Multivariate Approximation and Interpolation with Applications

Home > What's on > Workshops > Multivariate Approximation and Interpolation with Applications

Multivariate Approximation and Interpolation with Applications

 06 - 10 Sep 2010
 15 South College Street, Edinburgh EH8 9AA

Organiser

Name
Institution
Davydov, OlegUniversity of Strathclyde
Goodman, TimUniversity of Dundee

Approximation theory has evolved from classical work by Chebyshev, Weierstrass and Bernstein into an area that combines a deep theoretical analysis of approximation with insights leading to the invention of new computational techniques. Such invaluable tools of modern computation as orthogonal polynomials, splines, finite elements, Bézier curves, NURBS, radial basis functions, wavelets and subdivision surfaces have been developed and analysed with the prominent help of ideas coming from approximation theory.

The workshop is devoted to the approximation of functions of two or more variables. This area has many challenging open questions and its wide variety of applications includes problems of computer aided design, mathematical modelling, data interpolation and fitting, signal analysis and image processing.

The focus will be on the following research topics:
• Approximation and interpolation with multivariate polynomials, splines, and radial basis functions.
• Linear and non-linear subdivision.
• Adaptive and multiresolution methods of approximation.
• Shape preserving approximation.

Scientific Advisory Group

Carl de Boor (University of Wisconsin-Madison, USA)
Mira Bozzini (University of Milan, Italy)
Paolo Costantini (University of Siena, Italy)
Nira Dyn (University of Tel Aviv, Israel)
Mariano Gasca (University of Zaragoza, Spain)
Kurt Jetter (University of Stuttgart-Hohenheim, Germany)
Tom Lyche (University of Oslo, Norway)
Alistair Watson (University of Dundee, UK)

PDF file of full report.

Arrangements

Participation
Participation is by invitation only. People interested in participating should contact the organisers. The workshop will begin on Monday 6 September and finish on Friday 10 September 2010.

UK Visas
If you are travelling from overseas you may require an entry visa. A European visa does not guarantee entry to the UK. Please use this link to the UK Visas site to find out if you need a visa and, if so, how to apply for one. If you do require a visa, ICMS can (on request) provide a signed invitation letter.

Venue
The workshop will be held at 15 South College Street, Edinburgh. All lectures will be held in the Newhaven Lecture Theatre. To view this room and a list of the visual equipment available click here. In addition, two blackboards have recently been installed and a PC outputting to a high definition projector and audio system. Follow this link for a map showing the location of 15 South College Street. This map may also prove useful. 

Accommodation
ICMS will arrange en-suite rooms in hotels/ university accommodation nearby for those who request this. Accommodation is typically about 10 to 20 minute walk from 15 South College Street. Participants making their own arrangements may claim back the cost, with original receipts, up to a maximum of £55.00 per night bed and breakfast for a maximum of six nights. A list of Edinburgh accommodation of various sorts and prices is available here. Sections 3-4 are particularly relevant.

Wireless Access
The workshop venue, 15 South College Street, has wireless access throughout. On arrival at Registration you will be given instructions and a code for accessing the wireless network. For those without laptops, there will also be a couple of computers available for you to check your emails.

Travel
Information about travel to the UK and Edinburgh is available here. Lothian buses charge 1.20 GBP for a single journey and 3.00 GBP for a day ticket. Please note that the exact fare is required and no change is given.

A taxi directly from the airport will cost approximately 18.00 to 20.00 GBP to the city centre for a one-way journey.

From the airport, you can take the airport bus Airlink to the city centre. It will cost you 3.50 GBP for a single ticket, or 6.00 GBP return. More details on the airport bus Airlink is available here. The bus will take you to Waverley Bridge (next to Waverley Railway Station).

If travelling by train, please note that Edinburgh has two railway stations - Waverley Railway Station being the main station and closest to the workshop venue at 15 South College Street. If you alight at Edinburgh Waverley, the workshop venue is an easy 10 minute walk over North and South Bridge map. The second railway station is called Haymarket and is at the West End of the city centre. 

Please note that it is your responsibility to have adequate travel insurance to cover medical and other emergencies that may occur on your trip.

Catering
Morning and afternoon refreshments will be provided on each day of the workshop.

