Diplomanden- und Doktorandenseminar
Numerische Mathematik SS 2005

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Weitere Vorträge siehe auch:

• Kolloquium der Arbeitsgruppe Numerik, Modellierung, Differentialgleichungen
• Seminar des Fachgebiets Optimierung bei partiellen Differentialgleichungen (Gruppe Tröltzsch)

• Diplomanden- und Doktorandenseminar WS 2004/05
• Diplomanden- und Doktorandenseminar SS 2004
• Diplomanden- und Doktorandenseminar WS 2003/04
• Diplomanden- und Doktorandenseminar SS 2003
• Diplomanden- und Doktorandenseminar WS 2002/03

Abstracts zu den Vorträgen:

Abstract:
In a first part we give a short overview of results and open problems concerning the controllability of partial differential equations, with focus on the Navier-Stokes equations modelling the flow of incompressible viscous fluids. In a second part we investigate the particular problem of controlling Navier-Stokes equations between two infinite rotating coaxial cylinders. We prove that it is possible to move from a given Couette flow, that is a special stationary solution, to another one, by controlling the rotation velocity of the outer cylinder.

Literature:

• R. Glowinski and J.-L. Lions: Exact and approximate controllability for distributed parameter systems. Acta numerica, 1994, 269--378.
• R. Glowinski and J.-L. Lions: Exact and approximate controllability for distributed parameter systems, Acta numerica, 1995, 159--333.
• M. Schmidt and E. Trélat: Controllability of Couette flows, Preprint 12-2005, Institute of Mathematics, Technische Universität Berlin, 2005.
• (and the references therein)

Abstract:
We perform an index analysis on a model of an integrated circuit with some semiconductor devices modelled as distributed elements.

Existing index criteria for finite dimensional DAEs describing lumped circuits are generalised to abstract circuit DAEs of this kind. For finite DAEs the correlation between the perturbation index and the tractability index has been established. We present simulation some results concerning the expected correlation in the infinite dimensional case.

Introductory literature:

• Connection between the tractability index and the perturbation index:
  Balla, März: An unified approach to linear differential algebraic equations and their adjoint equations, 2001.
• Abstract DAEs:
  Lamour, März, Tischendorf: PDAEs and Further Mixed Systems as Abstract Differential Algebraic Systems, 2001.
• Circuit DAEs coupled with semiconductor PDEs:
  Bodestedt: Index of coupled systems in circuit simulations, 2004. Licentiate dissertation, Lund university.

Abstract:
The treatment of a pathogenic disease process can be interpreted as the optimal control of a dynamic system. This system describes the innate immune response and consists of both ordinary differential equations and delay differentail equations (DDEs). An introduction to optimal control problems with constant time delays is given as well as the idea of a proof of the minimum principle for this type of problem. Optimal solutions are shown for several variations of the model. I will also discuss second order sufficient condition for the non-delayed case of the studied model and show that there are still lots of interesting open questions concerning time-delayed optimal control problems.

Literature:

• A. Asachenkov, G. Marchuk, R. Mohler, S. Zuev: Disease Dynamics, Systems & Control: Foundation & Application Series, Birkhäuser, Boston, MA, 1994
• L. Göllmann: Numerische Behandlung optimaler Steuerprozesse mit konstanten Verzögerungen unter gemischten Steuer-Zustandsbeschränkungen, Dissertation am Institut für numerische und instrumentelle Mathematik der Universität Münster, 1996
• R.F. Stengel, R. Ghigliazza, N. Kulkarni, O. Laplace: Optimal control of innate immune response, Optimal Control Applications and Methods 2002, 23: 91-104
• Vanderbei, R. J., Shanno, D. F.: An Interior-Point Algorithm for Nonconvex Nonlinear Programming, Computational Optimization and Applications, 13:231-252, 1999

Abstract:
Second order systems of differential-algebraic equations arise naturally in industrial applications.
We will introduce some application classes of second order differential-algebraic systems. Among others we will consider multibody systems, electrical circuits and micro-electro mechanical systems. In addition we give examples of coupled systems consisting of differential-algebraic equations and partial differential equations.

Abstract:
Dynamic Iteration is a method for the simulation of coupled systems that allows for different solvers for each subsystem. It has been a well understood but little used method for ODEs. For DAEs, it requires the solution of additional linear systems in order to work properly. Recently, industry has become interested because it would probably allow to solve the DAE systems that arise in circuit simulation more efficiently. We give a short introduction to Dynamic itaration and we show its application to curcuit DAEs. Furthermore, a method will be presented to circumvent the additional linear systems needed for convergence of the method.

Suggested literature:

• @techreport{AliBG03 ,author={Ali,G., Bartel,A., G\"unther,M.},title={Analysis of Coupled Systems in Chip Design},institution={Institut f\"ur Wissenschaftliches Rechnen und Mathematische Modellbildung, Universit\"at Karlsruhe},standort={hardcopy}}
• @article {ArnG01 ,author={Arnold,M. and G{\"u}nther,M.},title={Preconditioned Dynamic Iteration for Coupled Differential-Algebraic Systems},journal={BIT},volume={41},number={1},pages={001--025},year={2001}}
• @techreport{Ebe04 ,author={Ebert, F.}, title={Convergence of relaxation methods for coupled systems of {ODE}s and {DAE}s}, type={Preprint}, number={176},institution={DFG Research Center {\sc Matheon}, Technische Universit\"at Berlin}, year={2004}, url={https://www.math.tu-berlin.de/\~\,docserv/toc.cgi}}

Abstract:
The distance to controllability for a linear control system is the distance (in the 2-norm) to the nearest uncontrollable system. I will present two algorithms based on methods of Gu and Burke-Lewis-Overton that estimate the distance to uncontrollability to any prescribed accuracy. The first method requires O(n^4) operations on average, which is an improvement over previous methods which have complexity O(n^6), where n is the size of the system. The second method requires O(n^3/tol) operations with tol denoting the accuracy with which we want to retrieve the distance to uncontrollability. The second method is well-suited to estimate the distance to uncontrollability with a few digits of precision. Numerical experiments indicate that the new methods are reliable in practice.

Keywords: distance to uncontrollability, complex controllability radius, trisection, real eigenvalue extraction, shifted inverse iteration, shift-and-invert Arnoldi, Sylvester equation, Kronecker product.