace transform in the complex plane. By requiring the Laplace transform to be analytic on some
problem dependent complex half-plane, these modes can be
suppressed. The resulting algorithm computes a finite number of coefficients of a series expansion of the Laplace transform, thereby providing an approximation to the exact boundary condition. The resulting error decays super-algebraically with the number of coefficients, so relatively few additional degrees of freedom are
sufficient to reduce the error to the level of the discretization error in the interior of the computational domain. The approach shows good results for the Schroedinger and the drift-diffusion equation
but, in contrast to the one-dimensional case, exhibits instabilities for the wave and Klein-Gordon equation. Numerical examples are shown that demonstrate the good performance in the former and the instabilities in the latter case.
Group-velocity matched cross-phase modulation between a fundamental soliton and a dispersive wave-packet has been previously suggested for optical switching applications similar to an optical transistor. Moreover, the nonlinear interaction in the resulting group-velocity horizon can be exploited for adiabatic compression of the soliton down into the few-cycle regime. Here we show that both mechanisms can be combined. In such a transient compressor, parameters of the dispersive wave may then serve to actively control the soliton compression and adjust the pulse duration in the presence of disturbances. While a certain amount of control is already enabled by the delay between soliton and dispersive wave, the means of controlling the compression process are substantially enhanced by additionally manipulating the chirp of the dispersive wave. Moreover, controlling the chirp of the dispersive wave also enables correction for limitations of the compression scheme due to a self-frequency shift of the soliton or for uncompensated dispersion in the scheme. This substantially widens the practicality of the compression scheme and other applications of the highly efficient nonlinear interaction at the group-velocity horizon.
We discuss analytical and numerical methods for the optimization of optoelectronic devices by performing optimal control of the PDE governing the carrier transport with respect to the doping profile. First, we provide a cost functional that is a sum of a regularization and a contribution, which is motivated by the modal net gain that appears in optoelectronic models of bulk or quantum- well lasers. Then, we state a numerical discretization, for which we study optimized solutions for different regularizations and for vanishing weights.
We show that many couplings between parabolic systems for processes in solids can be formulated as a gradient system with respect to the total free energy or the total entropy. This includes Allen-Cahn, Cahn-Hilliard, and reaction-diffusion systems and the heat equation. For this, we write the coupled system as an Onsager system $(X,\Phi,K)$ defining the evolution $\dot{U} = -K(U)D\Phi(U)$. Here $\Phi$ is the driving functional, while the Onsager operator $K(U)$ is symmetric and positive semidefinite. If the inverse $G = K^{-1}$ exists, the triple $(X,\Phi,G)$ defines a gradient system. Onsager systems are well suited to model bulk-interface interactions by using the dual dissipation potential $\Psi^*(U,\Xi) = 1/2 <\Xi,K(U)\Xi>$. Then, the two functionals $\Phi$ and $\Psi^*$ can be written as a sum of a volume integral and a surface integral, respectively. The latter may contain interactions of the driving forces in the interface as well as the traces of the driving forces from the bulk. Thus, capture and escape mechanisms like thermionic emission appear naturally in Onsager systems, namely simply through integration by parts.
We show the existence of solutions to a system of elliptic PDEs, that was recently introduced to describe the electrothermal behavior of organic semiconductor devices. Here, two difficulties appear: (i) the elliptic term in the current-flow equation is of $p(x)$-Laplacian-type with discontinuous exponent $p$, which limits the use of standard methods, and (ii) in the heat equation, we have to deal with an a priori $L^1$ term on the right hand side describing the Joule heating in the device. We prove the existence of a weak solution under very weak assumptions on the data. Our existence proof is based on Schauder’s fixed point theorem and the concept of entropy solutions for the heat equation. Here, the crucial point is the continuous dependence of the entropy solutions on the data of the problem.
We show the existence of solutions to a system of elliptic PDEs, that was recently introduced to describe the electrothermal behavior of organic semiconductor devices. Here, two diffculties appear: (i) the elliptic term in the current-flow equation is of p(x)-Laplacian type with discontinuous exponent p, which limits the use of standard methods, and (ii) in the heat equation, we have to deal with an a priori L1 term on the right hand side describing the Joule heating in the device. We prove the existence of a weak solution under very weak assumptions on the data. Our existence proof is based on Schauder’s fixed point theorem and the concept of entropy solutions for the heat equation. Here, the crucial point is the continuous dependence of the entropy solutions on the data of the problem.
We show that the spectrum of linear delay differential equations with
large delay splits into two different parts. One part, called the
strong spectrum, converges to isolated points when the delay parameter
tends to infinity. The other part, called the pseudocontinuous spectrum,
accumulates near criticality and converges after rescaling to a set
of spectral curves, called the asymptotic continuous spectrum. We
show that the spectral curves and strong spectral points provide a
complete description of the spectrum for sufficiently large delay
and can be comparatively easily calculated by approximating expressions.
When simulating isolated resonators, the application of transparent boundary conditions causes the approximated spectrum to be polluted with spurious solutions. Distinguishing these artificial solutions from solutions with a physical meaning is often difficult and requires a priori knowledge of the spectrum or the expected field distribution of resonant states. We present an implementation of the pole condition that distinguishes between incoming and outgoing waves by the location of the poles of their Laplace transform as transparent boundary condition. This implementation depends on one tuning parameter. We will use the sensitivity of the computed solutions to perturbations of this parameter as a means to identify spurious solutions. To obtain global statements, we will combine this technique with a convergence monitor for the boundary condition.
The perturbation and ADAE index of a degenerated hyperbolic system modelling a heat exchanger
(2007)
The heat exchanger in a heat pump can be modelled by the zero Mach-number limit of the Euler equations of compressible fluid flow. This system turns out to be a coupled hyperbolic/parabolic equation with coupled, time-dependent boundary conditions. Using the theory of abstract differential-algebraic equations it is shown that the frozen coefficient system has ADAE index 1. Moreover, the much stronger result is proven that the system has time-perturbation index one and space-perturbation index two even in the case of time-dependent boundary conditions. The results are stated in terms of the original physical variables. The estimates agree well with numerical experiments.