Buffet lunch will be provided in the Chapterhouse on the ground floor of 15 South College Street on Monday 6 September only.

The workshop dinner will take place on the evening of Thursday 9 September in The Magnum Restaurant, 1 Albany Street, Edinburgh, EH1 3PY.

The workshop grant will cover the cost of this catering for workshop participants.

Coach Trip to Rosslyn Chapel
On Wednesday 8 September there will be a coach trip to Rosslyn Chapel from 15.00–19.00. This will be free of charge to participants. Registration for the excursion will take place on Monday.

At 14.45 we will assemble for a 3-minute walk to Appleton Tower on Crighton Street to board our coach, which will leave promptly at 15.00, for a 30 minute coach journey to Rosslyn Chapel. This extraordinary building dates back to 1446. The architecture, stone carvings and history draw visitors from all over the globe. A guided tour of the Chapel is arranged for 16.00 and will last approximately 30-40 minutes. There will then be time to wander round the Chapel grounds or visit the Exhibition Room and shop before boarding the coach [at 18.30] for a return to Edinburgh around 19.00. Please note that, on our return, there will be no access to the building at 15 South College Street. 

Registration
Registration will take place 08.30 - 09.30 on Monday 6 September at 15 South College Street, Edinburgh. 

Financial Arrangements
Due to budget restrictions, we ask that you cover the cost of your own travel if possible. Your individual financial arrangements will be laid out in your invitation and repeated in your final email, which will be sent shortly before the workshop. If we have agreed to pay some of your travel costs, you will be informed by email. Reimbursement will take place after the workshop and will involve payment directly into your bank account. At Registration you will be given an expenses claim form and this should be submitted to ICMS, with original receipts. It would be helpful if you could bring your bank details to the workshop. In addition to the bank account number, participants from the USA and Canada will require their bank’s routing number, those from the UK will be asked for the bank sort code, and those from Europe and the rest of the world, their IBAN and SWIFT/BIC code. We cannot reimburse any item without a signed claim form and original receipts. 

Unless otherwise stated in your invitation email, there will be a registration fee of 30.00 GBP for the workshop. We ask that this is paid in advance by using this credit or debit card payment form. The form should be printed out, completed and faxed back (as email is not a secure way of sending credit card information). The fax number is on the form. If it is not possible for you pay in advance, you may print out the credit/debit card form above and bring the completed form along to Registration. We prefer not to handle cash at Registration.

 

Programme

Provisional Programme

Monday 6 September

08.30 - 09.30

Registration and coffee in the Chapterhouse, Ground Floor, 15 South College Street

 

Morning chairperson: Larry Schumaker

09.30 - 10.15

Nira Dyn (Tel Aviv University)
Geometric subdivision schemes

10.15 - 11.00

Johannes Wallner (TU Graz)
Multivariate data in manifolds: subdivision and derived operations

11.00 - 11.30

Coffee/Tea

11.30 - 12.15

Michael Floater (University of Oslo)
A piecewise polynomial approach to analyzing interpolatory subdivision

12.15 - 13.00

Joachim Stöckler (University of Dortmund)
Construction of frames for multivariate subdivision schemes

13.00 - 14.00

Lunch provided in the Chapterhouse (ground floor)

 

Afternoon chairperson: Carl de Boor

14.00 - 14.45

Amos Ron (University of Wisconsin-Madison)
Non linear approximation using surface splines and Gaussians

14.45 - 15.30

Thomas Hangelbroek (Vanderbilt University)
Approximation and interpolation on manifolds with kernels

15.30 - 16.00

Coffee/Tea

16.00 - 16.45

Milvia Rossini (University of Milan)
The detection and recovery of discontinuity curves from scattered data

 

Tuesday 7 September

 

Morning chairperson: Tom Lyche

09.00 - 09.45

Larry Schumaker (Vanderbilt University)
Spline spaces on TR-meshes with hanging vertices

09.45 - 10.30

Günther Nürnberger (University of Mannheim)
Local Lagrange interpolation by splines on tetrahedral partitions

10.30 - 11.00

Coffee/Tea

11.00 - 11.45

Paul Sablonnière (INSA de Rennes)
C1 and C2 bivariate LB-splines and associated interpolants

