Numerik-Oberseminar SS 2003

DAE-Day (Workshop zu Differentiell-algebraischen Gleichungen),
Freitag, 13. Juni 2003, 14:00-17:15 in MA 309

Interessenten sind herzlich eingeladen!

Rückblick:

• Numerik-Oberseminar WS 2002/03
• Numerik-Oberseminar SS 2002
• Numerik-Oberseminar WS 2001/02
• Numerik-Oberseminar SS 2001

• Diplomanden- und Doktorandenseminar des Fachgebiets Numerische Mathematik (Gruppe Mehrmann)
• Seminar des Fachgebiets Optimierung bei partiellen Differentialgleichungen (Gruppe Tröltzsch)

Abstracts zu den Vorträgen:

Abstract:
When GMRES is applied to streamline-diffusion upwind Petrov Galerkin (SUPG) discretized convection-diffusion problems, its residual norms typically exhibit an initial period of slow convergence followed by a fast linear decrease. Several approaches were made to understand this behavior. However, the existing analyses are soley based on the matrix of the discretized system and they do not take into account any influence of the right hand side (boundary conditions). Therefore they can not explain the length of the initial period of slow convergence which is right hand side dependent.
We will concentrate on a model problem with a constant velocity field parallel to one of the axes and with Dirichlet boundary conditions. Replacing the eigendecomposition of the system matrix by the simultaneous diagonalization of the matrix blocks, we show how the initial period of slow convergence is related to the boundary conditions. We also discuss general implications of our findings.

Abstract:
Several real Lie and Jordan algebras, along with their associated automorphism groups, can be elegantly expressed in the quaternion tensor algebra. The resulting insight into structured matrices leads to a class of simple Jacobi algorithms for the correponding n-by-n structured eigenproblems. These algorithms have many desirable properties, including parallelizability, ease of implementation, and strong stability.

Abstract:
Matrix factorizations permeate numerical linear algebra. Indeed, without too much exaggeration, one could almost define numerical linear algebra to be the study of matrix factorizations and their effective computation. Many of the standard factorizations, e.g. QR, LU, SVD, Polar Decomposition, .... apply to all (or almost all) real or complex n-by-n matrices. But what if one applies one of these standard factorizations to an element A of a matrix group G? When, if ever, can we expect the factors of A to also be elements of G? For example, when will the polar factors of a symplectic matrix also be symplectic? In this talk we consider various questions of this type for matrix groups associated with scalar products, e.g. the real and complex symplectic, complex orthogonal, and real pseudo-orthogonal groups.

Abstract:
An algorithm is developed for passivity preserving model reduction of linear time invariant systems. Implementation schemes are developed for both medium scale (dense) and large scale (sparse) applications. The algorithm is based upon interpolation at selected spectral zeros of the original transfer function to produce a reduced transfer function that has the specified roots as its spectral zeros. These interpolation conditions are satisfied through the computation of a basis for a selected invariant subspace of a certain blocked matrix which has the spectral zeros as its spectrum. Explicit interpolation is avoided and passivity of the reduced model is established, instead, through satisfaction of the necessary conditions of the Positive Real Lemma. It is also shown that this procedure indirectly solves the associated controllability and observability Riccati equations and how to select the interpolation points to give maximal or minimal solutions of these equations. From these, a balancing transformation may be constructed to give a reduced model that is balanced as well as passive and stable.

Abstract:
Surface-acoustic wave (SAW) filters are used for frequency filtering in telecommunications. The goal of the presented work is the numerical computation of so-called ''dispersion diagrams'', which gives the functional relation between ''excitation frequency'' and a complex surface-wave propagation constant. The mathematical model is governed by two main points, the underlying periodic structure and the indefinite coupled field problem due to the used piezoelectric materials. Floquet-Bloch theory allows to restrict an infinite periodically-perturbed computation domain to one reference cell by introducing quasi-periodic boundary conditions. Due to the Bloch-ansatz the dispersion relation between ''excitation frequency'' and the ''propagation constants'' of the SAW is described by a parameter depending eigenvalue problem. Two different methods of extracting the frequency-dependent eigenvalue problem are presented. Reducing the problem only to unknowns on the periodic boundary leads to a small-sized dense quadratic eigenvalue problem of the form (z^2A + zB + A^T)x=0, where A is non-hermitian and complex and B is complex-symmetric. The second method implies a large-dimensioned generalized non-hermitian eigenvalue problem which is solved by an Arnoldi method. The extracted eigenvalue problems have the special structure that eigenvalues occur in pairs t and 1/t. The influence of periodic perturbations in the computation geometry is shown in numerical experiment.

