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Zonotopes With Large 2D Cuts
(2009)
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.
The timetable is the essence of the service offered by any provider
of public transport'' (Jonathan Tyler, CASPT 2006). Indeed, the
timetable has a major impact on both operating costs and on passenger
comfort. Most European agglomerations and railways use periodic timetables in which operation repeats in regular intervals. In contrast, many North and South American municipalities use trip timetables in which the vehicle trips are scheduled individually subject to frequency constraints. We compare these two
strategies with respect to vehicle operation costs. It turns out that
for short time horizons, periodic timetabling can be suboptimal; for
sufficiently long time horizons, however, periodic timetabling can
always be done in an optimal way'.
We study a diffusion model of phase field type, consisting of
a system of two partial differential equations encoding the balances
of microforces and microenergy; the two unknowns are the order parameter
and the chemical potential. By a careful development of uniform estimates and
the deduction of certain useful boundedness properties, we
prove existence and uniqueness of a global-in-time
smooth solution to the associated initial/boundary-value problem;
moreover, we give a description of the relative $\omega$-limit set.
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.
Den kürzesten Weg in einem Graphen zu finden ist ein klassisches Problem der Graphentheorie. Über einen Vortrag zu diesem Thema beim Tag der Mathematik 2007 von R. Borndörfer kam ich in Kontakt mit dem Konrad-Zuse-Zentrum (ZIB), das sich u.a. mit Wegeoptimierung beschäftigt. Ein Forschungsschwerpunkt dort ist im Rahmen eines Projekts zur Chipverifikation das Zählen von Lösungen, das, wie wir sehen werden, eng mit dem Zählen von Wegen zusammenhängt.
Anhand von zwei Fragen aus der Graphentheorie soll diese Facharbeit unterschiedliche Lösungsmethoden untersuchen. Wie bestimmt man den kürzesten Weg zwischen zwei Knoten in einem Graphen und wie findet man alle möglichen Wege?
Nach einer Einführung in die Graphentheorie und einer Konkretisierung der Probleme wird zunächst für beide eine Lösung mit auf Graphen basierenden Algorithmen vorgestellt. Während der Algorithmus von Dijkstra sehr bekannt ist, habe ich für das Zählen von Wegen einen eigenen Algorithmus auf der Basis der Tiefensuche entwickelt.
Im zweiten Teil der Arbeit wird das Konzept der ganzzahligen Programmierung vorgestellt und die Lösungsmöglichkeiten für Wegeprobleme, die sich darüber ergeben.
In den letzten Jahren ist die Bedeutung computerunterstützter Darstellungen von Mathematik
im Lern- als auch im Forschungsbereich stark gestiegen. Obwohl bereits massive Anstrengungen
unternommen werden, die Mathematikausbildung im Ingenieursbereich durch Neue
Medien zu unterstützen und auszubauen[Mum], gibt es dennoch nur eine geringe Zahl von
Projekten, die auf die Studenten der Mathematik und Physik abzielen. Ferner beschränkten
sich viele Projekte nur auf die Verwaltung von Dokumenten, die den Lernenden — obgleich
teilweise aufgelockert durch eingestreute aktive Inhalte—zu einem passiven Konsumenten des
Lerninhaltes machen und die darum selten geeignet sind, die eigentständige, selbstgesteuerte
Auseinandersetzung mit der Materie zu fördern.
Wir präsentieren in diesem Artikel das Konzept des „Virtuellen Labors“, welches die Metapher
eines Laborpraktikums innerhalb eines Computernetzwerkes nachbildet und damit das zweite
Standbein der universitären Ausbildung in die Neuen Medien abbildet. Das Labor „Cinderella“
[KRG04] zur Untersuchung euklidischer und nicht-euklidischer Geometrie mag hier als ein
Beispiel dienen.
Wir werden im ersten Teil ein didaktisches Anforderungsprofil an derartige Labore definieren,
dann in einem zweiten Teil die sich daraus untersuchenden Konsequenzen für die Software-
Architektur darstellen und im dritten Teil ein am DFG-ForschungzentrumMATHEON der Berliner
Universitäten entwickeltes Labor für Elemente der statistischen Mechanik präsentieren.
This paper provides a generic formulation for rolling stock planning problems in the context of intercity passenger traffic. The main contributions are a graph theoretical model and a Mixed-Integer-Programming formulation that integrate all main requirements of the considered Vehicle-Rotation-Planning problem (VRPP). We show that it is possible to solve this model for real-world instances provided by our industrial partner DB Fernverkehr AG using modern algorithms and computers.
The classical approach to investigating polynomial eigenvalue problems is linearization, where the
polynomial is converted into a larger matrix pencil with the same eigenvalues. For any polynomial there are infinitely
many linearizations with widely varying properties, but in practice the companion forms are typically used. However,
these companion forms are not always entirely satisfactory, and linearizations with special properties may sometimes
be required.
In this paper we develop a systematic approach to generating large classes of linearizations for matrix polynomials.
Given a polynomial P, we show how to simply construct two vector spaces of pencils that generalize the companion
forms of P, and prove that almost all of these pencils are linearizations for P. Eigenvectors of these pencils are
shown to be closely related to those of P. A distinguished subspace is then isolated, and the special properties of
these pencils are investigated. These spaces of pencils provide a convenient arena in which to look for structured
linearizations of structured polynomials, as well as to try to optimize the conditioning of linearizations, issues to be
addressed in further work.
