Numerik-Oberseminar WS 2001/02

Interessenten sind herzlich eingeladen!

Rückblick: Das Numerik-Oberseminar SS 2001

Abstracts zu den Vorträgen:

Fredi Tröltzsch, TU Berlin
Eine Methode der Momentansteuerung bei der optimalen Kühlung von Profistählen
Di 30.10.2001, 16:15 Uhr in MA 742

Abstract:
Im Vortrag wird die Aufgabe behandelt, gewalzte Profilstähle in Wasser-Luft-Kühlstrecken optimal gesteuert abzukühlen. Die Steuerungsvariablen entsprechen den Intensitäten regelbarer Wasserdüsen, und der Abkühlungsprozess wird durch ein Anfangsrandwertproblem bei einer quasilinearen Wärmeleitgleichung beschrieben. Außerdem sind Schranken an die Steuerungen und an die Stahltemperatur vorgegeben. Die numerische Behandlung des Problems ergibt eine hochdimensionale nichtlineare Optimierungsaufgabe, welche nur mit enormem Zeitaufwand numerisch gelöst werden kann. Im Vortrag wird gezeigt, wie mit Hilfe einer Methode der Momentansteuerung (instantaneous control) die Rechenzeit drastisch gesenkt werden kann. Als interessanter Nebeneffekt dieser Technik ergibt sich die Möglichkeit der Berücksichtigung von diskreten Steuerungsvariablen.

Abstract:
The structured pseudospectra arise naturally when studying the spectral properties of a model with structured uncertainties. Here we investigate the stability of the spectra of infinite Toeplitz and Laurent matrices under the perturbation localized in some block. The main result is that the spectrum of a Toeplitz matrix with rational symbol is not affected by such perturbation. Also some results on approximation of the structured pseudospectrum of a infinite Toeplitz matrix with the structured pseudospectra of its finite sections will be presented.

Abstract:
The quadratic eigenvalue problems (QEP):

(t^2 M+tC+K) x = 0,

where M, C, K are n*n complex matrices, the nonzero x is the right eigenvector associated with the eigenvalue t, arise in many applications, including the finite element analysis of automobile brakes, earthquake engineering, the analysis of conservative and non-conservative structural systems, and least square problems with quadratic constraints. The standard way of dealing with the problems concerned with QEPs is to reformulate it into an equivalent and ``effective'' generalized eigenvalue problem (GEP), by means of which the definitions and evaluation expressions of the normwise backward errors and conditioning numbers for eigenvalues and eigenvectors of QEPs are discussed.

Abstract:
This is an elementary short course that was designed for the SIAM conference on Linear Algebra in Control Signals and Systems.
It gives an overview over the state-of-the-art in numerical methods for the solution of matrix equations of Lyapunov and Riccati type. It also discusses the application of these methods to H_2 and H_oo control.

Abstract:
In this talk we will consider the numerical solution of indefinite linear systems. Such systems arise in a variety of applications. Examples include linearized Navier-Stokes equations, saddle-point and linear least-squares problems, and constrained optimization. Our focus is on block two-by-two systems known as KKT systems in optimization or Stokes systems in fluid dynamics. We will present a new class of preconditioners for solving such systems, which is based on splittings of the (1,1) block of the matrix. Different splittings lead to different preconditioners, and we will show some theoretical results about the preconditioned matrices, e.g., their eigenvalue distributions.
Results related to the eigenvalue and eigenvector structure of the preconditioned matrices will allow us to discuss the convergence behavior of Krylov subspace methods such as GMRES for these matrices. In addition we will show how to solve the preconditioned linear systems by a fixed point iteration. Numerical examples will include applications in constrained optimization (arising in surface parameterization).
The talk is based on joint work in progress with Eric de Sturler.

Abstract:
Es soll ein Überblick über wichtige Resultate aus der Analysis des instationären, inkompressiblen Navier-Stokes-Problems gegeben werden. Ausgehend von der druckfreien schwachen Formulierung werden Existenz, Einzigkeit und Regularität schwacher Lösungen im zwei- und dreidimensionalen Fall untersucht. Insbesondere wird auch der Zusammenhang zwischen Glattheit der Lösung, Bedingungen an die Kompatibilität der Daten und parabolischer Glättungseigenschaft beleuchtet.

Abstract:
The talk will overview some results in optimal control (or possibly with several controls and several cost functionals in a Nash' noncooperative setting) of flows governed by the Navier-Stokes system (or possibly by the Oberbeck-Bussinesq system). A special attention will be paid to a convexity of the overall cost functional(s), which requires special physical conditions like smallness of Reynolds' number and which guearantees, e.g. sufficiency of the maximum principle or existence of a Nash equilibria.

Abstract:
During the production of many goods the working precision of the machine is very important. But the manufacturing of such machines cannot be done with the required precision. For obtaining the necessary working precison, the machine has to be calibrated, i.e., the control has to be adapted such that it realizes the precise movement of the tool. Alternatively, as it will be done in our case, one can also identify the actual parameters of the already manufactured machine. Then the model has to be adapted to the actual situation. Therefore, the control which is computed with respect to the model will be adapted such that the precise movement of the tool is realized.
For calibration, one first needs a model of the machine that contains all possible or necessary perturbations of parameters but not too many. Otherwise, a calibration would not be possible, not be unique, or not be efficient.
The talk presents a modelling strategy of arbitrary hybrid kinematics that includes all necessary parameters which have to be identified. The model of an hybrid kinematic is defined by a set of nonlinear implicit equations. By solving these nonlinear equations, one can compute the position of the endeffector by given control or vice versa. These computed positions can be compared with the actual positions of the endeffector, and with the help of these comparisons one can identify the actual parameters.