Kolloquium der Arbeitsgruppe Modellierung, Numerik, Differentialgleichungen SS 2005

Interessenten sind herzlich eingeladen!

Weitere Vorträge siehe auch:

• Diplomanden- und Doktorandenseminar Numerische Mathematik
• Seminar des Fachgebiets Optimierung bei partiellen Differentialgleichungen

Rückblick: (auf das Numerik-Oberseminar, den Vorläufer dieses Kolloquiums)

• Numerik-Oberseminar WS 2004/05
• Numerik-Oberseminar SS 2004
• Numerik-Oberseminar WS 2003/04
• Numerik-Oberseminar SS 2003
• Numerik-Oberseminar WS 2002/03
• Numerik-Oberseminar SS 2002
• Numerik-Oberseminar WS 2001/02
• Numerik-Oberseminar SS 2001

Abstracts zu den Vorträgen:

Abstract:
Two solution approaches for optimal control problems subject to DAE systems will be discussed.
The indirect approach is based on the evaluation of necessary conditions and exploits a local minimum principle, which is derived for semi-explicit index-2 differential-algebraic equations, mixed control-state constraints, and pure state constraints. The local minimum principle is based on necessary optimality conditions for general infinite optimization problems. The structure of the optimal control problem under consideration is exploited and allows to obtain more regular representations for the multipliers involved.
The direct approach is based on a discretization of the optimal control problem. Strategies for the mandatory calculation of gradients within the numerical solution of the discretized problem are discussed.
Particular areas of application are mechanical multibody systems in Gear-Gupta-Leimkuhler formulation and the simulation of test-drives.

Abstract:
Es ist bekannt, dass in turbulenten Strömungen durch Wandrauigkeiten eine Widerstandsreduktion erzielt werden kann (shark skin effect). Kann man auch in schleichenden Strömungen eine Widerstandsreduktion erreichen? Diese Frage wird beantwortet im Falle von schleichenden Filmströmungen. Durch Bodenwelligkeiten werden Rezirkulationsgebiete erzeugt, die wie flüssige Gleitlager reagieren. Die mathematisch- analytische Behandlung des Problems erfordert unkonventionelle Vorgehensweisen.

Nach Vorstellung der Methoden werden einige Beispiele gezeigt und experimentell getestet. Schließlich wird ein erster Schritt für eine Optimierungsaufgabe vorgestellt.

Abstract:
We present a functional version of the operator theoretic technique of the commutant lifting, providing constructions of contractive analytic Toeplitz operators whose symbols satisfy certain interpolation conditions. Also, we give applications to interpolation problems of Nevanlinna-Pick and Caratheodory-Fejer type over analytic polyhedra.

Abstract:
Degenerate parabolic equations are widely used to describe diverse physical phenomena in such fields as combustions, biology, chemistry, metallurgy, medicine, and fluid mechanics. One representative example is the well-known Navier--Stokes equations. These systems consist of partial differential, ordinary differential, and algebraic equations and are often called partial differential algebraic equations (PDAEs).

First a class of general linear PDAEs with diffusion-, convection-, and reactions-terms plus appropriate boundary conditions (Dirichlet, Neumann or mixed) is stated. The PDAE is transformed into a variational problem and this so-called weak problem distinguishes between the operator of the time derivative and the operator with the space derivatives. The second one should satisfy a Gårding-type inequality.

The perturbation index is defined in three different settings. First this index is considered for the variational problem in the Hilbert space V. As in the theory of (conformal) finite elements a perturbation index can be defined in some subspace V_h of V. If the basis-functions of V_h are given, the variational problem can be transformed into a MOL-DAE. For this problem a third perturbation index is defined. Various examples (parabolic, hyperbolic and mixed parabolic-hyperbolic systems) illustrate these perturbations indices.

Abstract:
We discuss numerical methods for higher order DAEs and nonlinear eigenvalue problems. The motivation for the work comes from an industrial problem of reducing the noise emission in German high speed trains. The vibration analysis of the rail track led to an extremely difficult structured quadratic eigenvalue problem (with palindromic structure). The industrial packages were not able to generate any correct digit for the eigenvalues in double precision.

We show how an analysis of the structures, the construction of normal forms and of new linearization techniques allows to solve this problem.

Understanding the mathematics behind the problem makes the difference, but moreover, the problem motivated the development of a whole new theory and of matrix theoretic techniques.