The well-known Kalman-Yakubovich-Popov Lemma establishes an equivalence between dissipativity and the solvability of a linear matrix inequality. In this paper we strengthen this result by showing the equivalence of dissipativity to the solvability of a so-called Lur'e equation, which mainly is a linear matrix inequality with a rank minimizing condition. Finally, we apply the result to standard systems to obtain the well-known result about the solvability of the algebraic Riccati equation.
We consider a coupled system of two elliptic PDEs, where the elliptic term in the first equation shares the properties of the $p(x)$-Laplacian with discontinuous exponent, while in the second equation we have to deal with an a~priori $L^1$ term on the right hand side. Such a system of equations is suitable for the description of various electrothermal effects, in particular those, where the non-Ohmic behavior can change dramatically with respect to the spatial variable. We prove the existence of a weak solution under very weak assumptions on the data and also under general structural assumptions on the constitutive equations of the model. The main difficulty consists in the fact that we have to overcome simultaneously two obstacles - the discontinuous variable exponent (which limits the use of standard methods) and the $L^1$ right hand side of the heat equation. Our existence proof based on Galerkin approximation is highly constructive and therefore seems to be suitable also for numerical purposes.
The long standing problem is discussed of how to deflate the part associated with the eigenvalue infinity in a structured matrix pencil using structure preserving unitary transformations. We derive such a deflation procedure and apply this new technique to symmetric, Hermitian or alternating pencils and in a modified form to (anti)-palindromic pencils. We present a detailed error and perturbation analysis of this and other deflation procedures and demonstrate the properties of the new algorithm with several numerical examples.
Classical stability properties of solutions
that are well-known for ordinary differential
equations (ODEs) are generalized to differential-algebraic equations (DAEs).
A new test equation is derived for the analysis of numerical methods applied
to DAEs with respect to the stability of the numerical approximations.
Morevover, a stabilization technique is developed to improve the stability of classical DAE integration methods. The stability regions for these stabilized discretization methods are determined and it is shown that they much better reproduce the stability properties known for the ODE case
than in the unstabilized form.
Movies that depict the stability regions for several methods are included for interactive use.
The Lang-Kobayashi model is a system of delay differential equations (DDEs) describing the dynamics of a semiconductor laser under delayed optical feedback. In this paper, we study the stability of so called external cavity modes (ECMs), which are harmonic oscillations corresponding to stationary lasing states. We focus on experimentally relevant situations, when the delay is large compared to the internal time scales of the laser. In this case, both the number of ECMs and the number of critical eigenvalues grows to infinity. Applying a newly developed asymptotic description for the spectrum of linearized DDEs with long delay, we are able to overcome this difficulty and to give a complete description of the stability properties of all ECMs. In particular, we distinguish between different types of weak and strong instabilities and calculate bifurcation diagrams that indicate the regions with different stability properties and the transitions between them.
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.
Chimera states are particular trajectories
in systems of phase oscillators with non-local coupling
that display a spatio-temporal pattern of coherent and incoherent motion.
We present here a detailed analysis
of the spectral properties for such trajectories.
First, we study numerically their Lyapunov spectrum
and its behavior for an increasing number of oscillators.
The spectra demonstrate the hyperchaotic nature of the chimera states
and show a correspondence of the Lyapunov dimension
with the number of incoherent oscillators.
Then, we pass to the thermodynamic limit equation
and present an analytic approach
to the spectrum of a corresponding linearized evolution operator.
We show that in this setting, the chimera state is neutrally stable
and that the continuous spectrum coincides with the limit
of the hyperchaotic Lyapunov spectrum obtained for the finite size systems.
Lyapunov and exponential dichotomy spectral theory is extended
from ordinary differential equations (ODEs) to nonautonomous
differential-algebraic equations (DAEs). By using orthogonal
changes of variables, the original DAE system is transformed into
appropriate condensed forms, for which concepts such as Lyapunov
exponents, Bohl exponents, exponential dichotomy and spectral
intervals of various kinds can be analyzed via the resulting
underlying ODE. Some essential differences between the spectral
theory for ODEs and that for DAEs are pointed out. Numerical
methods for computing the spectral intervals associated with
Lyapunov and Sacker-Sell (exponential dichotomy) spectra are
derived by modifying and extending those methods proposed for ODEs. Perturbation theory and error analysis are discussed, as
well. Finally, some numerical examples are presented to illustrate
the theoretical results and the properties of the numerical
methods.
Three properties of matrices: the spark, the mutual incoherence and the restricted isometry property have recently been introduced in the context of compressed sensing. We study these properties for matrices that are Kronecker products and show how these properties relate to those of the factors. For the mutual incoherence we also
discuss results for sums of Kronecker products.
A new concept is introduced for the adaptive finite element discretization of partial differential equations that have a sparsely
representable solution. Motivated by recent work on compressed sensing, a recursive mesh refinement procedure is presented that uses linear programming to find a good approximation to the sparse solution on a given refinement level. Then only those parts of the mesh are refined that belong to nonzero expansion coefficients. Error estimates for this procedure are refined and the behavior of the procedure is demonstrated via some simple elliptic model problems.
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.
Pseudo-Boolean problems generalize SAT problems by allowing linear constraints and a linear objective function. Different solvers, mainly having their roots in the SAT domain, have been proposed and compared,for instance, in Pseudo-Boolean evaluations. One can also formulate Pseudo-Boolean models as integer programming models. That is,Pseudo-Boolean problems lie on the border between the SAT domain and the integer programming field.
In this paper, we approach Pseudo-Boolean problems from the integer programming side. We introduce the framework SCIP that implements constraint integer programming techniques. It integrates methods from constraint programming, integer programming, and SAT-solving: the solution of linear programming relaxations, propagation of linear as well as nonlinear constraints, and conflict analysis. We argue that this approach is suitable for Pseudo-Boolean instances containing general linear constraints, while it is less efficient for pure SAT problems. We present extensive computational experiments on the test set used for the Pseudo-Boolean evaluation 2007. We show that our approach is very efficient for optimization instances and competitive for feasibility problems. For the nonlinear parts, we also investigate the influence of linear programming relaxations and propagation methods on the performance. It turns out that both techniques are helpful for obtaining an efficient solution method.
In the case of the equidistant discretization of the Airy differential equation (\discrete
Airy equation") the exact solution can be found explicitly. This fact is used
to derive a discrete transparent boundary condition (TBC) for a Schroedinger
equation with linear varying potential, which can be used in \parabolic equation"
simulations in (underwater) acoustics and for radar propagation in the troposphere.
