Abstracts (in arrival order)
  1. ON THE EXTENSION OF THE SHEPARD-BERNOULLI OPERATORS TO HIGHER DIMENSIONS
    F. DELL'ACCIO and F. DI TOMMASO (University of Calabria)

    We present special extensions to the bivariate case of the Shepard-Bernoulli 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) 471-483.
    [6] R. Caira, F. Dell.Accio, Shepard-Bernoulli operators, Math. Comp. 76 (2007) 299-321.

  2. 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 high-order 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), 82-100.
    [4] A. Sommariva and M. Vianello, Computing approximate Fekete points by QR factorizations of Vandermonde matrices, Comput. Math. Appl. 57 (2009), 1324-1336.

  3. 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. Vigo-Aguiar et al. (Eds.), Proceedings of the International Conference on Computational and Mathematical Methods in Science and Engineering, vol. 1, 2009, pp. 258--269.
    [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. 125--134.
    [3] G. E. Fasshauer, Meshfree Approximation Methods with MATLAB, World Scientific Publishers, Singapore, 2007.

  4. LOBACHEVSKY SPLINE FUNCTIONS FOR LANDMARK-BASED 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 landmark-based 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. Vigo-Aguiar et al. (Eds.), Proceedings of the International Conference on Computational and Mathematical Methods in Science and Engineering, vol. 1, 2009, pp. 73-79.
    [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. 970-973.
    [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.

  5. 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, 80-84.
    [2] V.V. Khlobystov and T.N. Popovicheva, Interpolation and identification problems, Cybernetics and Systems Analysis, Vol. 42 3 (2006), 392-397.
    [3] P.M. Prenter, Lagrange and Hermite Interpolation in Banach spaces, J. Approx. Theory 4 (1971), 419-432.
    [4] A. Torokhti and P. Howlett, Computational Methods for modelling of nonlinear systems, Elsevier, Amsterdam, 2007.


  6. 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 exponential-like 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 Rosenbrock-type methods, Appl. Numer. Math., 59(3-4) (2009) 568-581.


  7. 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 well-known imaging techniques in terms of sampling operators. We give some examples using the Blackman-Harris 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 Blackman-Harris kernels we can get kernels with properties suitable different applications.

    References (partial list)
    [1] A. Kivinukk, G. Tamberg. On Blackman-Harris windows for Shannon sampling series. Sampling Theory in Signal and Image Processing, 6 (2007) 87-108.
    [2] A. Kivinukk, G. Tamberg. Interpolating generalized Shannon sampling operators, their norms and approximation properties. Sampling Theory in Signal and Image Processing, 8 (2009) 77-95.
    [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.


  8. 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 time-dependent 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 time-integrators shows that exponential integrators perform quite well.

  9. MATHEMATICAL MODELING OF MITOCHONDRIAL SWELLING
    S. EISENHOFER, F. TOKOS, 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 Ca2+-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 Ca2+-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 Ca2+-induced permeability transition in energized liver mitochondria discriminates between inhibitor mechanisms, J. Biol. Chem., 283 (2008) 665-76
    [2] S. Eisenhofer, B. A. Hense, F. Tokos and H. Zischka: Modeling the volume change in mitochondria, Int. J. Biomath. Biostat. 1 (2010) 53-62
    [3] S. Eisenhofer, F. Tokos, 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) 31942-48
    [5] A. V. Pokhilko, F. I. Ataullakhanov and E. L. Holmuhamedov: Mathematical model of mitochondrial ionic homeostasis: three modes of Ca2+ transport, J. Theor. Biol., 243 (2006) 152-69

  10. MODELLING THE DYNAMICS OF GLUCOSE, INSULIN AND PANCREATIC β-CELL 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 β-cell cycle for the development of diabetes mellitus. We thus investigate the interplay between glucose, insulin and the β-cell 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 β-cell cycle and study the influence of insulin on the proliferation of β-cells.

    References
    [1] G.M. Grodsky: A threshold distribution hypothesis for packet storage of insulin and its mathematical modelling, The Journal of Clinical Investigation, 51:2047-2059, 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, 1-9, 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 4:605-619, 2000.

  11. 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.

  12. FINDING PATTERS IN HIGH DIMENSIONAL REAL-WORLD DATA USING EXAMPLE OF CHILDHOOD DIABETES DATA
    M. HAGEN, W. ZU CASTELL ((Helmholtz Center Munich)

    In the baby-diabetes project huge amount of high-dimensional data is being collected. The data ranges from time-series of antibody-concentrations to categorial data, like gene-snips. 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.