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Veröffentlichungen am Institut für Mathematik seit 1990

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Bomze, Immanuel M.; Eichfelder, Gabriele;
Copositivity detection by difference-of-convex decomposition and [omega]-subdivision. - In: Mathematical programming : Series A, Series B ; a publication of the Mathematical Programming Society.. - Berlin : Springer, ISSN 1436-4646, Bd. 138 (2013), 1/2, S. 365-400

We present three new copositivity tests based upon difference-of-convex (d.c.) decompositions, and combine them to a branch-and-bound algorithm of [omega]-subdivision type. The tests employ LP or convex QP techniques, but also can be used heuristically using appropriate test points. We also discuss the selection of efficient d.c. decompositions and propose some preprocessing ideas based on the spectral d.c. decomposition. We report on first numerical experience with this procedure which are very promising.
Berger, Thomas;
Zero dynamics and funnel control of general linear differential-algebraic systems. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2013. - Online-Ressource (PDF-Datei: 44 S., 438 KB). . - (Preprint. - M13,04)
Philipp, Friedrich; Trunk, Carsten
The numerical range of non-negative operators in Krein spaces. - In: Linear algebra and its applications : LAA.. - New York, NY : American Elsevier Publ., Bd. 438 (2013), 5, S. 2542-2556

We define and characterize the Krein space numerical range $W(A)$ and the Krein space co-numerical range $W_{\rm co}(A)$ of a non-negative operator $A$ in a Krein space. It is shown that the non-zero spectrum of $A$ is contained in the closure of $W(A)\cap W_{\rm co}(A)$.
Behrndt, Jussi; Luger, Annemarie; Trunk, Carsten
On the negative squares of a class of self-adjoint extensions in Krein spaces. - In: Mathematische Nachrichten. - [S.l.] : Wiley-VCH, ISSN 1522-2616, Bd. 286 (2013), 2/3, S. 118-148

A description of all exit space extensions with finitely many negative squares of a symmetric operator of defect one is given via Krein's formula. As one of the main results an exact characterization of the number of negative squares in terms of a fixed canonical extension and the behaviour of a function $\tau$ (that determines the exit space extension in Krein's formula) at zero and at infinity is obtained. To this end the class of matrix valued $\mathcal D_\kappa^{n\times n}$-functions is introduced and, in particular, the properties of the inverse of a certain $\mathcal D_\kappa^{2\times 2}$-function which is closely connected with the spectral properties of the exit space extensions with finitely many negative squares is investigated in detail. Among the main tools here are the analytic characterization of the degree of non-positivity of generalized poles of matrix valued generalized Nevanlinna functions and some extensions of recent factorization results.
Philipp, Friedrich; Strauss, Vladimir; Trunk, Carsten
Local spectral theory for normal operators in Krein spaces. - In: Mathematische Nachrichten. - [S.l.] : Wiley-VCH, ISSN 1522-2616, Bd. 286 (2013), 1, S. 42-58

Sign type spectra are an important tool in the investigation of spectral properties of selfadjoint operators in Krein spaces. It is our aim to show that also sign type spectra for normal operators in Krein spaces provide insight in the spectral nature of the operator: If the real part and the imaginary part of a normal operator in a Krein space have real spectra only and if the growth of the resolvent of the imaginary part (close to the real axis) is of finite order, then the normal operator possesses a local spectral function defined for Borel subsets of the spectrum which belong to positive (negative) type spectrum. Moreover, the restriction of the normal operator to the spectral subspace corresponding to such a Borel subset is a normal operator in some Hilbert space. In particular, if the spectrum consists entirely out of positive and negative type spectrum, then the operator is similar to a normal operator in some Hilbert space. We use this result to show the existence of operator roots of a class of quadratic operator polynomials with normal coefficients.
Eichfelder, Gabriele;
Ordering structures in vector optimization and applications in medical engineering. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2013. - Online-Ressource (PDF-Datei: 33 S., 397,8 KB). . - (Preprint. - M13,01)