We propose different strategies for the discrete TBC and show an efficient implementation.
Finally a stability proof for the resulting scheme is given. A numerical
example in the application to underwater acoustics shows the superiority of the new
discrete TBC.
We give an exposition of recent results on regularity and Fredholm properties for first-order one-dimensional hyperbolic PDEs. We show that large classes of boundary operators cause an effect that smoothness increases with time. This property is the key in finding regularizers
(parametrices) for hyperbolic problems. We construct regularizers for periodic problems for dissipative first-order linear hyperbolic PDEs and show that these problems are modeled by Fredholm operators of index zero.
The classical singular value decomposition for a matrix $A\in\Cmn$ is a
canonical form for $A$ that also displays the eigenvalues
of the Hermitian matrices $AA^\ast$ and $A^\ast A$. In this paper, we develop
a corresponding decomposition for $A$ that provides the Jordan canonical forms
for the complex symmetric matrices $AA^T$ and $A^TA$. More generally, we consider
the matrix triple $(A,G_1,G_2)$, where $G_1\in\CC{m}, G_2\in\CC{n}$
are invertible and either complex symmetric and complex skew-symmetric, and we
provide a canonical form under transformations of the form
$(A,G_1,G_2)\mapsto(X^T A Y, X^T G_1X, Y^T G_2Y)$, where $X,Y$ are nonsingular.
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.
The paper is devoted to Schroedinger operators on bounded intervals of the real axis with dissipative boundary conditions. In the framework of the Lax-Phillips scattering theory the asymptotic behaviour of the phase shift is investigated in detail and its relation to the spectral shift is discussed, in particular, trace formula and Birman-Krein formula are verified directly. The results are used for dissipative Schroedinger-Poisson systems.
We study the dynamics of a ring of unidirectionally coupled autonomous
Duffing oscillators. Starting from a situation where the individual
oscillator without coupling has only trivial equilibrium dynamics,
the coupling induces complicated transitions to periodic, quasiperiodic,
chaotic, and hyperchaotic behavior. We study these transitions in
detail for small and large numbers of oscillators. Particular attention
is paid to the role of unstable periodic solutions for the appearance
of chaotic rotating waves, spatiotemporal structures and the Eckhaus
effect for a large number of oscillators. Our analytical and numerical
results are confirmed by a simple experiment based on the electronic
implementation of coupled Duffing oscillators.
We examine robustness of exponential dichotomies of boundary value problems for general linear first-order one-dimensional hyperbolic systems. The boundary conditions are supposed to be of types ensuring smoothing solutions in finite time, which includes reflection boundary conditions. We show that the dichotomy survives in the space of continuous functions under small perturbations of all coefficients in the differential equations.
Resolving thin conducting sheets for shielding or even skin layers inside by the mesh of numerical methods like the finite element method (FEM) can be avoided by using impedance transmission conditions (ITCs). Those ITCs shall provide an accurate approximation for small sheet thicknesses $d$, where the accuracy is best possible independent of the conductivity or the frequency being small or large -- this we will call robustness. We investigate the accuracy and robustness of popular and recently developed ITCs, and propose robust ITCs which are accurate up to $O(d^2)$.
Three families of transmission conditions of different order are proposed for thin conducting sheets in the eddy current model. Resolving the thin sheet by a finite element mesh is often not possible. With these transmission conditions only the middle curve, but not the thin sheet itself, has not to be resolved by a finite element mesh. The families of transmission conditions are derived by an asymptotic expansion for small sheet thicknesses $\eps$, where each family results from a different asymptotic framework. In the first asymptotic framework the conductivity remains constant, scales with $1/\eps$ in the second and with $1/\eps^2$ in the third. The different asymptotics lead to different limit conditions, namely the vanishing sheet, a non-trivial borderline case, and the impermeable sheet, as well as different transmission conditions of higher orders. We investigated the stability, the convergence of the transmission conditions as well as their robustness. We call transmission conditions robust, if they provide accurate approximation for a wide range of sheet thicknesses and conductivities. We introduce an ordering of transmission conditions for the same sheet with respect to the robustness, and observe that the condition derived for the $1/\eps$ asymptotics is the most robust limit condition, contrary to order 1 and higher, where the transmission conditions derived for the $1/\eps^2$ asymptotics turn out to be most robust.
The efficient and reliable computation of guided modes in photonic crystal wave-guides is of great importance for designing optical devices. Transparent boundary conditions based on Dirichlet-to-Neumann operators allow for an exact computation of well-confined modes and modes close to the band edge in the sense that no modelling error is introduced. The well-known super-cell method, on the other hand, introduces a modelling error which may become prohibitively large for guided modes that are not well-confined. The Dirichlet-to-Neumann transparent boundary conditions are, however, not applicable for all frequencies as they are not uniquely defined and their computation is unstable for a countable set of frequencies that correspond to so called Dirichlet eigenvalues. In this work we describe how to overcome this theoretical difficulty introducing Robin-to-Robin transparent boundary conditions whose construction do not exhibit those forbidden frequencies. They seem, hence, well suited for an exact and reliable computation of guided modes in photonic crystal wave-guides.
Uncertainty is inevitable when solving science and engineering application problems. In the face of
uncertainty, it is essential to determine robust and risk-averse solutions. In this work,
we consider a class of PDE-constrained optimization problems in which the PDE coefficients
and inputs may be uncertain. We introduce two approximations for minimizing the
conditional value-at-risk for such PDE-constrained optimization problems. These approximations are based
on the primal and dual formulations of the conditional value-at-risk. For the primal problem,
we introduce a smooth approximation of the conditional value-at-risk in order to utilize
derivative-based optimization algorithms and to take advantage of the convergence properties
of quadrature-based discretizations. For this smoothed conditional value-at-risk, we prove
differentiability as well as consistency of our approximation. For the dual problem, we
regularize the inner maximization problem, rigorously derive optimality conditions, and demonstrate
the consistency of our approximation. Furthermore, we propose a fixed-point iteration that takes
advantage of the structure of the regularized optimality conditions and provides a means of calculating
worst-case probability distributions based on the given probability level. We conclude with numerical
results.
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.
In this paper, we propose and investigate numerical methods based on QR factorization for computing all or some Lyapunov or Sacker-Sell spectral intervals for
linear differential-algebraic equations.
