Preliminary abstracts of the talks of the invited speakers of the "1st Dolomites Workshop on Constructive
Approximation and Applications".
B. Bojanov: Interpolation by bivariate
polynomials
We discuss some recent results on interpolation by polynomials
in two variables using the classical point value table as well as
interpolation based on Radon projections.
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L. Bos:
Near Optimal Points for Polynomial Interpolation in Several
Variables
We discuss some nodal sets for Lagrange polynomial interpolation in
several variables, including the recently introduced so-called Padua
points for a square in $R^2$ that have been shown to have optimal rate
of growth of the Lebesgue constant. We also discuss some numerical
applications. This includes joint work with M. Caliari, S. de Marchi,
M. Vianello and Y. Xu.
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M. Bozzini: Polyharmonic B-splines: an approximation method for scattered data of extra-large size
Recently polyharmonic B-splines close to a gaussian were studied.
In this joint work with L. Lenarduzzi and M. Rossini we present a fast method exploiting these functions from a very large sample of scattered data
corrupted by noise and
eventually with outliers. Some real examples will be shown.
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C. Brezinski: The professional life of Walter
Gautschi
In this talk, I will review the most important results obtained by Walter
Gautschi
in the domains of ordinary differential equations, computation of special
functions,
interpolation, continued fractions, Padè approximation, convergence
acceleration,
Gauss-type quadratures, Fèjer quadratures, Chebyshev-type
quadratures, and
orthogonal polynomials.
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M. Buhmann: Radial basis function
interpolation
We consider radial basis function approximation by interpolation in
any dimension. The existence and properties of the radial basis
function interpolation depend not only on the choice of radial basis
functions but also in some circumstances on the location of the data
points. We will consider these aspects of radial basis functions
especially for the celebrated multiquadric radial basis function and
for the Gaussian kernels.
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C. de Boor: GC_n-sets
A $GC_n$-set, as introduced by Chung and Yao in 1977, is a set $X$
in $R^d$ correct for interpolation from $\Pi_{\le n}$ with the additional
geometric condition that, for each $x\in X$, the set $X\backslash
x$ lies
in the union of at most $n$ hyperplanes. The talk will translate to $R^d$
what
is known about such sets in the plane, with special attention to the
Gasca-Maeztu conjecture that, for $d=2$, any $GC_n$ set must have (at
least
one set of)$n+1$ collinear points.
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G. Fasshauer: On Choosing Optimal Shape
Parameters for RBF Approximation
Many radial basis functions contain a free parameter that can be tuned by
the user in order to
obtain a good balance between accuracy and stability. This dependence is
known in the literature as
the uncertainty or trade-off principle. The most popular
strategy for choosing an
optimal shape parameter is the leave-one-out cross validation
algorithm proposed by Rippa
[1] in the setting of scattered data interpolation.
We will report on extensions of this approach that can be applied in the
setting of RBF
pseudospectral methods for the solution of PDEs. Alternative strategies
are investigated that
include both the use of multiple shape parameters and more stable basis
functions.
Rippa, S.,
An algorithm for selecting a good value for the parameter $c$ in radial
basis function interpolation,
{\it } {\bf 11}, 193-210.
[1] Rippa, S.,
An algorithm for selecting a good value for the parameter c
in radial basis function interpolation,
,
Adv. in Comp. Math., 11, pp. 193-210.
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A. Iske: Multiscale Flow Simulation by Adaptive
Particle Methods
Particle models have provided very flexible discretization schemes
for the numerical simulation of multiscale phenomena in time-dependent
evolution processes. This talk reports on recent advances concerning
the design and analysis of adaptive particle methods for flow simulation.
To this end, basic tools from approximation, including scattered data
reconstruction by polyharmonis splines, are first discussed, before the
construction of adaptive multiscale algorithms is explained, and selected
of their computational aspects are addressed.
The good performance of the resulting numerical algorithms is demonstrated
in comparison with state-of-the-art simulation methods, where test case
scenarios from real-world applications are utilized.
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J. Levesley: Stabilising the Lattice Boltzmann
Method using Ehrenfests' Steps
The lattice-Boltzmann method (LBM) and its variants have emerged as
promising, computational efficient and increasingly popular
numerical methods for modelling complex fluid flow. However, it is
acknowledged that the method can demonstrate numerical instabilities, e.g.
in the vicinity of shocks.