11.45 - 12.30

Hendrik Speleers (Katholieke Universiteit Leuven)
Constructing a normalized basis for splines on Powell-Sabin triangulations

12.30 - 14.00

Lunch break

 

Afternoon chairperson: Nira Dyn

14.00 - 14.45

Malcolm Sabin (Numerical Geometry Ltd)
Geometric precision

14.45 -15.30

Costanza Conti (University of Florence)
Non-stationary subdivision schemes and their reproduction properties

15.30 - 16.00

Coffee/Tea

16.00 - 16.45

Tomas Sauer (University of Giessen)
Hermite subdivision and factorization

 

Wednesday 8 September

 

Morning chairperson: Mike Powell

09.00 - 09.45

Robert Schaback (University of Göttingen)
Bases for spaces of translates of kernels

09.45 - 10.30

Rick Beatson (University of Canterbury)
Preconditioning radial basis function interpolation problems

10.30 - 11.00

Coffee/Tea

11.00 - 11.45

Joe Ward (Texas A&M University)
A new paradigm for RBF and SBF error estimates

11.45 - 12.30

Kerstin Hesse (University of Sussex)
Smoothing approximation on the sphere from noisy scattered data

15.00 - 19.00

Coach trip to Rosslyn Chapel, Midlothian. See further details above.

 

Thursday 9 September

 

Morning chairperson: Carla Manni

09.00 - 09.45

Ulrich Reif (University of Darmstadt)
Multivariate polynomial interpolation on approximation

09.45 - 10.30

Elena Berdysheva (University of Hohenheim)
Bernstein-Durrmeyer operators with general weight functions

10.30 - 11.00

Coffee/Tea

11.00 - 11.45

Juan Manuel Peña (Universidad de Zaragoza)
Accuracy, stability and applications to CAGD

11.45 - 12.30

Kai Hormann (University of Lugano)
On the Lebesgue constant of barycentric rational interpolation

12.30 - 14.00

Lunch break

 

Afternoon chairperson: Kurt Jetter

14.00 - 14.45

Jesús Carnicer (Universidad de Zaragoza)
Remarks on the progressive iteration approximation property

14.45 - 15.30

Brad Baxter (University of London)
Exponential functionals of Brownian motion and approximation theory: a surprising link

15.30 - 16.00

Coffee/Tea

19.00 -

Workshop Dinner - The Magnum Restaurant, 1 Albany Street, Edinburgh 

 

Friday 10 September

 

Afternoon chairperson: Alistair Watson

09.00 - 09.45

Carla Manni (University of Roma "Tor Vergata")
Isogeometric analysis beyond NURBS

09.45 - 10.30

Maria Lucia Sampoli (University of Siena)
On a class of surfaces for geometric modeling

10.30 - 11.00

Coffee/Tea

11.00 - 11.45

Shai Dekel (Tel-Aviv University & GE-Healthcare)
Anisotropic representations and function spaces

11.45 - 12.30

Yuliya Babenko (Kennesaw State University)
Sharp asymptotics of the error of adaptive approximation and interpolation by some classes of splines

12.30 - 14.00

Lunch break

 

Afternoon chairperson: Martin Buhmann

14.00 - 14.45

Peter Binev (University of South Carolina)
Greedy algorithms for the Reduced Basis Method

14.45 - 15.30

Bin Han (University of Alberta)
Multivariate nonhomogeneous framelets and directional representation

15.30 - 16.15

Coffee/Tea

 