Furthermore, a perturbation and error analysis for these methods is presented. We
investigate how errors in the data and in the numerical integration affect the
accuracy of the approximate spectral intervals. Although we need to integrate
numerically some differential-algebraic systems on usually very long
time-intervals, under certain assumptions, it is shown that the error of the
computed spectral intervals can be controlled by the local error of numerical
integration and the error in solving the algebraic constraint.
Some numerical examples are presented to illustrate the theoretical results.
We propose a model reduction method for positive systems that ensures the positivity of the reduced-order model. In the standard as well as in the descriptor case, for continuous-time and discrete-time systems, our approach is based on constructing diagonal solutions of Lyapunov inequalities. These are linear matrix inequalities (LMIs), which are shown to be feasible. Positivity and stability are preserved and an error bound in the $\mathcal{H}_\infty$-norm is provided.
We propose a model reduction method for positive systems that ensures the positivity of the reduced model. For both, continuous-time and discrete-time systems, our approach is based on constructing diagonal solutions of Lyapunov inequalities. These are linear matrix inequalities (LMIs), which are shown to be feasible. Stability is preserved and an error bound in the $\mathcal{H}_\infty$-norm is provided.
For selfadjoint matrices in an indefinite inner product, possible canonical forms are identified that arise when the matrix is subjected to a selfadjoint generic rank one perturbation. Genericity is understood in
the sense of algebraic geometry. Special attention is paid to the perturbation
behavior of the sign characteristic. Typically, under such a perturbation,
for every given eigenvalue, the largest Jordan block of the eigenvalue is
destroyed and (in case the eigenvalue is real) all other Jordan blocks
keep their sign characteristic. The new eigenvalues, i.e., those eigenvalues of
the perturbed matrix that are not eigenvalues of the original matrix,
are typically simple, and in some cases information is provided about their sign
characteristic (if the new eigenvalue is real). The main results are proved by using
the well known canonical forms of selfadjoint matrices in an indefinite inner product, a version of the Brunovsky
canonical form and on general results concerning rank one perturbations.
Motivated by the analysis of passive control systems, we undertake a detailed perturbation analysis of Hamiltonian matrices that have eigenvalues on the imaginary axis. We construct minimal Hamiltonian perturbations that move and coalesce eigenvalues of opposite sign characteristic to form multiple eigenvalues with mixed sign characteristics, which are then moved from the imaginary axis to specific locations in the complex plane by small Hamiltonian perturbations. We also present a numerical method to compute upper bounds for the minimal perturbations that move all eigenvalues of a given Hamiltonian matrix outside a vertical strip along the imaginary axis.
Perturbation of Purely Imaginary Eigenvalues of Hamiltonian Matrices under Structured Perturbations
(2007)
We discuss the perturbation theory for purely imaginary eigenvalues of Hamiltonian matrices under Hamiltonian and non-Hamiltonian perturbations. We
show that there is a substantial difference in the behavior under these perturbations. We also discuss the perturbation of real eigenvalues of real
skew-Hamiltonian matrices under structured perturbations and use these results to analyze the properties of the URV method of computing the
eigenvalues of Hamiltonian matrices.
Persistence of rogue waves in extended nonlinear Schrödinger equations: Integrable Sasa-Satsuma case
(2012)
We present the lowest order rogue wave solution of the Sasa-Satsuma equation (SSE) which is one of the integrable extensions of the nonlinear Schrödinger equation (NLSE). In contrast to the Peregrine solution of the NLSE, it is significantly more involved and contains polynomials of fourth order rather than second order in the corresponding expressions. The correct limiting case of Peregrine solution appears when the extension parameter of the SSE is reduced to zero.
Periodic Solutions to Dissipative Hyperbolic Systems. II: Hopf Bifurcation for Semilinear Problems
(2013)
We consider boundary value problems for semilinear hyperbolic systems of the type
$$
\partial_tu_j + a_j(x,\la)\partial_xu_j + b_j(x,\la,u) = 0, \; x\in(0,1), \;j=1,\dots,n
$$
with smooth coefficient functions $a_j$
and $b_j$
such that
$b_j(x,\la,0) = 0$ for all $x \in [0,1]$, $\la \in \R$, and $j=1,\ldots,n$.
We state conditions for Hopf bifurcation, i.e.,
for existence, local uniqueness (up to phase shifts), smoothness and smooth dependence
on $\la$
of time-periodic solutions bifurcating from the zero stationary solution. Furthermore,
we derive a formula which determines the bifurcation direction.
The proof is done by means of a Liapunov-Schmidt reduction procedure.
For this purpose, Fredholm properties of the linearized
system and implicit function
theorem techniques are used.
There are at least two distinguishing features of Hopf bifurcation theorems for hyperbolic PDEs in comparison with those for parabolic PDEs or for ODEs:
First, the question if a non-degenerate time-periodic solution depends smoothly on the system parameters
is much more delicate. And second,
a sufficient amount of dissipativity is needed in the system, and a priori
it is not clear how to verify this in terms of the data of the PDEs and of the boundary conditions.
Periodic Solutions to Dissipative Hyperbolic Systems. I: Fredholm Solvability of Linear Problems
(2013)
This paper concerns linear first-order hyperbolic systems in one space dimension of the type
$$
\partial_tu_j + a_j(x,t)\partial_xu_j + \sum\limits_{k=1}^nb_{jk}(x,t)u_k = f_j(x,t),\; x \in (0,1),\; j=1,\ldots,n,
$$
with periodicity conditions in time and reflection boundary conditions in space. We state a kind of dissipativity condition (depending on the coefficients $a_j$ and $b_{jj}$ and the boundary reflection coefficients), which implies Fredholm solvability of the problem, i.e., either there is a nontrivial solution to the homogeneous problem (in this case the space of such solutions has finite dimension) or the nonhomogeneous problem is uniquely solvable for any right-hand side (in this case the solution depends continuously on the right-hand side). In particular, under those conditions no small denominator effects occur.
Our results work for many non-strictly hyperbolic systems, but they are new even in the case of strict hyperbolicity.
Finally, in the case that all coefficients $a_j$ are $t$-independent, we show that the solutions are $C^\infty$-smooth if the data are $C^\infty$-smooth.