We propose a simple and novel technique to stabilise the lattice-Boltzmann
method by monitoring the difference between microscopic and macroscopic
entropy.
Populations are returned to their equilibrium states if a threshold value
is exceeded. We coin the name Ehrenfests' steps for this procedure in
homage to the vehicle that we use
to introduce the procedure, namely, the Ehrenfests' idea of coarse
graining.
The one-dimensional shock tube for a compressible isothermal fluid is a
standard benchmark test for hydrodynamic codes. We observe that, of all
the LBMs considered in the
numerical experiment with the one-dimensional shock tube, only the method
which includes Ehrenests' steps is capable of suppressing spurious
post-shock oscillations.
We can aso compare our new method with smoothed particle hydrodynamic
simulations, another of the commonly used simulation techniques for
complex fluid flow.
The work has as coauthors R. Brownlee and A. Gorban.
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L. Montefusco: Numerical aspects in surface
reconstruction with Radial Basis Functions
The problem of
reconstructing surfaces from scattered data using Radial Basis Functions
(RBF) is a widely investigated inverse problem. Hence, a particular
attention must be paid to the numerical aspects involved in its solution.
In fact, it is well known that for very large and highly unevenly
distributed sets of data points the matrices of the resulting linear
systems can be very poorly conditioned and the instability grows as
the regularity of the RBF increases. In this joint work with Giulio
Casciola and Serena Morigi we will present
some different strategies for circumventing this problem while
still maintaining a good level of the reproducing quality of
the reconstruction. A first proposal is concerning with a
local approach to the reconstruction using a partition of unity
strategy that allows us to decompose the large original data set
into smaller subsets with a consequent reduction of numerical
problems. This local control on the reconstructed surface let us adapt
the reconstruction to the shape of the data (cf. [1]).
Nevertheless, the intrinsic nature of the RBF can produce
numerical instabilities even for small data sets if the data are
unevenly distributed. To afford the latter problem we propose
a metric regularization approach based on anisotropic
RBF which can be very efficient in case of particular
data distributions.
[1] G. Casciola, D. Lazzaro, L.B. Montefusco and S.Morigi,
Fast surface
reconstruction and hole filling using Radial Basis Functions,
Numerical Algorithms,
Vol.39, pp.289--305 (2005)
[2] G. Casciola, D. Lazzaro, L.B. Montefusco and
S. Morigi, Shape preserving surface reconstruction using locally
anisotropic RBF Interpolants, Computer and Mathematics with
Applications, to appear.
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T. Sauer: Geometric lattices: construction and
error
The explicit construction of point sets $\Xi$ which allow for unique
interpolation
by $\Pi_n$, the polynomials of total degree at most $n$, is still an
important problem in multivariate interpolation. Many such construction
emerge from the \emph{geometric condition} introduced in the
now classical paper by Chung and Yao. This geometric condition corresponds
to the algebraic property that all \emph{Lagrange fundamental polynomials}
$\ell_\xi$, $\xi \in \Xi$, defined by
$\ell_\xi \left( \xi' \right) = \delta_{\xi,\xi'}$, $\xi,\xi' \in \Xi$,
can
be factorized into linear polynomials, a requirement that is \emph{always}
satisfied in the univariate case but seldom in several variables.
The topic of the talk is to point out that such configurations provide
very simple formulas for the error $f - L_n f$ of the interpolation
operator
$L_n$ applied to a sufficiently smooth function which are ruled by
few geometric quantities and to introduce another method for the
construction of such lattices which is, surprisingly, based on univariate
Haar spaces.
This is joint work with Jesús Carnicer and Mariano Gasca.
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R. Schaback: Kernel Methods
Kernels are valuable tools in various fields of Numerical Analysis,
including approximation,
interpolation, meshless methods for solving partial differential
equations, neural networks, and Machine Learning.
This contribution explains why and how kernels are applied in these
disciplines. It uncovers the links between them,
as far as they are related to kernel techniques. It addresses non-expert
readers and focuses on practical guidelines
for using kernels in applications.
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I. H. Sloan: Radial basis functions and
polynomials: a hybrid approximation on the
sphere
Many researchers have discussed approximation by radial basis functions on
a sphere, using scattered data.
