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CAA: Stable kernelbased approximation and applications
Kernelbased approximation have become particularly popular in the last decades
especially with the use of RBF (Radial Basis Functions). Our research is addressed
primarily to the analysis and construction of stable bases and their use
in image reconstruction from Radon data and meshless approximations.
People: S. De Marchi (coordinator), A. Martinez, E. Perracchione and D. Poggiali (Padova)
Collaborators: M. Buhmann (Giessen),
R. Cavoretto (Torino), A. De Rossi (Torino), B. Haasdonk (Stuttgart),
A. Iske (Hamburg), M. Rossini (Milano), G. Santin (Stuttgart), R. Schaback (Goettingen), H. Wendland (Bayreuth).
Papers
 Image Reconstruction from Scattered Radon Data by Weighted Positive Definite Kernel Functions
draft, with A. Iske and G. Santin, Calcolo 55(2) (2018), https://doi.org/10.1007/s1009201802476.
 Convergence rate of the dataindependent Pgreedy algorithm in kernelbased approximation
by G. Santin and B. Haasdonk, Dolomites Res. Notes on Approx. 10 (2017), pp. 6878.
 A rescaled method for RBF approximation
by S. De Marchi, A. Idda and G. Santin
Springer Proceedings on Mathematics and Statistics, Vol. 201 (2017), pp.3959.

Partition of unity interpolation using stable kernelbased techniques
by R. Cavoretto, S. De Marchi, A. De Rossi, E. Perracchione
and G. Santin
Appl. Numer. Math. 116 (2017), pp. 95107
online, http://dx.doi.org/10.1016/j.apnum.2016.07.005
 Kernelbased Image Reconstruction from Scattered Radon Data
by S. De Marchi, A. Iske and A. Sironi
Dolomites Res. Notes on Approx. Vol 9 (2016), special issue for the workshop "Kernelbased methods and function approximation", pp. 1931.
available as Hamburger Beitraege
zur Angewandten Mathematik 201611

Approximation of Eigenfunctions in Kernelbased Spaces
by G. Santin and R. Schaback
Adv. Comput. Math. 42(4) (2016), pp. 973993.

RBF approximation of large datasets by partition of unity
and local stabilization
by R. Cavoretto, S. De Marchi, A. De Rossi, E. Perracchione
and G. Santin
Proceedings CMMSE (2015), Vol. IIIIIIIV, pp. 317326.

Fast computation of orthonormal basis for RBF spaces through Krylov space methods
by S. De Marchi and G. Santin
BIT Numerical Mathematics 55(4) (2015), pp. 949966.

A new stable basis for radial basis function interpolation
by S. De Marchi and G. Santin
J. Comp. Appl. Math., Vol. 253 (2013), pp. 113.

Stability of KernelBased Interpolation
by S. De Marchi and R. Schaback
Adv. Comput. Math. Vol. 32(2), 2010, p. 155161
Examples and more:
these are examples and figures illustrating the results of the paper "Stability of KernelBased Interpolation".

Nonstandard Kernels and their Applications
by S. De Marchi and R. Schaback
Dolomites Res. Notes on Approx. (DRNA) Vol. 2, (2009), pp. 1643.

Univariate Radial Basis Functions with Compact Support Cardinal Functions
by L. Bos and S. De Marchi
East J. Approx., Vol. 14(1) 2008, pp. 6980.

NearOptimal Dataindependent Point Locations for Radial Basis Function Interpolation
by with R. Schaback and H. Wendland
Adv. Comput. Math. 23(3) (2005), pp. 317330.
Papers on applications
 Rational RBFbased partition of unity method for efficiently
and accurately approximating 3D objects
by E. Perracchione, arXiv preprint arXiv:1802.01842, 2018. To appear in Comput. Appl. Math.

Optimal selection of local
approximants in RBFPU interpolation using bivariate LOOCV
arXiv preprint 1703.04282  R. Cavoretto, A. De Rossi and
E. Perracchione
J. Sci. Comput., to appear

Approximating basins of attraction for dynamical systems
via stable radial bases
preprint  R. Cavoretto, S. De Marchi, A. De Rossi, E. Perracchione
and G. Santin
AIP Conference Proceedings, 1738, 390003 (2016); doi:10.1063/1.4952177
online
Submitted

Fast and Stable Rational RBFbased Partition of Unity interpolation
by S. De Marchi, A. Martinez and E. Perracchione (Dec. 2017).

