Abstracts (in arrival order)
 ON THE EXTENSION OF THE SHEPARDBERNOULLI OPERATORS TO HIGHER DIMENSIONS
F. DELL'ACCIO and F. DI TOMMASO (University of Calabria)
We present special extensions to the bivariate case of the ShepardBernoulli
operators introduced in [6]
and of several others univariate combined Shepard operators [5]. These new
interpolation operators are realized by using bivariate three point
extensions of univariate expansions formulas in combination with
bivariate Shepard operators and do not require special partitions of the
node's convex hull.
We study properties of the new operators and provide
applications to the scattered data interpolation problem.
References (partial list)
[5] Gh. Coman, R.T. Trimbitas, Combined Shepard univariate operators, East
J. Approx. 7 (2001) 471483.
[6] R. Caira, F. Dell.Accio, ShepardBernoulli operators, Math. Comp. 76
(2007) 299321.
 LOCATING GOOD POINTS FOR MULTIVARIATE POLYNOMIAL APPROXIMATION
L. BOS (University of Verona), S. DE MARCHI, A. SOMMARIVA and M. VIANELLO (University of Padua)
We present a survey on (Weakly) Admissible Meshes and corresponding
Discrete Extremal Sets. These provide new computational tools for
polynomial least squares and interpolation on multidimensional compact
sets, with different applications such as numerical cubature, digital
filtering, spectral and highorder methods for PDEs.
References (partial list)
[1] L. Bos, J.P. Calvi, N. Levenberg, A. Sommariva and M. Vianello,
Geometric Weakly Admissible Meshes, Discrete Least Squares Approximations
and Approximate Fekete Points, Math. Comp., under revision.
[2] L. Bos, S. De Marchi, A. Sommariva and M. Vianello, Computing
multivariate Fekete and Leja points by numerical linear algebra, 2009,
submitted.
[3] J.P. Calvi and N. Levenberg, Uniform approximation by discrete least
squares polynomials, J. Approx. Theory 152 (2008), 82100.
[4] A. Sommariva and M. Vianello, Computing approximate Fekete points by
QR factorizations of Vandermonde matrices, Comput. Math. Appl. 57 (2009),
13241336.
 A FAST INTERPOLATION ALGORITHM USING THE PARTITION OF UNITY METHOD ON THE SPHERE
R. CAVORETTO and A. DE ROSSI (University of Turin)
A fast algorithm for the spherical interpolation of large scattered data sets is proposed.
It is a generalization and an extension of the algorithms presented in the papers [1,2].
The solution scheme involves the partition of unity method (PUM) on the sphere and uses
spherical radial basis functions as local approximants [3].
The associated algorithm is implemented and optimized by applying a spherical nearest
neighbour searching procedure. More specifically, this technique is mainly based on the
partition of the sphere in a suitable number of spherical
zones, the construction of a certain number of subdomains such that the sphere is
contained in the union of the subdomains, with some mild overlap among
the subdomains, and finally the employment of an efficient spherical zone searching procedure.
Computational cost and storage requirements of the spherical algorithm are analyzed as well.
Numerical results show the good accuracy of the spherical PUM method and the high efficiency
of the algorithm.
References
[1] R. Cavoretto and A. De Rossi, A spherical interpolation algorithm using zonal basis functions,
in: J. VigoAguiar et al. (Eds.), Proceedings of the International Conference on Computational
and Mathematical Methods in Science and Engineering, vol. 1, 2009, pp. 258269.
[2] A. De Rossi, Spherical interpolation of large scattered data sets using zonal basis functions,
in: M. Daehlen, K. M rken, L.L. Schumaker (Eds.), Mathematical Methods for Curves and Surfaces,
Nashboro Press, Brentwood, TN, 2005, pp. 125134.
[3] G. E. Fasshauer, Meshfree Approximation Methods with MATLAB, World Scientific Publishers,
Singapore, 2007.
 LOBACHEVSKY SPLINE FUNCTIONS FOR LANDMARKBASED REGISTRATION OF MEDICAL IMAGES
G. ALLASIA, R. CAVORETTO and A. DE ROSSI (University of Turin)
We propose the use of a class of spline functions, called Lobachevsky splines,
for landmarkbased registration (see, e.g., [2,4]).
We recall the analytic expressions of the Lobachevsky splines and some of their properties,
reasoning in the context of probability theory (see [1,3]).
These functions have simple analytic expressions and compact support, but they were never
used in the image registration context. Numerical results show accuracy of Lobachevsky's
splines, comparing them with compactly supported radial basis functions (RBFs), such as
Wendland's functions [5].
References
[1] G. Allasia, Scattered multivariate interpolation by a class of spline functions, in:
J. VigoAguiar et al. (Eds.), Proceedings of the International Conference on
Computational and Mathematical Methods in Science and Engineering, vol. 1, 2009, pp. 7379.
[2] R. Cavoretto and A. De Rossi, A local IDW transformation algorithm for medical image registration,
in: T.E. Simos, G. Psihoyios, Ch. Tsitouras (Eds.), Proceedings of the International
Conference of Numerical Analysis and Applied Mathematics, AIP Conference Proceedings, vol. 1048,
Melville, New York, 2008, pp. 970973.
