_____________________________ INVITED LECTURES _________________________
Walter Gautschi
Department of Computer Sciences Purdue University, West Lafayette, USA.
"Orthogonal Polynomials and Quadrature"
Abstract:
Various concepts of orthogonality on the real line are
reviewed that arise in connection with quadrature rules.
Orthogonality relative to a positive measure and Gauss-type
quadrature rules are classical. More recent types of orthogonality
include orthogonality relative to a sign-variable measure of
interest in connection with Gauss-Kronrod quadrature, and power
orthogonality for Tur\'an-type quadrature. Relevant questions of
numerical computation are also considered.
"Gauss Quadrature for Rational Functions"
Abstract:
Gauss-type quadrature rules are studied that are exact for a
mixture of polynomials and rational functions, the latter being
selected so as to simulate poles that may be present in the
integrand. The underlying theory is presented as well as methods
for constructing such rational Gauss formulae. Applications are
given to the computation of generalized Fermi-Dirac and
Bose-Einstein integrals.
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Gene Golub
Computer Science Dept, Stanford University, Stanford, USA
"Bounds for the Entries of Matrix Functions with Applications to
Preconditioning"
Abstract:
Let $A$ be a symmetric
matrix and let $f$ be a smooth function defined on an interval
containing the spectrum of $A.$ The entries of the matrix $f(A)$ can
be expressed as Riemann--Stieltjes integrals of $f$ with respect to a
suitable measure. By approximating these integrals with Gauss--type
quadrature rules one obtains bounds or estimates for the entries of
$f(A).$ These quadrature rules can be evaluated by means of the
Lanczos process. Explicit bounds and estimates are obtained after a
single Lanczos step for a wide class of functions $f.$ More refined
approximations or tighter bounds can be obtained by taking a few
Lanczos steps. In this talk, a few choices of $f$ of interest for
preconditioning are considered. A new result on the decay of entries
of analytic functions of band matrices is proved, which justifies in
many cases the use of banded approximations to $f(A)$. The
effectiveness of this approach will be illustrated by numerical
examples. This work is in collaboration with Michele Benzi
(Los Alamos National Laboratory).
"Inverting Shape From Moments"
Abstract:
Computation of certain numerical quadratures on polygonal regions of
the plane and the reconstruction of these region from their moments
can be viewed as dual problems. In this talk, we discuss this idea
and address the inverse problem of reconstructing a region in the
complex plane from a finite number of its complex moments. The
numerical computations involved in the algorithms can be very
ill-conditioned. We have managed to derive inversion algorithms,
based on matrix pencils, with improved stability, and have come to
recognize when the problem will be ill-conditioned. We will also
briefly discuss an application to a geophysical inversion problem.
(Joint work with Peyman Milanfar and Jim Varah)
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Wolfram Koepf
HTWK Leipzig, Department IMN, Germany
"Software for the Algorithmic Work with Orthogonal Polynomials and Special
Functions I and II"
Abstract:
In the last decade major steps towards an algorithmic treatment of
orthogonal polynomials and special functions (OP \& SF) have been made,
notably Zeilberger's brilliant extension of Gosper's algorithm on
algorithmic definite hypergeometric summation.
By implementations of these and other algorithms symbolic computation has
the potential to change the daily work of everybody who uses orthogonal
polynomials or special functions in research or applications.
It can be expected that symbolic computation will also play an
important role in on-line versions of major revisions of existing
formula books in the area of OP \& SF.
It this couple of talks I will present software in Maple, Mathematica and
Reduce of those algorithmic techniques, in particular of Gosper's,
Zeilberger's, and Petkov\v{s}ek's algorithms and their q-analogoues.
Some implementational details are discussed.
The main emphasis, however, is given to on-line demonstrations of
these algorithms using our Maple implementations (jointly with
Harald B\"oing) covering many examples from the field of OP \& SF.
The use of CAOP, a World Wide Web version of the Askey-Wilson scheme
developed by Ren\'e Swarttouw, is presented as well, and it's
implementation is discussed.
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Yvon Maday
Universit\'e Pierre et Marie Curie, Paris, France
"The Basic spectral element and mortar elements methods for elliptic
problems "
Abstract:
The spectral element method for elliptic problem is a high order
approximation method derived from the variational formulation of the
problem. It combines the domain decomposition techniques through bricks
(may be curved but nevertheless regular) with the high order of precision
of the polynomial approximation to define an efficient way for the
numerical simulation of the phenomenon.
The numerical analysis of this type of method relies heavily on basic
properties of the Legendre and more generally the Jacoby family of
orthogonal polynomials.
We shall present in this part the main ingredient that make understand why
spectral methods work so well.
The basic spectral element method is limited to domain decompositions that
satisfy the general statement of finite element type that requires that the
intersection of two subdomains is either empty or a common vertex or a
common edge. The mortar element method allows for more flexibility in the
decomposition and allows even for coupling spectral and finite element
methods. The basics of this mortar element method will be presented in the
spectral framework.
"The spectral element methods for resolution of the Stokes and Navier-Stokes
problems"
Abstract:
The extension of the spectral element method from the Laplace equation to
the Stokes problem in velocity pressure formulation involves a
compatibility condition between the discrete pressure space and the
discrete velocity space. This condition is known as the (L.B.B) inf-sup
condition. In the spectral context, we cannot avoid the analysis of this
compatibility condition and the technique required for the evaluation of
its dependancy in the degree of the polynomial approximation involves a lot
of nice pieces of analysis. We shall sketch the basics of the theory that
is currently available with a particular emphasis to the most fundamental
properties.
The extension of the method to the resolution of the Navier-Stokes problem
will be finally developped so as the current algorithms of resolution.
Numerical results will finally illustrate the previous analysis.
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Marko Petkovsek
University of Ljubljana. Ljubljana, Slovenia
"Linear Operators and Compatible Polynomial Bases I and II"
Abstract:
In the first part of my talk, I will review the algorithms for finding
``nice'' explicit solutions of linear recurrences with polynomial
coefficients. These include solutions which are polynomials, rational
functions, hypergeometric or q-hypergeometric terms, interlacings of
hypergeometric terms, and $m$-sparse solutions.
In the second part, I will look at power series and, more generally,
polynomial series which solve linear operator equations. An operator $L$
and a basis $\cal B$ for the polynomial space are "compatible" if the
corresponding recurrence for the coefficients of a series $y$ in the kernel
of $L$ is of finite order.
Satisfactory compatibility properties are exhibited not only by the powers
and the falling powers, but also by certain families of orthogonal
polynomials, such as the Gegenbauer polynomials. Combined with the algorithms
mentioned in the first part, this gives the possibility of finding series
solutions which have ``nice'' coefficients with respect to a selected
polynomial basis.
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Doron Zeilberger
Department of Mathematics. Temple University, Philadelphia, USA.
"The Unreasonable Power of Orthogonal Polynomials in Combinatorics I and II
Abstract:
Great Mathematics is destined to be eventually used,
often in quite a different place from where it was first
created. A case in point are the q-analogues of the classical
orthogonal polynomials, preached, and largely developed, by
George Andrews, Dick Askey, and their students, that lead
to the brilliant solution of the Refined Alternating Sign
Matrix Conjecture.