Numerik-Oberseminar SS 2002

DAE-Day (Workshop zu Differentiell-algebraischen Gleichungen), Freitag, 12. Juli 2002, 9:00-16:45 in MA 407

Interessenten sind herzlich eingeladen!

Rückblick:

• Numerik-Oberseminar WS 2001/02
• Numerik-Oberseminar SS 2001

Abstracts zu den Vorträgen:

Raphael Loewy, Technion, Haifa, Israel
Spaces of matrices containing a nonzero matrix of bounded rank
Di 30.4.2002, 16:15 Uhr in MA 742

Abstract:
Let k,n be positive integers and let S(n,R) denote the set of all n-by-n real symmetric matrices. We consider the problem of finding the smallest positive integer m such that every m-dimensional subspace of S(n,R) contains a nonzero matrix of rank bounded above by k. This problem is open in general. We consider several examples where the solution is known and the case n=k-2, where much has yet to be done. If time permits we consider also subspaces of S(n,R) containing a nonzero matrix with a multiple eigenvalue.

Abstract:
For adjoint calculations, parameter estimation, and similar purposes one may need to reverse the execution of a computer program. The simplest option is to record a complete execution log and then to read it backwards. This requires massive amounts of storage. Instead one may generate the execution log piecewise by restarting the ``forward'' calculation repeatedly from suitably placed checkpoints. The basic structure of the resulting reversal schedules is illustrated. Different strategies are analysed with respect to the resulting temporal and spatial complexity on serial and parallel machines. For serial machines known optimal compromises between operations count and memory requirement are explained. They are illustrated by means of an optimal control problem based on Navier-Stokes-Equation. For program execution reversal on multi-processors the new challenges and demands on an optimal reversal schedule are described. We present parallel reversal schedules that are provably optimal with regards to the number of concurrent processes and the total amount of memory required. They are applied to calculated derivative information for the optimal steering of a Formula One car.

Abstract:
In den vergangenen Jahrzehnten wurden differential-geometrische Methoden zur Steureung und Regelung nichtlinearer Strecken entwickelt. Genannt seien das Verfahren der exakten Linearisierung durch Rückführung sowie flachheitsbasierte Regelungskonzepte.
Bei beiden Ansätzen spielen Lie-Ableitungen und Lie-Klammern eine wichtige Rolle. Diese speziellen Ableitungen wurden bisher per Hand oder symbolisch mittels Computeralgebrasystemen wie mathematica oder Maple berechnet. Bereits bei einfachen Regelungsproblemen kann man sehr komplizierte symbolische Ausdrücke für den Regler erhalten. Für den Entwurf nichtlinearer Beobachter wurden Gradienten von Lie-Ableitungen bzw. zusützlich auch iterierte Lie-Klammern benötigt.
Bei großen Beobachtungsindizes sind ebenfalls erhebliche Rechenzeiten bzw. unhandlich große Ausdrücke zu erwarten.
Für die Berechnung von nichtlinearen Reglern und nichtlinearen Beobachtern wurden bisher symbolische Methoden eingesetzt. Für einfache Beispiele bereitet dies kaum Probleme, während man bei umfangreicheren Anwendungen schnell an die Grenzen hinsichtlich Rechenzeit und Speicherplatzbedarf stößt. Im Vortrag soll ein grundsätzlich anderer Zugang beschrieben werden. Es wird gezeigt, wie sich regelungstheoretisch relevante Ableitungen mit Hilfe des Automatischen Differenzierens berechnen lassen.

Abstract:
While being beautiful on their own Hamiltonian matrices are present in a variety of applications such as linear-quadratic optimization or H_oo design. Computing eigenvalues and invariant subspaces constitutes one of the most important and challenging tasks associated with these matrices. For this purpose a new software library has been developed. It is primarily based on the so-called BMX method and refined by several new techniques. The mathematical aspects behind are to be described in this talk, with special emphasis on balancing, irreducible forms and block representations of symplectic orthogonal transformations. It will be shown how these ingredients let the software outperform all earlier implementations of Hamiltonian eigensolvers with respect to efficiency and reliability.

Abstract:
Between the flying magnetic head and the harddisk we have a thin air film and it is possible to model the thin film flow based on the compressible Navier-Stokes equation. The basic discretisation principles of the mathematical model (direct and inverse problems) were demonstrated.
Some aspects of grid generation and resulting difficulties were discussed and validated by a comparison of the numerical results with those of the white light measurement technique.

Abstract:
Model and order reduction of dynamical systems has increasing impact on simulation and optimization of technical and physical processes like active vibration control of large flexible space structures, optimal control of fluid flow, circuit simulation and VLSI chip design, etc. In this talk, we will survey model reduction methods based on the use of the matrix sign function. The underlying model reduction methods are mostly based on truncation methods. We will discuss how the sign function iteration can be used to efficiently and reliably implement these methods.

Abstract:
Zur Analyse von gekoppelten Systemen von PDEs und DAEs werden abstrakte differential-algebraische Systeme (ADAS), d.h. differential-algebraische Systeme in Hilberträumen eingeführt. Basierend auf der endlich dimensionalen Theorie der Entkopplung von DAEs wird ein analoges Konzept für unendlich-dimensionale Systeme vorgestellt. Dies führt zu einer Index-Definition für die abstrakten Systeme und Informationen zu geeigneten Anfangs- und Randwertbedingungen für die eindeutige Lösbarkeit der gekoppelten Systeme.

Abstract:
Resolvent estimates for second order differential operators are of vital interest when studying the numerical solution of parabolic problems, especially if smoothing estimates or time discretisations come into play. So it is no surprise that over the last two decades such estimates have been proved with an increasing generality. While the multidimensional case attracts the main attention and poses major difficulties, in one space dimension people are ambitious to handle general meshes that go beyond quasi-uniformity and have no unnecessary logarithmic factors in the bounds. The talk reviews the developement over the past years including recent new results.

Abstract:
In diesem Vortrag wird das H-unitaere Prokrustesproblem formuliert und untersucht. Es handelt sich dabei um die Aufgabe, zu gegebenen Tupeln von Vektoren eines pseudounitaeren Raumes eine Isometrie zu finden, mit deren Hilfe eine Transformation im Sinne einer optimalen Ausgleichung ausgefuehrt werden kann (least squares problem). Die erforderlichen Grundlagen ueber indefinite Skalarprodukte und die zugehoerigen H-Polarzerlegungen werden zunaechst erarbeitet und im Anschluss daran zur Loesung des Problems verwendet. Ausserdem wird fuer einen haeufig in der Praxis auftretenden Spezialfall ein vereinfachtes Verfahren zur Berechnung von H-Polarzerlegungen angegeben, was auf einfachen Orthogonalisierungsverfahren und der Konstruktion von Orthogonalkomplementen beruht.

Abstract:
When considering the unmanageable complexity of computer systems, Dijkstra recently made the following observations:

(i)

When exhaustive testing is impossible --i.e., almost always-- our trust can only be based on proof (be it mechanized or not).

(ii)

A program for which it is not clear why we should trust it, is of dubious value.

(iii)

A program should be structured in such a way that the argument for its correctness is feasible and not unnecessarily laborious.

(iv)

Given the proof, deriving a program justified by it, is much easier than, given the program, constructing a proof justifying it. Over the last decade, our research in the area of the implementation of high performance sequential and parallel linear algebra libraries has made the wisdom of the above observations painfully obvious.