We describe the appearance and stability of spatio-temporal periodic
patterns (rotating waves) in unidirectional rings of coupled oscillators
with delayed couplings. We show how delays in the coupling lead
to a splitting of each rotating wave into several new ones. The appearance
of rotating waves is mediated by Hopf bifurcations of the symmetric
equilibrium.
We also conclude that the coupling delays can be effectively
replaced by increasing the number of oscillators in the chain.
The phenomena are shown for Stuart-Landau
oscillators as well as for coupled FitzHugh-Nagumo systems interacting
via excitatory chemical synapses.
Passivation of LTI systems
(2007)
In this paper we consider a passivation procedure for linear time-invariant systems.
This procedure is based on the spectral properties of related Hamiltonian matrices.
We also present a structure-preserving algorithm for computing the imaginary
eigenvalues and the corresponding eigenvectors of Hamiltonian matrices.
Numerical examples are given.
In this paper we formulate a boundary layer approximation
for an Allen--Cahn-type equation involving a small parameter $\eps$. Here, $\eps$ is related to the thickness of the boundary layer and we are interested in the limit when $\eps$ tends to $0$ in order to derive nontrivial boundary conditions. The evolution of the system is written as an energy balance formulation of the L^2-gradient flow with the corresponding Allen--Cahn energy functional. By transforming the boundary layer to a fixed domain we show the convergence of the solutions to a solution of a limit system. This is done by using concepts related to Gamma- and Mosco convergence. By considering different scalings in the boundary layer we obtain different boundary conditions.
In large-area Organic Light-Emitting Diodes (OLEDs) spatially inhomogeneous luminance at high power due to inhomogeneous current flow and electrothermal feedback can be observed. To describe these self-heating effects in organic semiconductors we present a stationary thermistor model based on the heat equation for the temperature coupled to a p-Laplace-type equation for the electrostatic potential with mixed boundary conditions. The p-Laplacian describes the non-Ohmic electrical behavior of the organic material. Moreover, an Arrhenius-like temperature dependency of the electrical conductivity is considered.
We introduce a finite-volume scheme for the system and discuss its relation to recent network models for OLEDs. In two spatial dimensions we derive a priori estimates for the temperature and the electrostatic potential and prove the existence of a weak solution by Schauder's fixed point theorem.
A mathematical model is set up that can be useful for controlled voltage excitation in time-dependent electromagnetism.
The well-posedness of the model is proved and an associated optimal control problem is investigated. Here, the control
function is a transient voltage and the aim of the control is the best approximation of desired electric and magnetic fields in
suitable $L^2$-norms.
Special emphasis is laid on an adjoint calculus for first-order necessary optimality conditions.
Moreover, a {peculiar attention is devoted to propose a formulation for which the computational complexity of the finite element solution method is substantially reduced}.
We prove an optimal regularity result for elliptic operators $-\nabla \cdot \mu \nabla:W^1,q_0 \rightarrow W^-1,q$ for a $q>3$ in the case when the coefficient function $\mu$ has a jump across a $C^1$ interface and is continuous elsewhere. A counterexample shows that the $C^1$ condition cannot be relaxed in general. Finally, we draw some conclusions for corresponding parabolic operators.
An optimal control problem is studied for a quasilinear Maxwell equation of nondegenerate parabolic
type. Well-posedness of the quasilinear state equation, existence of an optimal control, and weak G\^ateaux-differentiability of the control-to-state mapping are proved. Based on these results, first-order necessary optimality conditions and
an associated adjoint calculus are derived.
Several classes of optimal control of electromagnetic fields are considered. Special emphasis is
laid on a non-standard $H$-based formulation of the equations of electromagnetism in multiply connected conductors. By this technique, the Maxwell equations can be solved with reduced computational complexity. While the magnetic field $H$ in the conductor is obtained from an elliptic equation
with the $\curl \sigma^{-1} \curl$ operator, an elliptic equation with the $\div \mu \nabla$ operator is set up for a potential $\psi$ in the isolator.
Both equations are coupled by appropriate interface conditions. In all problems, the
electrical current is controlled in the conducting domain. Several types of control functions are discussed. In particular, the problem of sparse optimal control is investigated in a package of electrical wires. For all problems, the associated sensitivity
analysis is performed.
In this paper a general form of the infinite-horizon linear quadratic control problem is considered. We will discuss quadratic cost functionals which involve not only the state and input-variables but also derivatives of the state and input-variables of arbitrary order under constraints given by linear systems of higher order. We will examine two results that relate the linear quadratic control problem to an optimality system, which is given through a para-Hermitian matrix polynomial. The results can be applied to general rectangular descriptor systems (see Subsection 6.1) to obtain results which so far were only known for quadratic descriptor systems. Also we will see that the notion of dissipativity (when introduced in the proper way) is equivalent to the solvability of the linear quadratic control problem.
In a strong constant electric field, a dielectric particle
immersed in a weakly conducting fluid exhibits spontaneous rotations. This phenomenon is known under the name of the Quincke effect. In the original setup the particle was suspended on a silk thread and performed torsional oscillations of remarkably high amplitude.
We derive the governing equations for this experiment, and
ascertain that onset of oscillations from the quiescent state corresponds to the supercritical Hopf bifurcation.
For the case of a soft thread, we characterize the regime of large-scale torsional relaxation oscillations: explicit estimates are derived for their period and amplitude,
effects of bifurcation delay are described. In a stronger electric field, these relaxation oscillations yield to small-scale erratic rotations of the pendulum.
In a strong constant electric field, a dielectric particle immersed in a weakly conducting fluid exhibits spontaneous rotations. This phenomenon is known under the name of the Quincke effect. In the original setup the particle was suspended on a silk thread and performed torsional oscillations of remarkably high amplitude. We derive the governing equations for this experiment, and ascertain
that onset of oscillations from the quiescent state corresponds to the supercritical Hopf bifurcation.
For the case of a soft thread, we characterize the regime of large-scale torsional relaxation oscillations:
explicit estimates are derived for their period and amplitude, effects of bifurcation delay are described. In a stronger electric field, these relaxation oscillations yield to small-scale erratic rotations of the pendulum.
We prove a necessary and sufficient criterion for the exponential
stability of periodic solutions of delay differential equations with
large delay. We show that for sufficiently large delay the Floquet
spectrum near criticality is characterized by a set of curves, which we
call asymptotic continuous spectrum, that is independent on the
delay.