Abstract:
Although general linear methods were originally introduced as a unifying framework to relate concepts such as consistency, stability and convergence, there now seem to be prospects that practical methods within this wide class can be found.

This talk will survey some of the theoretical ideas behind general linear methods and will introduce some special methods and families of methods within this large family. In particular, methods satisfying the inherent RK stability property will be considered in detail. We will discuss how to construct methods with the IRKS property and explore some issues related to their practical implementation.

Abstract:
Adjoint flow solvers are one of the components required in order to establish an efficient numerical optimization capability for aerodynamic shape designs. The development of adjoint flow solvers and their application to shape designs of airfoils and 3D aircraft configurations are presented. Furthermore, it will be demonstrated that the adjoint approaches can handle multi-point designs, multi-constraints as well as multi-disciplinary optimization (MDO) problems.

Abstract:
Control oriented flow models must combine simplicity and ample dynamic range. The dimensionality of computational fluid dynamics models (O(104) at the very low end) is a serious if not insurmountable impediment: This applies to the use of practical analytical nonlinear design methods, to the ability to estimate a large number of flow states from few and typically noisy sensor readings, and to the real time computations necessary for feedback implementation. Seeking low order models, the issue of an ample dynamic envelope becomes critical in the context of feedback control, where transients and the encounter with deviations from designated target states or orbits, are in the essence. The talk reviews several new enablers to address the tension between these two constraints. These enablers include `sub-grid' estimation of turbulence and pressure representations, proper orthogonal decomposition (POD) modes from multiple operating points and tunable models, actuation models, physics based identification of low energy modes that play important dynamic roles, invariant manifold reductions and interpolated models. The invariant manifold that defines the model's dynamic envelope must be respected - but can also be exploited - in observer and control design, such as by a restriction to slow drift in the system's periodic behavior, enabling the use of simplifying dynamic phasor models.

Abstract:
Der Vortrag beschäftigt sich mit numerisch zugänglichen Näherungen an die zeitabhängige Mehrteilchen-Schrödinger-Gleichung. Derartige Näherungen werden aus dem Dirac-Frenkel-Variationsprinzip erhalten, das Bewegungsgleichungen auf einer Approximationsmannigfaltigkeit liefert. Diese nichtlinearen, partiellen oder gewöhnlichen Differentialgleichungen bilden ein nicht-kanonisches Hamilton-System und erhalten daher eine symplektische 2-Form und die Gesamtenergie. Sie liefern Näherungen, die zumindest auf kurzen Zeitintervallen nahe der Bestapproximation an die Wellenfunktion auf der Mannigfaltigkeit liegen. Strukturerhaltende Zeitintegrationsverfahren können durch "variationelles Splitting" erhalten werden. Wichtige Beispiele variationeller Approximationen sind das zeitabhängige Multikonfigurations-Hartree-Verfahren (MCTDH) und die variationelle Propagation Gauß'scher Wellenpakete.

Abstract:
In diesem Vortrag werden Verallgemeinerungen einer klassischen Integralungleichung von Hardy und Littlewood betrachtet. Diese Verallgemeinerungen hängen mit symmetrischen Operatoren in einem Hilbertraum zusammen und erlauben die Herleitung diverser neuer und alter Integral- und Reihenungleichungen.

Abstract:
We introduce the convex combinatorial optimization problem, a far reaching generalization of the standard linear combinatorial optimization problem. We show that it is strongly polynomial time solvable over any edge-guaranteed family, and discuss several applications.

Abstract:
In this talk we consider the numerical computation of the photonic band structure of periodic materials such as photonic crystals. We first discuss how this calculation may be formulated as a Hermitian nonlinear eigenvalue problem. Numerical methods for nonlinear eigenvalue problems are usually based on Newton's method or are extensions of techniques for the standard eigenvalue problem. We present a variation on existing methods which has its derivation in methods for bifurcation problems, where bordered matrices are used to compute critical points in singular systems. This new approach has several advantages over the current methods. First, in our numerical calculations the new variation is more robust than existing techniques, having a larger domain of convergence. Second, the linear systems remain Hermitian and are nonsingular as the method converges. Third, the approach provides an elegant and efficient way of both thinking about the problem and organising the computer solution so that only one linear system needs to be factorised at each stage in the solution process. Finally, first and higher order derivatives are calculated as a natural extension of the basic method, and this has advantages in the electromagnetic problem discussed here, where the band structure is plotted as a set of dispersion curves. (This is joint work with Dr Chris Poulton, and appeared in JCP, 204 (2005) pp.65-81.)