Presentations

Babenko, Yuliya
Sharp asymptotics of the error of adaptive approximation and interpolation by some classes of splines
View Abstract
First we shall briefly present a general scheme for obtaining the asymptotic estimates for the error of interpolation and approximation by splines in various settings (bivariate as well as multivariate). Then we shall introduce our recent results on sharp asymptotics of the interpolation and approximation error in some special cases and discuss possibilities for removal of some technical restrictions. In particular, we will present the sharp asymptotics of the error of approximation by interpolating harmonic splines and the error of asymmetric (sign sensitive) approximation by linear splines of C2 functions on I2.
Baxter, Brad
Exponential functionals of Brownian motion and approximation theory: a surprising link
View Abstract
Exponential functionals of Brownian motion and certain Levy processes, arising in finance and mathematical physics, can be expressed as divided differences, via the Hermite-Genocchi formula. This observation greatly simplifies the resultant formulas, providing new opportunities for approximation theory. As an application, we derive a new bound on a certain correlation coefficient via properties of divided differences.
Beatson, Rick
Preconditioning radial basis function interpolation problems
View Abstract
Radial basis function interpolation is now a well established and useful technique for scattered data interpolation. Unfortunately, the usual formulation of the RBF interpolation problem with a globally supported basic function Phi is often very ill conditioned. This ill conditioning can be viewed as a consequence of a bad choice of basis. In this paper we will discuss a much better choice of basis. The choice considered is based in part upon the mean value coordinates recently introduced by Floater. Theoretical and numerical results will be presented showing the desirable properties of the preconditioner. This is joint work with Oleg Davydov and Jeremy Levesley.
Berdysheva, Elena
Bernstein-Durrmeyer operators with general weight functions
View Abstract
Binev, Peter
Greedy algorithms for the Reduced Basis Method
View Abstract
The Reduced Basis Method is a technique used for rapid solving of a large family of parametric PDEs. We consider this method as approximating the elements of a compact set in a Hilbert space via an appropriate n-dimensional subspace. We introduce a greedy procedure for finding such an approximation and analyse its rate of convergence.
Carnicer, Jesús
Remarks on the progressive iteration approximation property
View Abstract
The progressive iteration approximation property is a useful tool for solving interpolation problems by curves. A drawback of the method is that the convergence speed might be slow. We mention some possibilities for accelerating the convergence related with the convergence analysis of Richardson iterations and preconditioning. We also propose least squares techniques in order to improve the performance of the method.
Conti, Costanza
Non-stationary subdivision schemes and their reproduction properties
View Abstract
Non-stationary subdivision schemes have proven to be efficient iterative algorithms to construct special classes of curves ranging from polynomials or trigonometric curves to conic sections or spirals. The aim of this talk is to establish the algebraic conditions that fully identify the exponential polynomials reproduction properties of a given convergent, univariate, binary, non-stationary subdivision scheme. A subdivision scheme is said to possess the property of reproducing exponential polynomials if, for any initial data uniformly sampled from some exponential polynomial function, the scheme yields the same function in the limit. The importance of this property is due to the fact that several functions obtained as combinations of exponential polynomials (such as conic sections, spirals or special trigonometric and hyperbolic functions) are of great interest in graphical and engineering applications. Since the space of exponential polynomials trivially includes standard polynomials, the results in this work extend the theory recently developed in [1] to the non-stationary context. As the symbol of the scheme changes from level to level and the parametrization plays a crucial role in this kind of study, the proofs of the non-stationary case are often significantly more difficult and intricate than in the stationary case, and much of the results previously obtained can not be straightforwardly generalized but require a complete reformulation. To illustrate the potentialities of these simple but very general algebraic conditions we will consider affine combinations of known subdivision symbols with the aim of creating new non-stationary subdivision schemes with enhanced reproduction properties.

[1] N. Dyn, K. Hormann, M.A. Sabin, Z. Shen, Polynomial reproduction
by symmetric subdivision schemes, J. Approx. Theory, 155, pp. 28-42, 2008. 1
Dekel, Shai
Anisotropic representations and function spaces
View Abstract
We present a framework of anisotropic representations and function spaces based on multilevel ellipsoid covers. The highly anisotropic locally and scale-wise varying structure of the dilations considered here prevents us from using Fourier analysis techniques. Recently, we have been studying the Hardy spaces over this setup, thereby generalizing previous known results.
Dyn, Nira
Geometric subdivision schemes
View Abstract
Subdivision schemes are efficient computational methods for the design, representation and approximation of 2D and 3D curves, and of surfaces of arbitrary topology in 3D. Subdivision schemes generate curves/surfaces from discrete data by repeated refinements. While these methods are simple to implement, their analysis is rather complicated.

First we discuss briefly linear schemes and in particular the linear 4-point scheme. Then we introduce the notion of geometric subdivision schemes, and suggest two different ideas for the extension of the linear 4-point scheme to a geometric 4-point scheme. Properties of these two geometric 4-point schemes are discussed and their performance is demonstrated by examples.