According to the Helmholtz decomposition,
the irrotational parts of the momentum balance equations of
the incompressible Navier-Stokes equations are balanced by the
pressure gradient.
Unfortunately, nearly all mixed methods for incompressible flows
violate this fundamental property, resulting in the well-known
numerical instability of poor mass conservation.
The origin of this problem is the
lack of $L^2$-orthogonality between discretely divergence-free velocities
and irrotational vector fields.
In order to cure this,
a new variational crime using divergence-free
velocity reconstructions is proposed.
Applying lowest order Raviart-Thomas velocity reconstructions
to the nonconforming Crouzeix-Raviart element
allows to construct
a cheap flow discretization for general 2d and 3d
simplex meshes that possesses the same
advantageous robustness properties like divergence-free flow
solvers.
In the Stokes case,
optimal a-priori error estimates for the velocity gradients
and the pressure are derived. Moreover, the discrete velocity
is independent of the continuous pressure.
Several detailed linear and nonlinear
numerical examples illustrate the theoretical findings.
Grad-div stabilization has been proved to be a very useful tool in discretizations
of incompressible flow problems. Standard error analysis for inf-sup stable conforming pairs of
finite element spaces predicts that the stabilization parameter should be optimally chosen
to be $\mathcal O(1)$. This paper revisits this choice for the Stokes equations on the basis
of minimizing the $H^1(\Omega)$ error of the velocity and the $L^2(\Omega)$ error of the pressure.
It turns out, by applying a refined error analysis, that the optimal parameter choice is more subtle
than known so far in the literature. It depends on the used norm,
the solution, the family of finite
element spaces, and the type of mesh. Depending on the situation, the
optimal
stabilization parameter might range from being very small to very large.
The analytic results
are supported by numerical examples.
In this paper we introduce a new method for the computation of KKT matrices that arise from solving constrained, nonlinear optimization problems. This method requires updating of null-space factorizations after a low rank modification. The update procedure has the advantage that it is significantly cheaper than a re-factorization of the system at each new iterate. This paper focuses on the cheap update of a rectangular LU decomposition after a rank-1 modification.
Two different procedures for updating the LU factorization are presented in detail and compared regarding their costs of computation and their stability. Moreover we will introduce an extension of these algorithms which further improves the computation time. This turns out to be an excellent alternative to algorithms based on orthogonal transformations.
We study in detail Schroedinger-type operators on a bounded interval of the real axis with dissipative boundary conditions. The characteristic function of such operators is computed, its minimal self-adjoint dilation is constructed and the generalized eigenfunction expansion for the dilation is developed. The problem is motivated by semiconductor physics.
Classical gradient systems have a linear relation between rates and driving forces. In generalized gradient systems we allow for arbitrary relations derived from general non-quadratic dissipation potentials. This paper describes two natural origins for these structures.
A first microscopic origin of generalized gradient structures is given by the theory of large-deviation principles. While Markovian diffusion processes lead to classical gradient structures, Poissonian jump processes give rise to cosh-type dissipation potentials.
A second origin arises via a new form of convergence, that we call EDP-convergence. Even when starting with classical gradient systems, where the dissipation potential is a quadratic functional of the rate, we may obtain a generalized gradient system in the evolutionary -limit. As examples we treat (i) the limit of a diffusion equation having a thin layer of low diffusivity, which leads to a membrane model, and (ii) the limit of difusion over a high barrier, which gives a reaction-diffusion system.
Recent research has shown that
in some practically relevant situations like multi-physics flows[11]
divergence-free mixed finite elements may have a significantly
smaller discretization error than standard non-divergence-free
mixed finite elements. In order to judge the overall performance of
divergence-free mixed finite elements, we
investigate linear solvers for the saddle point linear systems arising in $((P_k)^d,P_{k-1}^{disc})$ Scott-Vogelius finite element implementations of the incompressible Navier-Stokes equations. We investigate both direct and iterative solver methods.
Due to discontinuous pressure elements in the case of Scott-Vogelius elements, considerably more solver strategies seem to deliver promising results than in the case of standard mixed finite elements like
Taylor-Hood elements. For direct methods, we extend recent preliminary work using sparse banded solvers on the penalty method formulation to finer meshes, and discuss extensions. For iterative methods, we test augmented Lagrangian and H-LU preconditioners with GMRES, on both full and statically condensed systems.
Several numerical experiments are provided that show these classes of solvers are well suited for use with Scott-Vogelius elements, and could deliver an interesting overall performance in several applications.
In this paper a high-order finite element method with curvilinear elements is proposed for the simulation of plasmonic structures. Most finite element packages use low order basis functions and non-curved elements, which is very costly for demanding problems such as the simulation of nano-antennas. To enhance the performance of finite elements, we use curvilinear quadrilateral elements to calculate the near-field from an impinging plane wave with
second order absorbing boundary conditions. The magnetic field amplitude on the surface of one object is compared with a computation based on a multiple multipole expansion. Moreover, the convergence behavior of p-FEM with absorbing boundary conditions motivate an adaptive strategy of polynomial degree enhancement and enlargement of the domain.
We study the approximation of Wasserstein gradient structures
by their finite-dimensional analog. We show that simple
finite-volume discretizations of the linear Fokker-Planck
equation exhibit the recently established entropic gradient-flow
structure for reversible Markov chains. Then we reprove the
convergence of the discrete scheme in the limit of vanishing
mesh size using only the involved gradient-flow structures.
In particular, we make no use of the linearity of the equations
nor of the fact that the Fokker-Planck equation is of second order.
We introduce a new operator for stabilizing error that arises from the weak enforcement of mass conservation in finite element simulations of incompressible flow problems. We show this new operator has a similar positive effect on velocity error as the well-known and very successful grad-div stabilization operator, but the new operator is more attractive from an implementation standpoint because it yields a sparser block structure matrix. That is, while grad-div produces fully coupled block matrices (i.e. block-full), the matrices arising from the new operator are block-upper triangular in two dimensions, and in three dimensions the 2,1 and 3,1 blocks are empty. Moreover, the diagonal blocks of the new operator's matrices are identical to those of grad-div. We provide error estimates and numerical examples for finite element simulations with the new operator, which reveals the significant improvement in accuracy it can provide. Solutions found using the new operator are also compared to those using usual grad-div stabilization, and in all cases, solutions are found to be very similar.
