Graduate course on SPECIAL TOPICS IN NUMERICAL LINEAR ALGEBRA organized in the context of the UCL/DEA in Mathematics and sponsored by the Inter University Pole of Attraction V/10/29 "Dynamical Systems and Control : Computation, Identification and Modelling"

Venue : Auditoire CESAME, Av. G. Lemaitre 4, B-1348 Louvain-la-Neuve
Click here for directions and a map

If students register for this course and request an evaluation, this can be arranged (in the form of a short report or additional presentation on one of the topics addressed in these seminars)

PS: This page will be updated with slides of the talks and related papers as soon as they become available

14:00-14:10 Introduction
                      (Paul Van Dooren, CESAME, UCL)

14:10-15:10 Positive Caratheodory interpolation on the polydisk Slides
                      (Hugo Woerdeman, Dept. Mathematics, William and Mary, USA)
Abstract
The positive Caratheodory interpolation problem in the Agler-Herglotz class on the polydisc is solved, along with a several variable version of the Naimark dilation theorem. In addition, the positive Caratheodory interpolation problem for general holomorphic functions is discussed and numerical results are presented
Related report : Positive Caratheodory interpolation on the polydisk

15:30-16:30 Rational approximation theory solving a telecom problem Slides
                      (Annie Cuyt, Dept. Mathematics, UIA)
Abstract
The telecom problem under consideration is the computation of the cell loss probabilities of a multiplexer. This problem could not be dealt with efficiently until we recently proposed the use of rational models. The problem is univariate in nature and I have to introduce Thiele interpolating continued fractions and partial Newton-Pade approximants. Then I build up the problem from easy to difficult and I discuss the ability of the new method to deal with all situations
Related report : Rational approximation theory solving a telecom problem

14:00-15:00 Positive extensions and Riesz-Fejer factorization for two-variable trigonometric polynomials Slides
                      (Hugo Woerdeman, Dept. Mathematics, William and Mary, USA)
Abstract
The autoregressive filter problem for bivariate stochastic processes is reduced to a finite positive definite matrix completion problem where the completion is required to satisfy additional low rank conditions. The autoregressive filter problem may also be interpreted as a two-variable positive extension problem for trigonometric polynomials where the extension is required to be the reciprocal of the absolute value squared of a stable polynomial. For the proof a specific two-variable Kronecker theorem is developed, as well as a two-variable Christoffel-Darboux formula. As a corollary of the main result a necessary and sufficient condition for the existence of a spectral Riesz-Fejer factorization of a two-variable trigonometric polynomial is given in terms of the Fourier coefficients of its reciprocal
Related papers: Positive extensions and Riesz-Fejer factorization for two-variable trigonometric polynomials
A numerical algorithm for stable 2D autoregressive filter design

15:15-16:15 Optimization problems over nonnegative trigonometric polynomials with interpolation constraints Slides
                      (Yvan Hachez and Yurii Nesterov, CESAME, UCL)
Abstract
Optimization problems over the cone of nonnegative trigonometric polynomials are described. We focus on linear constraints on the coefficients that represent interpolation constraints. For these problems, the complexity of solving the dual problem is shown to be almost independent of the number of constraints, provided that an appropriate preprocessing has been done. These results can be extended to other curves of the complex plane (real axis, imaginary axis), to nonnegative matrix polynomials and to interpolation constraints on the derivatives
Related paper: Optimization problems over nonnegative trigonometric polynomials with interpolation constraints

14:00-15:00 Orthogonal rational functions and diagonal plus semiseparable matrices Slides
                      (Marc Van Barel, Dept. Comp. Sc., KUL)
Abstract
We consider the vector space of all proper rational functions having possible poles in n given points, and give a bilinear form that defines an inner product in this space. We then study the properties of an orthonormal basis of the above space with respect to that inner product. We also design an efficient and stable algorithm to compute such a basis via a suitable recurrence relationship and indicate the relation with the inverse eigenvalue problem for diagonal plus symmetric semiseparable matrices
Extended abstract