A general sufficient condition for convergence is presented, which applies to the above two geometric schemes and to a new geometric 4-point scheme based on interpolation by circles. Finally the talk is concluded, with a sufficient condition for G1 limit curves.
Floater, Michael
A piecewise polynomial approach to analyzing interpolatory subdivision
View Abstract
The four-point interpolatory subdivision scheme of Dubuc and its generalizations to irregularly spaced data studied by Warren and by Daubechies, Guskov, and Sweldens are based on fitting cubic polynomials locally. In this talk we analyze the convergence of the scheme by viewing the limit function as the limit of piecewise cubic functions arising from the scheme. This allows us to recover the regularity results of Daubechies et al. in a simpler way and to obtain the approximation order of the scheme and its first derivative. If time permits we will discuss how these ideas might be applied to the multivariate setting.
Han, Bin
Multivariate nonhomogeneous framelets and directional representation
View Abstract
In this talk, we present some recent developments on multivariate nonhomogeneous framelets and their basic properties such as perfect reconstruction, stability, sparsity, approximation orders, and algorithmic aspects using filter banks. We demonstrate that stability of framelets has a close relation to subdivision schemes and algorithmic aspects of framelets such as perfect reconstruction property and sparsity can be completely understood in the discrete domain. Moreover, directional representation in high dimensions can be easily achieved using nonstationary nonhomogeneous tight framelets.
Hangelbroek, Thomas
Approximation and interpolation on manifolds with kernels
View Abstract
In this talk I will present very recent results for interpolation and approximation on compact Riemannian manifolds using kernels. I will discuss:
the rapid decay of Lagrange functions,
the L_p stability of bases for the underlying kernel spaces,
the uniform boundedness of Lebesgue constants,
the uniform boundedness of the L_2 projector in L_p,
and progress on specific problems on spheres and SO(3).

This is joint work with Fran Narcowich, Xingping Sun and Joe Ward.
Hesse, Kerstin
Smoothing approximation on the sphere from noisy scattered data
View Abstract
In geophysical applications, measured scattered data usually contains noise, and any approximation method should take this into account. In this talk we discuss the properties of a "smoothing approximation" of noisy scattered data on the sphere by a "hybrid approximant" that is the sum of a low to medium degree polynomial and a radial basis function approximant. This hybrid approximant is computed via solving a large linear system and has the property that it minimizes a certain functional which depends on a smoothing parameter λ > 0 that balances between fitting the data and getting a smooth solution. For λ → 0 , we obtain the interpolation scenario. A crucial question is how this smoothing parameter λ should be chosen depending on the level of the noise, and in this talk we discuss one a-posteriori strategy for choosing λ, namely Morozov's discrepancy principle. For λ chosen with Morozov's discrepancy principle, we give L2-error estimates in terms of powers of the mesh norm and in terms of the noise level. We show that Morozov's discrepancy principle is a valid parameter choice strategy, in that the L2-error tends to zero if the noise level and the mesh norm tend to zero. Numerical tests are presented that illustrate the theoretical work. This talk is about joint work in progress with Ian Sloan and Rob Womersley.
Hormann, Kai
On the Lebesgue constant of barycentric rational interpolation
View Abstract
Approximating a function with rational interpolation often gives excellent results, but it is generally hard to control the occurrence of poles. However, the particular class of barycentric rational interpolants is guaranteed to have no poles and arbitrarily high approximation power. We first give an introduction to this kind of interpolation and then study the corresponding Lebesgue constants, which turn out to have logarithmic growth in the case of equidistant interpolation points.
Manni, Carla
Isogeometric analysis beyond NURBS
View Abstract
In many problems governed by partial differential equations, such as solids, structures and fluids the standard analysis methods, Finite Elements Methods (FEM), are based on crude approximations of the involved geometry. On the other hand, the geometric approximation inherent in the mesh can lead to accuracy problems.