This manuscript is on the theory and numerical procedures of vector optimization w.r.t. various ordering structures, on recent developments in this area and, most important, on their application to medical engineering. In vector optimization one considers optimization problems with a vector-valued objective map and thus one has to compare elements in a linear space. If the linear space is the finite dimensional space R^m this can be done componentwise. That corresponds to the notion of an Edgeworth-Pareto-optimal solution of a multiobjective optimization problem. Among the multitude of applications which can be modeled by such a multiobjective optimization problem, we present an application in intensity modulated radiation therapy and its solution by a numerical procedure. In case the linear space is arbitrary, maybe infinite dimensional, one may introduce a partial ordering which defines how elements are compared. Such problems arise for instance in magnetic resonance tomography where the number of Hermitian matrices which have to be considered for a control of the maximum local specific absorption rate can be reduced by applying procedures from vector optimization. In addition to a short introduction and the application problem, we present a numerical solution method for solving such vector optimization problems. A partial ordering can be represented by a convex cone which describes the set of directions in which one assumes that the current values are deteriorated. If one assumes that this set may vary dependently on the actually considered element in the linear space, one may replace the partial ordering by a variable ordering structure. This was for instance done in an application in medical image registration. We present a possibility of how to model such variable ordering structures mathematically and how optimality can be defined in such a case. We also give a numerical solution method for the case of a finite set of alternatives.
Ilchmann, Achim; Wirth, Fabian
On minimum phase. - Ilmenau : Techn. Univ., Inst. für Mathematik, 2013. - Online-Ressource (PDF-Datei: 10 S., 270 KB). . - (Preprint. - M13,02)

Wir diskutieren Minimalphasigkeit von schwach-stabilen Transferfunktionen; letzteres sind rationale Funktionen, bei denen das Nennerpolynom Nullstellen in der abgeschlossenen linken komplexen Halbebene hat. Minimalphasigkeit wird hier mittels der Ableitung der Argumentfunktion der Transferfunktion definiert. Es wird dann mit Hilfe der Hurwitz-Reflektion gezeigt, daß jede schwach-stabile Transferfunktion eindeutig in ein Produkt von Allpass und minimalphasiger Funktion zerlegt werden kann. Das wesentliche Resultat ist, daß eine schwach-stabile Transferfunktion minimalphasig ist genau dann, wenn das Zählerpolynom der Transferfunktion schwach-stabil ist. Ein weiteres Resultat ist, daß die Nulldynamik einer minimalen Realisation asymptotisch stabil ist genau dann, wenn das Zählerpolynom der Transferfunktion Hurwitz ist. Insbesondere folgt aus asymptotisch stabiler Nulldynamik die Minimalphasigkeit, aber keineswegs umgekehrt. Abschließend zeigen wir, daß ein minimalphasiges System als kanonischer Repräsentant innerhalb der Äquivalenzklasse aller Systeme mit identischem Betragsverhalten interpretiert werden kann.
Ilchmann, Achim; Reis, Timo
. - Surveys in differential-algebraic equations ; 1. - Berlin [u.a.] : Springer, 2013. - VII, 231 S.. . - (Differential-algebraic equations forum, DAE-F) ISBN 3642349277 = 978-3-642-34927-0
- Literaturangaben

Reinhardt, Rüdiger; , ; ,
Nichtlineare Optimierung : Theorie, Numerik und Experimente. - Berlin : Springer Spektrum, 2013. - X, 383 S.. ISBN 3-8274-2948-X
- Literaturverz. S. [371] - 375

Die Grundlagen zur nichtlinearen Optimierung insbesondere ausgewählte Verfahren der nichtlinearen Optimierung werden als Lehrbuch für Studenten dargestellt. Zu den Verfahren werden Experimente beschrieben, die anhand der von R. Reinhardt entwickelten und von A. Hoffmann erweiterten Lehrsoftware EdOptLab unter Matlab von jedem Leser nachvollzogen werden können. So kann der Leser die Verfahren der Optimierung selbst erleben. Das System EdOptlab wird kostenlos zur Verfügung gestellt.

Berger, Thomas;
Bohl exponent for time-varying linear differential-algebraic equations. - In: International journal of control. - London : Taylor & Francis, ISSN 1366-5820, Bd. 85 (2012), 10, S. 1433-1451