The computation of guided modes in photonic crystal wave-guides is a key issue in the process of designing devices in photonic communications. Existing methods, such as the super-cell method, provide an efficient computation of well-confined modes. However, if the modes are not well-confined, the modelling error of the super-cell method becomes prohibitive and advanced methods applying transparent boundary conditions for periodic media are needed. In this work we demonstrate the numerical realization of a recently proposed Dirichlet-to-Neumann approach and compare the results with those of the super-cell method. For the resulting non-linear eigenvalue problem we propose an iterative solution based on Newton's method and a direct solution using Chebyshev interpolation of the non-linear operator. Based on the Dirichlet-to-Neumann approach, we present a formula for the group velocity of guided modes that can serve as an objective function in the optimization of photonic crystal wave-guides.
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.
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.
We consider linear differential-algebraic m-input m-output systems with positive
strict relative degree or proper inverse transfer function; in the single-input single-output case these
two disjoint classes make the whole of all linear DAEs without feedthrough term. Structural properties
- such as normal forms (i.e. the counterpart to the Byrnes-Isidori form for ODE systems), zero
dynamics, and high-gain stabilizability - are analyzed for two purposes: first, to gain insight into the
system classes and secondly, to solve the output regulation problem by funnel control. The funnel
controller achieves tracking of a class of reference signals within a pre-specified funnel; this means in
particular, the transient behaviour of the output error can be specified and the funnel controller does
neither incorporate any internal model for the reference signals nor any identification mechanism, it
is simple in its design. The results are illuminated by position and velocity control of a mechanical
system encompassing springs, masses, and dampers.
This work 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.
Here we construct with the Z-transformation
so-called discrete transparent
boundary conditions
for higher-order parabolic equations schemes.
These methods are Pade ``Parabolic'' approximations of
the one-way Helmholtz equation
and frequently used in integrated optics and (underwater) acoustics.
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.
The article investigates the relation between global solutions of hyperbolic balance laws and viscous balance laws on the circle. It is thematically located at the crossroads of hyperbolic and parabolic partial differential equations with one-dimensional space variable and periodic boundary conditions. The two equations are given by:
u_t+f(u)_x=g(u)
and
u_t+f(u)_x=e u_{xx}+g(u).
The main result of the paper corrects a result on the persistence of heteroclinic connections by Fan and Hale from 1995 when viscosity vanishes: The "Connection Lemma" states that a connection can only persist if the zero number of the source state is a multiple of the zero number of the target state. The "Cascading Theorem" then yields convergence of heteroclinic connections to a sequence of heteroclinic connections and stationary solutions in case of non-persistence.
In addition a full description of the connection problem of rotating waves on the parabolic attractor is given.
We study optimal control problems for general unstructured nonlinear differential-algebraic equations of arbitrary index.
In particular, we derive necessary conditions in the case of linear-quadratic control problems and extend them to the general nonlinear case.
We also present a Pontryagin maximum principle for general unstructured nonlinear DAEs in the case of restricted controls.
Moreover, we discuss the numerical solution of the resulting two-point boundary value problems and present a numerical example.
In this paper the numerical approximation of solutions of Itô stochastic differential
equations is considered, in particular for equations with a small parameter ? in the noise coex-
cient. We construct stochastic linear multi-step methods and develop the fundamental numerical
analysis concerning their mean-square consistency, numerical stability in the mean-square sense and
mean-square convergence. For the special case of two-step Maruyama schemes we derive conditions
guaranteeing their mean-square consistency. Further, for the small noise case we obtain expansions
of the local error in terms of the stepsize and the small parameter ?. Simulation results using several
explicit and implicit stochastic linear k-step schemes, k = 1; 2, illustrate the theoretical findings.
We investigate different dynamical regimes of neuronal network in the CA3 area of the hippocampus.
The proposed neuronal circuit includes two fast- and two slowly-spiking cells which are interconnected by means of dynamical synapses. On the individual level, each neuron is modeled by FitzHugh-Nagumo equations. Three basic rhythmic patterns are observed: gamma-rhythm in which the fast neurons are uniformly spiking, theta-rhythm in which the individual spikes are separated by quiet epochs, and theta/gamma rhythm with repeated patches of spikes. We analyze the influence of asymmetry of synaptic strengths on the synchronization in the network and demonstrate that
strong asymmetry reduces the variety of available dynamical states. The model network exhibits multistability; this results in occurrence of hysteresis in dependence on the conductances of individual connections. We show that switching between different rhythmic patterns in the network
depends on the degree of synchronization between the slow cells.
Model Reduction for a Class of Nonlinear Electrical Circuits by Reduction of Linear Subcircuits
(2010)
We analyze a model reduction approach for a class of electrical circuits containing nonlinear resistance.
In our approach
the linear subcircuits are extracted and replaced with linear passive reduced-order models. The resulting nonlinear
reduced-order model preserves admissibility and passivity. Moreover, we derive a-priori bounds for the error between
the input-output map of the original circuit equations and that of our reduced-order model. These bounds
are valid for all inputs. Since only linear subcircuits are reduced, our approach is effective when the number of nonlinear resistances is relatively small. The performance of our approach is illustrated numerically.
We show that elliptic second order operators $A$ of divergence type fulfill maximal parabolic regularity on distribution spaces, even if the underlying domain is highly non-smooth and $A$ is complemented with mixed boundary conditions. Applications to quasilinear parabolic equations with non-smooth data are presented.
Many problems in science and engineering require the evaluation of functionals of
the form $F(A) = u^T f(A)u$, where $A$ is a large symmetric matrix, $u$ a vector, and $f$ a nonlinear
function. A popular and fairly inexpensive approach to determining upper and lower bounds for
such functionals is based on first carrying out a few steps of the Lanczos procedure applied to $A$ with
initial vector $u$, and then evaluating pairs of Gauss and Gauss-Radau quadrature rules associated
with the tridiagonal matrix determined by the Lanczos procedure. The present paper extends this
approach to allow the use of rational Gauss quadrature rules.