Therefore, in recent years an analysis framework based on functions capable of representing exact geometry was developed, giving rise to isogeometric analysis, where the term isogeometric is due to the fact that the solution space for dependent variables is represented in terms of the same functions which represent the geometry. Isogeometric analysis so far developed is based on NURBS, and has revealed to be an effective tool for several problems of interest in applications.

Nevertheless, the rational model (NURBS) presents several drawbacks, both from the geometrical and the analytical point of view. Thus, our attention has been focused on overcoming some disadvantages of NURBS standards, analyzing isogeometric analysis schemes with different spaces and bases, still equipped with refinement properties, geometric features and classical algorithms as degree elevation, knot insertion, etc. The spaces we deal with admit a representation in term of functions which are a natural extension of polynomial B-splines.
Nürnberger, Günther
Local Lagrange interpolation by splines on tetrahedral partitions
View Abstract
The development of local Lagrange interpolation methods and local Hermite interpolation methods for splines is completely different. For tetrahedral partitions, the construction of local Lagrange interpolation sets is based on the decomposition of the partition into classes of tetrahedra with respect to common vertices, common edges and common faces, while the construction of local Hermite interpolation sets is based on minimal determining sets. A local Lagrange interpolation method for cubic C¹ splines on arbitrary tetrahedral partitions was developed by Hecklin, Nürnberger, Schumaker and Zeilfelder. The advantage of Lagrange interpolation algorithms is that only data values are needed but no derivatives. A special and important aim is to develop algorithms of low locality. For doing this, jointly with Schneider, we construct uniform noncube partitions which can be decomposed into few classes of tetrahedra. This implies that for cubic C¹ splines the method is 2-local and stable, and therefore has optimal approximation order. In addition, jointly with Schneider, we consider splines of higher smoothness, namely C² splines of degree nine on the same partition. While for cubic C¹ splines the partition is decomposed into classes of tetrahedra with respect to common vertices and common edges, for C² splines also common faces have to be considered. The resulting method is 3-local and stable. Our numerical and graphic results confirm the efficiency of the algorithms.
Peña, Juan Manuel
Accuracy, stability and applications to CAGD
View Abstract
Recent accurate and stable computational methods for Approximation Theory and Computer Aided Geometric Design are presented. The relationship with numerical methods for some structured classes of matrices related to positivity is analyzed. In fact, for these classes of matrices many computations can be performed with high relative accuracy.
New results on the localization of the smallest eigenvalue of these classes of matrices are presented and applied to some problems in Computer Aided Geometric Design.
Reif, Ulrich
Multivariate polynomial interpolation on approximation
View Abstract
In the first part of the talk, error estimates for polynomial interpolation on tensor product grids are discussed. In contrast
to univariate interpolation, constants may grow unboundedly as the spacing between interpolation points tends to zero.

In the second part of the talk, we present Bramble-Hilbert-type results for domains which are bounded by diffeomorphic images of graphs. Constants and approximating polynomials are specified explicitly.
Ron, Amos
Nonlinear approximation using surface splines and Gaussians
View Abstract
It is well-known that non-linear approximation has an advantage over linear schemes in the sense that it provides comparable approximation rates to those of the linear schemes, but to a larger class of approximands. This was established for spline approximations and for wavelet approximations. In the talk we extend those results to two other
cases:

- homogeneous radial basis function (surface spline) approximations,

- gaussian function approximations.

We discuss in this talk a new algorithm for approximating functions using dilated translates of the above cases. We show that our algorithm is suitably optimal in the sense that it provides approximation rates similar to other established nonlinear methodologies like spline and wavelet approximations.

The results are joint partly with Ron DeVore and partly with Thomas Hangelbroek.
Rossini, Milvia
The detection and recovery of discontinuity curves from scattered data
View Abstract
In this talk we discuss the detection of faults and gradient faults of a surface from a set of scattered and noisy data. In particular we address the problem of recovering the discontinuity curve by an admissible approximant, that is by a curve which respects the partition of the sample given by the real discontinuity curve.
Sabin, Malcolm
Geometric precision
View Abstract
In classical approximation theory we have some nice concepts and quantitative measures. When dealing with functions we can ask of an interpolation method what the precision set is: i.e. the set of those functions which it reproduces exactly from samples. This then leads to the measure called reproduction degree which is the highest degree of polynomial in the precision set.