Abstract:
In my talk I will discuss a new discretization concept for abstract control problems with control constraints which extends the common discrete approaches. Discretization only is applied to the state variables, which in turn implicitly yields a discretization of the control variables by means of the first order optimality condition. For discrete controls obtained in this way an optimal error estimate in terms of the state-discretization parameter is presented. Applied to control of partial differential equations combined with finite element discretization of the state the key features of the new control concept include

• decoupling of finite element grid and discrete active set
• numerical implementation requires only small additional overhead compared to commonly utilized methods (discretization of state AND of admissible controls)
• Approach applicable in 1,2 and 3 spatial dimensions, and also for Galerkin type discretization schemes of time dependent state equations
• Numerical analysis seriously simplifies compared to the common approach

Abstract:
Based on the Boussinesq-approximation mathematical models for the description of crystal melts, especially the Czochralski growth method and the zone melting technique will be discussed.
Due to the technological properties of the melting equipment free and forced convection are of interest. The numerical solution of the initial boundary value problem is done by a finite volume method in space and second order methods in time (CR, Euler backward, Adams-Bashforth). The solution method will be validated by the consideration of a special benchmark for crystal growth of Wheeler.
Some parameter studies of the zone melting technique wil be shown and compared to experimental results of crystal growth engineers.
For the minimization of interesting functionals an optimization problem is formulated and an adjoint problem will be constructed by the Lagrange parameter technique. An iteration method for the solution of the full optimization problem will be demonstrated and some concrete optimization problems for the Czochralski and zone melting technique will be solved and discussed.

Abstract:
The state feedback decoupling problem with stability for general proper systems described by (A, B, C, D) quadruples has been studied for a long time. But, there are no numerically verifiable solvability conditions and no numerically implementable methods for solving it in the existing literatures. Hence, it is still an unsolved problem from both theoretical and numerical points of view. In this talk, I will introduce some of our recent work, we have developed a numerically reliable method for solving this important unsolved problem. Our method consists of four different stages: Stage 1 is to reduce the underlying problem for a general proper system into: 1) a state feedback decoupling problem with stability for a reduced system without direct feedthrough matrix, ii) a simultaneous problem of state feedback disturbance decoupling and "unusual" state feedback decoupling with stability for a reduced system with nonsingular direct feedthrough matrix; Then in Stage 2, the simultaneous problem with stability produced in Stage 1 is reduced into an "unusual" state feedback decoupling problem with stability for a reduced system with nonsingular direct feedthrough matrix; A useful reduction property of the "unusual" state feedback decoupling problem with stability obtained in Stage 2 is derived in Stage 3 and a numerically reliable algorithm is developed for solving this unusual decoupling problem with stability; Finally, Stage 4 consists of the backtransformations of the results in Stages 1, 2 and 3 to the desired solutions for the original decoupling problem with stability. The main tool that we use is numerical linear algebra and our numerical method can be implemented in a numerically reliable way.

Abstract:
We describe a general procedure for reducing a class of convex optimization problems involving cones of generalized squares to semi-definite programming problems. We combine this technique with rather subtle results related to Hilbert's seventeenth problem to construct a sequence of semi-definite and linear programming relaxations for the problem of a global optimization of a homogeneous polynomial on the simplex and on the sphere. Using recent results of B. Reznick we rigorously estimate how close is the optimal value of the original problem and the optimal value of a linear programming and semi-definite programming relaxation in both cases. Numerical aspects of these approximations are briefly discussed.

Abstract:
Since the work of Kalman on optimal control in the 60s of the last century the symmetric Riccati equation has enjoyed lively interest. But also in noncooperative differential games Riccati equations appear, which loose the symmetry property. The talk analyzes the role of non-symmetric algebraic Riccati equations in the following game theoretic situations:
a) open-loop Nash games
b) open-loop Stackelberg games.
Starting point is the introduction to non-symmetric Riccati theory, which constitutes a connection between the invertibility of a Toeplitz operator and the existence of a strongly stabilizing solution to the Riccati equation. This connection will be applied to investigate the mentioned games. For Nash games it characterizes completely the existence of unique Nash equilibria in terms of the existence of a strongly stabilizing solution to the Nash Riccati equation. A condition for the existence of Stackelberg equilibria can be obtained. We will also present a value function type approach. For Nash games this method yields a (second) characterization of unique Nash equilibria. Finally, an existence result for solutions of a non-symmetric Riccati equation appearing in Stackelberg games can be deduced.