We describe an embedding of a quantum mechanically described structure into a macroscopic flow. The open quantum system is partly driven by an adjacent macroscopic flow acting on the boundary of the bounded spatial domain designated to quantum mechanics. This leads to an essentially non-selfadjoint Schroedinger-type operator, the spectral properties of which will be investigated.
In this work we consider the so-called Lur'e matrix equations that arise e.g. in model reduction and linear-quadratic infinite time horizon optimal control. We characterize the set of solutions in terms of deflating subspaces of even matrix pencils. In particular, it is shown that there exist solutions which are extremal in terms of definiteness. It is shown how these special solutions can be constructed deflating subspaces of even matrix pencils.
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.
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.
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.
Local existence, uniqueness and smooth dependence for nonsmooth quasilinear parabolic problems
(2009)
We prove local existence, uniqueness, Hölder regularity in space and time, and smooth dependence in Hölder spaces for a general class of quasilinear parabolic initial boundary value problems with nonsmooth data.
As a result the gap between low smoothness of the data, which is typical for
many applications, and high smoothness of the solutions, which is necessary
for the applicability of differential calculus to abstract formulations of the
initial boundary value problems, has been closed. The theory works for any
space dimension, and the nonlinearities are allowed to be nonlocal and to
have any growth. The main tools are new maximal regularity results [19, 20]
in Sobolev–Morrey spaces for linear parabolic initial boundary value problems
with nonsmooth data, linearization techniques and the Implicit Function Theorem.
Studying high-dimensional Hamiltonian systems with microstructure, it is an important and challenging problem to identify reduced macroscopic models that describe some effective dynamics on large spatial and temporal scales. This paper concerns the question how reasonable macroscopic Lagrangian and Hamiltonian structures can by derived from the microscopic system. In the first part we develop a general approach to this problem by considering non-canonical Hamiltonian structures on the tangent bundle. This approach can be applied to all Hamiltonian lattices (or Hamiltonian PDEs) and involves three building blocks: (i) the embedding of the microscopic system, (ii) an invertible two-scale transformation that encodes the underlying scaling of space and time, (iii) an elementary model reduction that is based on a Principle of Consistent Expansions. In the second part we exemplify the reduction approach and derive various reduced PDE models for the atomic chain. The reduced equations are either related to long wave-length motion or describe the macroscopic modulation of an oscillatory microstructure.
We consider the numerical solution of projected Lyapunov equations using Krylov subspace iterative methods. Such equations
play a fundamental role in balanced truncation model reduction of descriptor systems. We present generalizations of
the extended block and global Arnoldi methods to projected Lyapunov equations and compare these methods with the alternating direction implicit method with respect to performance on different examples.
A deflation strategy is also proposed to overcome possible breakdown in the
recurrence.
Transient analysis in industrial chip design leads to very large systems
of differential-algebraic equations (DAEs). The numerical solution of these
DAEs strongly depends on the so called index of the DAE. In general,
the higher the index of the DAE is, the more sensitive the numerical
solution will be to errors in the computation. So, it is advisable to use
mathematical models with small index or to reduce the index.
This paper presents an index reduction method that uses information
based on the topology of the circuit. In addition, we show that the
presented method retains structural properties of the DAE.
We explore the concept of passive-feedback lasers for direct signal
modulation at 40 Gbit/s. Based on numerical simulation and bifurcation
analysis, we explain the main mechanisms in these devices which are
crucial for modulation at high speed. The predicted effects are
demonstrated experimentally by means of correspondingly designed
devices. In particular a significant improvement of the modulation
bandwidth at low injection currents can be demonstrated.
The well known De Giorgi result on Hölder continuity for solutions of the Dirichlet problem is re-established for mixed boundary value problems, provided that the underlying domain is a Lipschitz domain and the border between the Dirichlet and the Neumann boundary part satisfies a very general geometric condition. Implications of this result for optimal control theory are presented.
We consider hybrid systems of differential-algebraic equations and present
a general framework for general nonlinear over- and underdetermined hybrid
systems that allows the
analysis of existence and uniqueness and the application of index reduction
methods for hybrid differential-algebraic systems.
A particular difficulty in the numerical simulation of hybrid systems is
(numerical) chattering, i.e., fast oscillations between modes of operations.
A regularization technique using sliding modes allows to regularize the
system behavior in the case of chattering.
Further, we show how chattering behavior during the numerical solution can
be prevented using sliding mode simulation. The advantage of the sliding mode
simulation is illustrated by numerical examples.
In this paper, we introduce a high order finite element (FEM) implementation using perfectly matched layer (PML) for the scattering by plasmonic structures inside layered media. The PML is proven to be very accurate and efficient by a comparative analysis with a commercial FEM software and the Multiple Multipole Program (MMP). A convergence analysis using hp-adaptive refinement inside the PML layer
shows that adaptive mesh refinement inside the PML layer is most efficient. Based on this convergence analysis an hp-strategy is proposed, which shows a remarkable error reduction for small additional computational costs.
In these notes we discuss two approaches to evolutionary Γ- convergence of gradient systems in Hilbert spaces. The formulation of the gradient system is based on two functionals, namely the energy functional and the dissipation potential, which allows us to employ Γ- convergence methods. In the first approach we consider families of uni- formly convex energy functionals such that the limit passage of the time-dependent problems can be based on the theory of evolutionary variational inequalities as developed by Daneri and Savar ́e 2010. The second approach uses the equivalent formulation of the gradient system via the energy-dissipation principle and follows the ideas of Sandier and Serfaty 2004.
We apply both approaches to rigorously derive homogenization limits for Cahn–Hilliard-type equations. Using the method of weak and strong two-scale convergence via periodic unfolding, we show that the energy and dissipation functionals Γ-converge. In conclusion, we will give specific examples for the applicability of each of the two approaches.
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.
We propose transmission conditions of order $1$, $2$ and $3$ approximating the shielding behaviour of thin conducting curved sheets for the magneto-quasistatic eddy current model in 2D. This model reduction applies to sheets whose thicknesses $\eps$ are at the order of the skin depth or essentially smaller. The sheet has itself not to be resolved, only its midline is represented by an interface. The computation is directly in one step with almost no additional cost. We prove the well-posedness w.r.t.~to the small parameter $\eps$ and obtain optimal bound for the modelling error outside the sheet of order $\eps^{N+1}$ for the condition of order $N$. We end the paper with numerical experiments involving high order finite elements for sheets with varying curvature.