In the geometric context things are much less clear. The precision set can be defined as the set of curves which an interpolation reproduces exactly from samples, but we only have the first two examples (straight lines and circles) from which to extrapolate an equivalent for the reproduction degree.

The talk raises this issue, looks at some candidates for the next member of the sequence of desirable-to-reproduce curves and rambles through some other issues which may be relevant.
Sablonnière, Paul
C1 and C2 bivariate LB-splines and associated interpolants
View Abstract
LB-splines are Lagrange splines with compact support. Given a set of data points X := {xα, α ∈ A} in the plane, the Lagrange splines Lα satisfy Lα(xβ)= δα, β for all (α, β) ∈ A2. Some of them have been studied in the literature. Here, we focus on C1 and C2 LB-splines on uniform and quasi-uniform criss-cross triangulations of the plane. Several types of LB-splines and associated Lagrange interpolation operators are presented. A quadratic one was introduced by Powell (1974) in the uniform case, other quadratics were described by the author at conferences in Chamonix (1994), Oujda (2009) and Avignon (2010). The other examples seem to be new. Those operators can often be expressed as linear combinations of quasi-interpolants. Finally, some error estimates and applications are given.
Sampoli, Maria Lucia
On a class of surfaces for geometric modeling
View Abstract
In many applications it is desirable to describe curves and surfaces parametrically. Most CAD/CAM systems represent curves and surfaces in polynomial or rational spline forms. Therefore many conventional geometric modeling operations have been designed to deal with polynomial/rational curves and surfaces.

In several applications it is required to consider the offsetting and more in general the convolution operation. Unfortunately the offset and the convolution of two polynomial/rational curves or surfaces are not necessarily rational. Therefore special families of curves and surfaces which are closed under the offset and the convolution operation have been studied.

It can be shown that these subsets of curves and surfaces can be characterized by some intrinsic geometric properties. For instance for the curve case PH (Pythagorean Hodograph) curves have been introduced. They are polynomial curves whose tangent modulus is a polynomial. An extension to surfaces can be given by LN (Linear Normals) surfaces, characterized by having a linear field of normal vectors. They possess rational offset and convolution surfaces along with many other interesting properties.

Aim of this talk is to give a survey of the results so far obtained in this context, presenting the general ideas behind the main results
Sauer, Tomas
Hermite subdivision and factorization
View Abstract
This talk reports on joint work with J.L. Merrien where we investigate factorization properties of Hermit subdivision schemes in one and several variables. Such factorizations take the place of "sum rules" and relate to preservation of particular polynomial vectors by the Hermite scheme. In the univariate case, "factorization plus contractivity" again gives a sufficient (though not necessary) condition for convergence of the Hermite scheme. In addition, some of the unavoidable multivariate problems will be described.
Schaback, Robert
Bases for spaces of translates of kernels
View Abstract
This talk will review some recent results concerning bases of spaces spanned by translates of symmetric positive definite kernels. Special attention is given to stability properties and generalized Lebesgue constants. Results are partially based on joint work with Stefano De Marchi, Stefan Müller and Maryam Pazoki.
Schumaker, Larry
Spline spaces on TR-meshes with hanging vertices
View Abstract
A constructive theory of C0 polynomial spline spaces defined on mixed meshes consisting of triangles and rectangles is developed. These meshes include triangulations with hanging vertices as well as T-meshes. In addition to dimension formulae, the construction of explicit basis functions and their supports and stability are discussed. The approximation power of the spaces is also treated. Joint work with Lujun Wang.
Speleers, Hendrik
Constructing a normalized basis for splines on Powell-Sabin triangulations
View Abstract
We will discuss the construction of a suitable B-spline representation for splines defined on triangulations with a Powell-Sabin refinement. In such a refinement every triangle of the triangulation is split into six sub-triangles. The presented basis functions have a local support, they are non-negative, and they form a partition of unity. The construction can be geometrically interpreted as determining a set of triangles that must contain a specific set of points. This B-spline representation allows a natural definition of control points and control polynomials associated with each vertex. These control polynomials locally mimic the shape of the spline surface, and they can be used in the design of smooth surfaces. A spline in such a representation can also be evaluated in a stable way using a sequence of simple convex combinations. We will illustrate the construction of this B-spline representation in more detail for the C1-continuous quadratic and C2-continuous quintic Powell-Sabin spline space.
Stöckler, Joachim
Construction of frames for multivariate subdivision schemes
View Abstract
We consider the construction of multivariate tight frames based on subdivision schemes. In a shift-invariant setting, the construction is closely related to the sum-of-squares (SOS) decomposition of non-negative trigonometric polynomials. I will give a short overview of recent results on SOS, and then include specific examples of tight frames for Loop and butterfly subdivision. Some remarks on extraordinary vertices in Loop subdivision will be added.
Wallner, Johannes
Multivariate data in manifolds: subdivision and derived operations
View Abstract
In the last years several authors have been studying subdivision schemes and multiscale operations derived from them which apply to manifold-valued data. Many properties of linear operations carry over to their nonlinear analogues which are defined in terms of the natural operations available for given geometric data. In some cases this transfer is challenging: For the question of convergence, there is still a big gap between the proven and the expected results; and regarding the existence of multiscale transforms, some fundamental obstructions are apparent. On the other hand smoothness, approximation and stability properties have been successfully investigated, as are irregular combinatorics. This presentation gives an overview on the current state of research in this area.
Ward, Joe
A new paradigm for RBF and SBF error estimates
View Abstract
Traditional error estimates for scattered data approximation using radial basis functions on Rd or spherical basis functions on Sd were derived in the context of reproducing kernel Hilbert spaces. The RKHS setting limited the quality of the approximation to primarily Hilbert space estimates and the classes of functions to which the estimates applied.