Usually there is no polynomial component in such approximations if, as
here, the kernel that generates
the radial functions is (strictly) positive definite. On the other hand,
the utility of polynomials
for approximating slowly varying components is well known - an extreme
case is the NASA model of
the earth's gravitational potential, which represents the potential by a
purely polynomial
approximation of high degree. In this joint work with Alvise Sommariva
we propose a hybrid
approximation, in which there is a radial basis functions component to
handle the rapidly
varying and localised aspects, but also a polynomial component to handle
the more slowly varying
and global parts. The convergence theory (including a doubled rate of
convergence for
sufficiently smooth functions) makes use of the "native space" associated
with the positive
definite kernel (with no polynomial involvement in the definition). A
numerical experiment
for a simple model with a geophysical flavour establishes the potential
value of the hybrid approach.
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H. Wendland: Recent Resuls on Meshless Symmetric
Collocation
Meshless collocation methods for the numerical solution of
partial
differential equations have recently become more and more
popular.
They provide a greater flexibility when it comes to adaptivity
and
time-dependent changes of the underlying region.
Radial basis functions or, more generally, (conditionally)
positive
definite kernels are one of the main stream methods in
the field of meshless collocation. In this talk, I will give a
survey of
well-known and recent results on meshless, symmetric collocation
for boundary
value problems using positive definite kernels. In particular, I
will address
the following topics
- Well-posedness of the problem, particularly for differential
operators with non-constant coefficients.
- Error analysis in Sobolev spaces.
- Stability analysis of the collocation matrix.
- Stabilization by smoothing.
- Examples.
I will refer to the previous results in [1], [2], [3], [5].
However, this talk is mainly based upon recent results from joint
work with
Francis J. Narcowich and Joseph D. Ward from Texas A&M
University, with
Christian Rieger from the University of Göttingen, and with
Peter Giesl from
the Technical University of Münich [4], [6], [7], [8].
Bibliography
-
G.E. Fasshauer, Solving partial differential equations by collocation with radial basis functions, in Surface Fitting
and Multiresolution Methods, A.L. Mehaute, C.Rabut, and
L.L. Schumaker, eds., Nashville, 1997, Vanderbilt University Press,
pp. 131-138.
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C. Franke and R. Schaback, Convergence order
estimates of meshless collocation methods using radial basis functions, Adv.
Comput. Math., 8 (1998), pp. 381-399.
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C. Franke and R. Schaback, Solving partial differential equations by collocation using radial basis functions, Appl. Math.
Comput., 93 (1998), pp. 73-82.
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P. Giesl and H. Wendland, Meshless collocation: Error
estimates with application to dynamical systems, Preprint, Göttingen/München, 2006.
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R. Lorentz, F. J. Narcowich, and J. D. Ward,
Collocation discretization of the transport equation with radial basis
functions, Appl. Math. Comput., 145 (2003), pp. 97-116.
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F.J. Narcowich, J. D. Ward, and H. Wendland, Sobolev bounds on functions with scattered zeros, with applications to radial
basis function surface fitting, Math. Comput., 74 (2005), pp. 643-763.
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H. Wendland, On the stability of meshless symmetric
collocation for boundary value problems. Preprint, Göttingen, 2006.
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H. Wendland and C. Rieger, Approximate interpolation
with applications to selecting smoothing parameters, Numer. Math., 101 (2005), pp.643-662.
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Y. Xu: A New Reconstruction Algorithm for Radon
Data
We discuss a new algorithm for reconstruction of images from Radon data.
The algorithm is called OPED as it is based on Orthogonal Polynomial
Expansion
on the Disk. OPED is fundamentally different from the filtered back
projection (FBP) method, the main algorithm currently being used in the
computer tomography (CT) and medical image,
OPED allows one to use fan geometry directly without the additional
procedures such as interpolation and rebinning. It reconstructs high
degree polynomials
exactly and converges uniformly for smooth functions without the
assumption that functions are band-limited. Our initial test indicates
that the algorithm is stable, provides high resolution
images, and has small global error.
[1] Y. Xu, A direct approach to the reconstruction of images
from Radon projections, Adv. in Applied Math., in print.
[2] Y. Xu, O. Tischenko and C. Heoschen A new reconstruction
algorithm for Radon Data, SPIE Proceedings of Medical Imaging, 2006,
in print.
[3] Y. Xu, O. Tischenko and C. Heoschen New tomographic
reconstruction algorithm, submitted.
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