RBFbased partition of unity method for elliptic PDEs: Adaptivity and stability issues via VSKs
by S. De Marchi, A. Martinez, E. Perracchione and M. Rossini (June 2017).
 Greedy Kernel Approximation for Sparse Surrogate Modelling
by B. Haasdonk and G. Santin (June 2017).
Posters

Rational stable RBFPU interpolation via VSKs
by S. De Marchi, A. Martinez and E. Perracchione
poster presented at the "Dolomites Research Week on Approximation 2017", Alba di Canazei (Italy), Sept. 2017.
 A rescaled method for RBF approximation
by S. De Marchi, A. Idda and G. Santin
poster presented at "4th Workshop on Constructive Approximation and Applications", Alba di Canazei (TN Italy), Sept. 2016.
 Kernel methods for Radon transform
by S. Guglielmo and G. Santin
poster presented at International CAE Conference 2013, Pacengo del Garda (Italy), Oct. 2013.

WSVD basis for RBF and Krylov subspaces
by S. De Marchi and G. Santin
poster presented at the "Dolomites Research Week on Approximation 2013", Alba di Canazei (Italy), Sept. 2013.

A New Stable Basis for RBF Approximation
by S. De Marchi and G. Santin
poster presented at the "Dolomites Research Week on Approximation 2012", Alba di Canazei (Italy), Sept. 2012.
Software
Books
Meshfree Approximation for MultiAsset European and American Option Problems
by Stefano De Marchi, Maddalena Mandarà and Anna Viero
ISBN: 9788854851511 (2012), pp. 92.
Lecture Notes
Four Lectures on radial basis functions by S. De Marchi and E. Perracchione (2015), pp. 112
Presentations
 S. De Marchi: Kernelbased Image
Reconstruction from scattered Radon data by (anisotropic) positive definite functions
Kernelbased methods and function approximation  Department of Mathematics, University of Torino (Italy), February 5th, 2016.
 S. De Marchi: On a new orthonormal basis for RBF native spaces and its fast computation
Colloqium at the Department of Mathematics, University of Torino (Italy), on June 11th, 2014.
 G. Santin: A fast algorithm for computing a truncated orthonormal basis for RBF native spaces
Multivariate Approximation, Verona 2930 November, 2013.
 S. De Marchi: On a new orthonormal basis for RBF native spaces
San Diego (USA), SIAM Annual meeting: July 8th, 2013.
PhD theses
Master theses
 Medical image reconstruction using kernel based methods, candidate: Amos Sironi,
University of Padua, A. Y. 201011.
 A new stable basis for RBF approximation, candidate: Gabriele Santin, University
of Padua, A. Y. 201112.
 Radial basis functions networks for ODEs: application to diabetes and insulin
therapy models, candidate: Giulia Antinori, University of Padua, A. Y. 201112.
 A Scilab radial basis functions toolbox, candidate: Anna Bassi, University of
Padua, A. Y. 201112.
 Reconstruction of medical images from Radon data in trasmission and emission
tomography, candidate: Davide Poggiali, University of Padua, A. Y. 201112.~demarchi/
 Kernelbased medical image reconstruction, candidate: Maria Angela Narduzzo,
University of Padua, A. Y. 201314.
 Kernelbased medical image reconstruction from Radon data, candidate: Silvia
Guglielmo, University of Padua, A. Y. 201314.
 A comparison of some RBF interpolation methods: theory and numerics, candidate:
Andrea Idda, University of Padova, A.Y. 201415.
 Numerical solution of PDEs on general surfaces by RBFs, candidate: Sara Carlino,
University of Padova, A.Y. 201516.
Degree theses
 Radial basis functions approximation for European call option price, candidate:
Maddalena Mandarà, University of Verona, A. Y. 200708.
 Meshfree approximation for multiasset American option problems, candidate: Anna
Viero, University of Verona, A. Y. 200708.