In recent years progress has been made in extending error estimates to far greater classes of functions such as those belonging to Sobolev spaces and Besov spaces at least in the case where no boundary is involved. This talk will highlight some of these developments.

Participants

Name
Institution
Gaelle, AndriamaroUniversity of Strathclyde
Yuliya, BabenkoKennesaw State University
Brad, BaxterUniversity of London
Rick, BeatsonUniversity of Canterbury
Elena, BerdyshevaUniversity of Hohenheim
Peter, BinevUniversity of South Carolina
Mira, BozziniUniversity of Milano Bicocca
Martin, BuhmannJustus-Liebig University
Jesús, CarnicerUniversidad de Zaragoza
Andrew, ChernihUniversity of New South Wales
Costanza, ContiUniversity of Florence
Oleg, DavydovUniversity of Strathclyde
Carl, de BoorUniversity of Wisconsin-Madison
Shai, DekelTel-Aviv University & GE-Healthcare
Nira, DynTel Aviv University
Michael, FloaterUniversity of Oslo
Tim, GoodmanUniversity of Dundee
Bin, HanUniversity of Alberta
Thomas, HangelbroekVanderbilt University
Kerstin, HesseUniversity of Sussex
Kai, HormannUniversity of Lugano
Kurt, JetterUniversity of Hohenheim
Tom, LycheUniversity of Oslo
Carla, ManniUniversity of Roma "Tor Vergata"
Jean-Louis, MerrienINSA de Rennes
Günther, NürnbergerUniversity of Mannheim
Juan Manuel, PeñaUniversidad de Zaragoza
Mike, PowellUniversity of Cambridge
Andrianarivo Fabien, RabarisonUniversity of Strathclyde
Christophe, RabutUniversity of Toulouse
Ulrich, ReifUniversity of Darmstadt
Martin, ReimersUniversity of Oslo
Amos, RonUniversity of Wisconsin-Madison
Milvia, RossiniUniversity of Milan
Malcolm, SabinNumerical Geometry Ltd
Paul, SablonnièreINSA de Rennes
Maria Lucia, SampoliUniversity of Siena
Tomas, SauerUniversity of Giessen
Robert, SchabackUniversity of Göttingen
Larry, SchumakerVanderbilt University
Hendrik, SpeleersKatholieke Universiteit Leuven
Joachim, StöcklerUniversity of Dortmund
Robert, TongNumerical Algorithms Group Ltd
Johannes, WallnerTU Graz
Joe, WardTexas A&M University
Alistair, WatsonUniversity of Dundee
Wee Ping, YeoUniversity of Strathclyde