[3] R. Cavoretto, Meshfree Approximation Methods, Algorithms and Applications, PhD Thesis,
University of Turin, 2010.
[4] J. Modersitzki, Numerical Methods for Image Registration, Oxford Univ. Press, Oxford, 2004.
[5] H. Wendland, Scattered Data Approximation, Cambridge Monogr. Appl. Comput. Math., vol. 17,
Cambridge Univ. Press, Cambridge, 2005.
 LAGRANGE INTERPOLATION ON ARBITRARILY DISTRIBUTED DATA IN BANACH SPACES
G. ALLASIA and C. BRACCO (University of Turin)
The classical Lagrange interpolation problem can be directly generalized to Banach spaces.
This wider setting is quite interesting, because many possible applications, for instance in nonlinear
system modelling and learning theory, can be considered. Here we approach the Lagrange interpolation
problem in Banach spaces by cardinal basis interpolation. Some error estimates are given and the results
of several numerical tests are reported in order
to show the approximation performances of the proposed interpolants.
Finally, some remarks about the application of the cardinal basis interpolants to learning theory are made.
References
[1] G. Allasia and C. Bracco, Two interpolation operators on irregularly distributed data in inner product spaces, Proceedings of the CMMSE09, 2009, 8084.
[2] V.V. Khlobystov and T.N. Popovicheva, Interpolation and identification problems, Cybernetics and Systems Analysis, Vol. 42 3 (2006), 392397.
[3] P.M. Prenter, Lagrange and Hermite Interpolation in Banach spaces, J. Approx. Theory 4 (1971), 419432.
[4] A. Torokhti and P. Howlett, Computational Methods for modelling of nonlinear systems, Elsevier, Amsterdam, 2007.
 MESHFREE EXPONENTIAL INTEGRATORS
M. CALIARI (University of Verona), A. OSTERMANN and S. RAINER (University of
Innsbruck)
In this work, we are concerned with ODEs arising from
a meshfree RBF approximation in space of some nonlinear PDEs with
significant advection. For the time evolution, we use a variable
step size exponential Rosenbrock integrator.
The collocation points for the radial basis functions are
automatically adapted to the solution by
measuring the interpolation errors at the corresponding check points
after each time step. The evaluations of the exponential operator
and of related exponentiallike operators
are approximated by polynomial
interpolation at purely imaginary symmetric Leja nodes.
Some numerical experiments in two space dimensions illustrate
the capability and the reliability of this innovative approach.
References (partial list)
[1] G. E. Fasshauer, Meshfree Approximation Methods with MATLAB, Interdisciplinary Mathematical Sciences  Vol. 6. World Scientific Publishers, Singapore, 2007.
[2] M. Caliari and A. Ostermann, Implementation of exponential Rosenbrocktype methods, Appl. Numer. Math., 59(34) (2009) 568581.
 ON APPLICATIONS OF SHANNON SAMPLING OPERATORS IN IMAGE PROCESSING
G. TAMBERG (Tallinn University of Technology)
In this work, we are concerned with some applications of generalized Shannon sampling operators in image processing. A systematic study of sampling operators for arbitrary kernel
functions was initiated at RWTH Aachen by P. L. Butzer and his students since 1977. The generalized sampling operators give a natural way to represent images as (continuous) functions.
Using sampling operators with appropriate kernel functions gives us an opportunity to represent many wellknown imaging techniques in terms of sampling operators.
We give some examples using the BlackmanHarris and Hann sampling operators, defined with bandlimited kernels.
Applications include image resizing, edge detection (Hann kernel gives Sobel operator), image enhancement. Choosing appropriate parameter values for BlackmanHarris kernels
we can get kernels with properties suitable different applications.
References (partial list)
[1] A. Kivinukk, G. Tamberg. On BlackmanHarris windows for Shannon sampling series. Sampling Theory in Signal and Image Processing, 6 (2007) 87108.
[2] A. Kivinukk, G. Tamberg. Interpolating generalized Shannon sampling operators, their norms and approximation properties. Sampling Theory in Signal and Image Processing, 8 (2009) 7795.
[3] R. L. Stens. Sampling with generalized kernels. In J. R. Higgins and R. L. Stens, editors, Sampling Theory in Fourier and Signal Analysis: Advanced Topics. Clarendon Press, Oxford, 1999.
 RADIAL BASIS FUNCTION APPROXIMATION FOR THE TIME DEPENDENT SCHROEDINGER EQUATION
K. KORMANN and E. LARSSON (Uppsala University)
We present ongoing work on the approximation of the timedependent Schroedinger
equation with radial basis functions. For the spatial coordinates, we consider
both collocation and Galerkin approaches. Then, we propagate the system in time
based on a standard ODE solver. We compare the performance of collocation and
Galerkin approximations and especially address the temporal stability of the
schemes. A numerical study with various timeintegrators shows that exponential
integrators perform quite well.