15:15-16:15 Positive definite Hankel matrices, positive real functions and related questions
                      (Yves Genin, CESAME, UCL)
Abstract
It has been known for decades that any Caratheory function defines an infinite nested set of nonnegative Toeplitz matrices and conversely. It is much less known that a similar property holds true for nonnegative definite Hankel matrices in a somewhat weaker sense. In the Hankel case, the duality in question only involves an appropriate subclass of the whole class of positive real functions. The properties of this subclass of functions, which encompasses in particular the so called lossless functions, will be discussed in some details. Furthermore and as an application, it will be shown how the Hankel matrix extension problem and some of its generalizations to the undefinite situation can be solved at this light in a straightforward manner
Related paper: Hankel matrices, positive real functions and related questions

14:00-15:00 Rational approximation and balanced truncation
                      (Jan Willems, Dept. El. Engrg. ESAT, KUL)
Abstract
We will deal with the ubiquitous finite-dimensional linear systems d/dt x=Ax+Bu, y=Cx+Du. After reviewing the construction of a balanced state representation for a controllable/observable system, we define what is meant by a balanced truncation, and point out the relation with the Hankel singular values. We subsequently discuss two important issues of the reduced model: its stability, and the H_infinity error bound. Both involve, in a somewhat curious way, the Hankel sigular values and their multiplicites. The theory that we discuss is not original work of the speaker. At best, the proofs are new. The proof of the error bound, in fact, is based on the theory of dissipative systems

15:15-16:15 Rational interpolation and model reduction Slides
                      (Antoine Vandendorpe and Paul Van Dooren, CESAME, UCL)
Abstract
We consider the problem of constructing a reduced order system of a given linear time-invariant system described by a (generalized) state-space model. We derive an explicit way of constructing such reduced order via Multipoint Pade techniques and discuss the complexity of this approach for large systems described by sparse models. We also show that this approach is in fact quite general. For SISO systems it essentially allows to construct any reduced order model, provided the interpolation points are appropriately chosen. For MIMO systems we need to extend Multipoint Pade to tangential interpolation, and we show that there then are mild constraints on the type of reduced order models that can be obtained in this manner
Related report : Model reduction via truncation : an interpolation point of view

14:00-15:00 Spectral factorizations and sums of squares representations via semidefinite programming Slides
                      (Hugo Woerdeman, Dept. Mathematics, William and Mary, USA)
Abstract
It is known that a matrix (trigonometric) polynomial that is nonnegative definite on the real axis (on the circle) allows a stable factorization. We shall explore the different possibilities how the stable spectral factor may be determined via semidefinite programming. In the dual formulation this leads to a question what possible barrier functions there exist for classes of positive semidefinite structured block matrices. The multivariable version, i.e. finding sums of squares representations, is also explored. Appropriate definitions for stability in this context also come to pass
Related papers: Spectral factorizations and sums of squares representations via semidefinite programming
Model Theory for rho-contractions

15:15-16:15 Commutant lifting and metric constrained interpolation Slides
                      (Rien Kaashoek, Dept. Mathematics, VU Amsterdam)
Abstract
The Sz-Nagy-Foias commutant lifting theorem has proved to be a useful tool in solving metric constrained interpolation problems appearing in mathematics and engineering. In this talk we use the classical Caratheodory-Schur extension problem to motivate this theorem and the underlying general lifting problem. A few other applications to metric constrained interpolation will be discussed too. Next, two new versions of the lifting theorem will be developed. The first is a relaxed version of the theorem, and the second a robust version. Also some new applications will be considered

14:00-15:00 Identification with rational functions Slides
                      (Patrick Van gucht and Adhemar Bultheel, Dept. Comp. Sc., KUL)
Abstract
In the last few decades of research in system identification, much effort has gone to including prior knowledge of the position of the poles in the rational model for a system. We show how the theory of orthogonal rational functions, (a generalization of the orthogonal polynomials), fit into this framework if we take a weight function depending on the input signal. Moreover, this technique can provide much more accurate results than the rational functions, used in other research, due to an optimal condition of the Jacobian in the approximation problem
Related paper: Using orthogonal rational functions for system identification

15:15-16:15 Asymptotic convergence rates of rational interpolation to exponential functions
                      (Alphonse Magnus, CESAME, UCL)
Abstract
Several ways to build rational approximations to various species of analytic functions are examined. Special emphasis is put on strong asymptotic estimates of the form $f(z) -R_n(z) \sim \sigma(z) \rho^n(z)$, when such estimates are available, and on approximation to exponentials of polynomials
Related report : Complex approximation and interpolation. ps-gz 183K