The adaptive weights smoothing (AWS) procedure was introduced in
Polzehl and Spokoiny (2000) in the context of image denoising. The
procedure has some remarkable properties like preservation of edges and
contrast, and (in some sense) optimal reduction of noise. The procedure
is fully adaptive and dimension free. Simulations with artificial images
show that AWS is superior to classical smoothing techniques especially
when the underlying image function is discontinuous and can be well
approximated by a piecewise constant function. However, the latter as-
sumption can be rather restrictive for a number of potential applications.
Here the AWS method is generalized to the case of an arbitrary local lin-
ear parametric structure. We also establish some important results about
properties of the AWS procedure including the so called "propagation
condition" and spatial adaptivity. The performance of the procedure is
illustrated by examples for local polynomial regression in univariate and
bivariate situations.
We formulate the static mechanical coupling of a geometrically exact Cosserat rod
to a nonlinearly elastic continuum. In this setting, appropriate coupling conditions have
to connect a one-dimensional model with director variables to a three-dimensional
model without directors.
Two alternative coupling conditions are proposed,
which correspond to two different configuration trace spaces.
For both we show existence of solutions of the coupled problems, using the direct
method of the calculus of variations. From the first-order optimality conditions
we also derive the corresponding conditions for the dual variables. These are
then interpreted in mechanical terms.
We consider the problem of utility maximization for small traders on incomplete
financial markets. As opposed to most of the papers dealing with this
subject, the investors’ trading strategies we allow underly constraints described
by closed, but not necessarily convex, sets. The final wealths obtained by trading
under these constraints are identified as stochastic processes which usually are
supermartingales, and even martingales for particular strategies. These strategies
are seen to be optimal, and the corresponding value functions determined
simply by the initial values of the supermartingales. We separately treat the
cases of exponential, power and logarithmic utility.
Under market frictions like illiquidity or transaction costs, contingent claims
can incorporate some inevitable intrinsic risk that cannot be completely hedged
away but remains with the holder. In general, they cannot be synthesized by
dynamical trading in liquid assets and hence not be priced by no-arbitrage arguments alone. Still, an agent can determine a valuation with respect to her
preferences towards risk. The utility indifference value for a variation in the
quantity of illiquid assets held by the agent is defined as the compensating variation
of wealth, under which her maximal expected utility remains unchanged.
In this work numerical methods for the solution of two classes of structured generalized eigenvalue problems, $Ax=\lambda Bx$, are developed. Those classes are the palindromic ($B=A^T$) and the even ($A=A^T$, $B=-B^T$) eigenvalue problems.
The spectrum of these problems is not arbitrary, rather do eigenvalues occur in pairs.
We will construct methods for palindromic and even eigenvalue problems that are of cubic complexity and that are guaranteed to produce eigenvalues that are paired to working precision.
At the heart of both methods is a new URV-type matrix decomposition, that simultaneously transforms three matrices to skew triangular form, i.e., to a form that is triangular with respect to the Northeast-Southwest diagonal.
The algorithm to compute this URV decomposition uses several other methods to reduce a single square matrix to skew triangular form: the skew QR factorization and the skew QRQ$^T$ decomposition. Moreover, a method to compute the singular value decomposition of a complex, skew symmetric matrix is presented and used.
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.
We study discrete curvatures computed from nets of curvature lines on a given smooth surface and prove their uniform convergence to smooth principal curvatures. We provide explicit error bounds, with constants depending only on properties of the smooth limit surface and the shape regularity of the discrete net.
We present Undercover, a primal heuristic for nonconvex mixed-integer nonlinear programming (MINLP) that explores a mixed-integer linear subproblem (sub-MIP) of a given MINLP. We solve a vertex covering problem to identify a minimal set of variables that need to be fixed in order to linearize each constraint, a so-called cover. Subsequently, these variables are fixed to values obtained from a reference point, e.g., an optimal solution of a linear relaxation. We apply domain propagation and conflict analysis to try to avoid infeasibilities and learn from them, respectively. Each feasible solution of the sub-MIP corresponds to a feasible solution of the original problem.
We present computational results on a test set of mixed-integer quadratically constrained programs (MIQCPs) and general MINLPs from MINLPLib. It turns out that the majority of these instances allow for small covers. Although general in nature, the heuristic appears most promising for MIQCPs, and complements nicely with existing root node heuristics in different state-of-the-art solvers.
We present Undercover, a primal heuristic for mixed-integer nonlinear programming (MINLP). The heuristic constructs a mixed-integer linear subproblem (sub-MIP) of a given MINLP by fixing a subset of the variables. We solve a set covering problem to identify a minimal set of variables which need to be fixed in order to linearise each constraint. Subsequently, these variables are fixed to approximate values, e.g. obtained from a linear outer approximation. The resulting sub-MIP is solved by a mixed-integer linear programming solver. Each feasible solution of the sub-MIP corresponds to a feasible solution of the original problem. Although general in nature, the heuristic seems most promising for mixed-integer quadratically constrained programmes (MIQCPs). We present computational results on a general test set of MIQCPs selected from the MINLPLib.