MATHEMATICAL MODELING
OF MITOCHONDRIAL SWELLING
S. EISENHOFER, F. TOîKOS, B. A. HENSE, F. FILBIR and H.
ZISCHKA (Helmholtz Center Munich)
The permeabilization of mitochondrial membranes accompanied
by mitochondrial swelling is a decisive event in apoptosis or necrosis
culminating in cell death. Swelling of mitochondria is measured by means of
decreasing light scattering values at a fixed wave length. Our aim is to
understand and describe the experimental results mathematically. For that we
introduce two approaches, which model the Ca^{2+}induced swelling from
different perspectives. The first model considers the swelling performance of
single mitochondria and translates this behavior to a whole population [2]. The
second one gives insight into the kinetics of mitochondrial swelling and shows
the total volume changes of a mitochondrial population [3]. Both models are in
accordance with the experimentally determined course of volume increase
throughout the whole swelling process. By that, the existence of a positive
feedback mechanism is confirmed and we obtain consistent parameter changes with
increasing Ca^{2+}concentrations. The second model can be adapted to
other inducing conditions or to mitochondria from other biological sources. To
take into account the spatial dependence, we extended this model and are now
working on a semilinear parabolic PDE modeling the swelling behavior of
mitochondria.
References
[1] S. V. Baranov, I. G. Stavroskaya, A. M. Brown, A. M.
Tyryshkin and B. S. Kristal: Kinetic model for Ca^{2+}induced
permeability transition in energized liver mitochondria discriminates between
inhibitor mechanisms, J. Biol. Chem., 283 (2008) 66576
[2] S. Eisenhofer, B. A. Hense, F. To—kos and H. Zischka: Modeling the
volume change in mitochondria, Int. J. Biomath. Biostat. 1 (2010) 5362
[3] S. Eisenhofer, F. To—kos, B. A. Hense, S. Schulz, F.
Filbir and H. Zischka: A mathematical model of mitochondrial swelling, BMC
Research Notes (2010) 3:67
[4] S. Massari: Kinetic analysis of the mitochondrial
permeability transition, J. Biol. Chem., 271 (1996) 3194248
[5] A. V. Pokhilko, F. I. Ataullakhanov and E. L.
Holmuhamedov: Mathematical model of mitochondrial ionic homeostasis: three
modes of Ca^{2+} transport, J. Theor. Biol., 243 (2006) 15269

MODELLING THE DYNAMICS OF GLUCOSE, INSULIN AND PANCREATIC
βCELL CYCLE
M.GALLENBERGER, W. ZU CASTELL, B. A. HENSE (Helmholtz Center Munich)
C. KUTTLER (Techinical University Munich)
Recent experimental results indicate the relevance of the βcell cycle for the
development of diabetes mellitus. We thus investigate the interplay between
glucose, insulin and the βcell cycle starting with a mathematical model
of ordinary differential equations. Modification of parameters can simulate type
1 as well as type 2 diabetes. Unlike to other existing models we describe
explicitly the βcell cycle and study the influence of insulin on the
proliferation of βcells.
References
[1] G.M. Grodsky: A threshold distribution hypothesis for packet storage of insulin
and its mathematical modelling, The Journal of Clinical Investigation, 51:20472059,
1972.
[2] L. Daukste, B. Basse, B.C. Baguley, and D.J.N. Wall: Using a stem cell and progeny model to illustrate the relationship between cell cycle times
of in vivo human tumour cell tissue populations, in vitro primary cultures and the cell lines derived from them,
Journal of Theoretical Biology, 19, 2009
[3] B. Topp, K. Promislow, G. De Vries, R.M. Miura, and D.T. Finegood, A Model of ?Cell Mass, Insulin, and Glucose Kinetics:
Pathways to Diabetes, Journal of Theoretical Biology, 206, issue 4:605619, 2000.
 MOUNTAIN ASCENT OPTIMIZATION
G. JAKLIC, T. KANDUC, S. PRAPROTNIK and E. ZAGAR (FMF, University of Ljubljana)
Finding an optimal curve on the surface is generally a difficult problem,
frequently encountered in civil engineering, road and railway
construction, etc. In this poster, the search for an optimal mountain
ascent (in the sense of energy consumption) will be considered.
Discrete terrain data are given. A smooth relief description is
constructed using macroelements. Energy functional, which depends on
terrain inclination and path length, is defined. Thus the problem
simplifies to the discrete problem of finding the shortest path on a mesh
of Bezier curves. Numerical results indicate that the resulting paths are
a good approximation of the natural routes. Some examples on real data
will be presented.

FINDING PATTERS IN HIGH DIMENSIONAL REALWORLD DATA USING EXAMPLE OF CHILDHOOD DIABETES DATA
M. HAGEN, W. ZU CASTELL ((Helmholtz Center Munich)
In the babydiabetes project huge amount of highdimensional data is being collected. The data ranges from timeseries of antibodyconcentrations to categorial data,
like genesnips. Biologists are interested in understanding why some children develop diabetes, while others do not, although they don't differ much in each single measured variable.
So far, analysis of the data has only been done univariately. We believe that looking at the data multivariately gives a deeper understanding of the data.
Presented on the poster is an overview of the work done, and an outlook on the steps ahead.