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Santin, Gabriele

Gabriele Santin

Ph.D.

Research assistant (former employee until 09/2019)
Institute of Applied Analysis and Numerical Simulation
Working Group Numerical Mathematics

Contact

Office Hours

after appointment

Subject

I work in the field of kernel-based approximation methods.
My current interest is in greedy algorithms for scalar and vectorial data, with applications to surrogate models.

Publications

Publlications:

2021
1. M. Kohr, S. E. Mikhailov, and W. L. Wendland, “Dirichlet and transmission problems for anisotropic Stokes and Navier-Stokes systems with L∞ tensor coefficient under relaxed ellipticity condition,” Discrete Contin. Dyn. Syst., vol. 41, no. 9, Art. no. 9, 2021, doi: 10.3934/dcds.2021042.
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  @article{MR4256469, author = {Kohr, Mirela and Mikhailov, Sergey E. and Wendland, Wolfgang L.}, doi = {10.3934/dcds.2021042}, fjournal = {Discrete and Continuous Dynamical Systems. Series A}, issn = {1078-0947}, journal = {Discrete Contin. Dyn. Syst.}, mrclass = {35Q30 (35J57 46E35 76D03 76D05 76D07)}, mrnumber = {4256469}, number = 9, pages = {4421--4460}, title = {Dirichlet and transmission problems for anisotropic {S}tokes and {N}avier-{S}tokes systems with L∞ tensor coefficient under relaxed ellipticity condition}, url = {https://doi.org/10.3934/dcds.2021042}, volume = 41, year = 2021 }
  Link
  https://doi.org/10.3934/dcds.2021042
2019
1. M. Köppel et al., “Comparison of data-driven uncertainty quantification methods for a carbon dioxide storage benchmark scenario,” Comput. Geosci., vol. 2, no. 23, Art. no. 23, 2019, doi: https://doi.org/10.1007/s10596-018-9785-x.
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  @article{koppel2017comparison, abstract = {A variety of methods is available to quantify uncertainties arising within the modeling of flow and transport in carbon dioxide storage, but there is a lack of thorough comparisons. Usually, raw data from such storage sites can hardly be described by theoretical statistical distributions since only very limited data is available. Hence, exact information on distribution shapes for all uncertain parameters is very rare in realistic applications. We discuss and compare four different methods tested for data-driven uncertainty quantification based on a benchmark scenario of carbon dioxide storage. In the benchmark, for which we provide data and code, carbon dioxide is injected into a saline aquifer modeled by the nonlinear capillarity-free fractional flow formulation for two incompressible fluid phases, namely carbon dioxide and brine. To cover different aspects of uncertainty quantification, we incorporate various sources of uncertainty such as uncertainty of boundary conditions, of conceptual model definitions and of material properties. We consider recent versions of the following non-intrusive and intrusive uncertainty quantification methods: arbitary polynomial chaos, spatially adaptive sparse grids, kernel-based greedy interpolation and hybrid stochastic Galerkin. The performance of each approach is demonstrated assessing expectation value and standard deviation of the carbon dioxide saturation against a reference statistic based on Monte Carlo sampling. We compare the convergence of all methods reporting on accuracy with respect to the number of model runs and resolution. Finally we offer suggestions about the methods� advantages and disadvantages that can guide the modeler for uncertainty quantification in carbon dioxide storage and beyond.}, author = {K\{"o}ppel, M. and Franzelin, F. and Kr\{"o}ker, I. and Oladyshkin, S. and Santin, G. and Wittwar, D. and Barth, A. and Haasdonk, B. and Nowak, W. and Pfl\{"u}ger, D. and Rohde, C.}, doi = {https://doi.org/10.1007/s10596-018-9785-x}, journal = {Comput. Geosci.}, number = 23, owner = {santinge}, pages = {339-354}, publisher = {Springer International Publishing}, title = {Comparison of data-driven uncertainty quantification methods for a carbon dioxide storage benchmark scenario}, url = {https://doi.org/10.1007/s10596-018-9785-x}, volume = 2, year = 2019 }
  Link
  https://doi.org/10.1007/s10596-018-9785-x
2. D. Seus, F. A. Radu, and C. Rohde, “A linear domain decomposition method for two-phase flow in porous media,” Numerical Mathematics and Advanced Applications ENUMATH 2017, pp. 603–614, 2019, doi: https://doi.org/10.1007/978-3-319-96415-7_55.
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  @article{seus2019linear, abstract = {This article is a follow up of our submitted paper (D. Seus et al, Comput Methods Appl Mech Eng 333:331--355, 2018) in which a decomposition of the Richards equation along two soil layers was discussed. A decomposed problem was formulated and a decoupling and linearisation technique was presented to solve the problem in each time step in a fixed point type iteration. This article extends these ideas to the case of two-phase in porous media and the convergence of the proposed domain decomposition method is rigorously shown."}, author = {Seus, David and Radu, Florin A. and Rohde, Christian}, doi = {https://doi.org/10.1007/978-3-319-96415-7_55}, editor = {Radu, Florin Adrian and Kumar, Kundan and Berre, Inga and Nordbotten, Jan Martin and Pop, Iuliu Sorin}, isbn = {978-3-319-96415-7}, journal = {Numerical Mathematics and Advanced Applications ENUMATH 2017}, pages = {603-614}, publisher = {Springer International Publishing}, title = {A linear domain decomposition method for two-phase flow in porous media}, url = {https://www.springerprofessional.de/a-linear-domain-decomposition-method-for-two-phase-flow-in-porou/16377432}, year = 2019 }
  Link
  https://www.springerprofessional.de/a-linear-domain-decomposition-method-for-two-phase-flow-in-porou/16377432
2018
1. T. Brünnette, G. Santin, and B. Haasdonk, “Greedy kernel methods for accelerating implicit integrators for parametric ODEs,” 2018. [Online]. Available: http://www.simtech.uni-stuttgart.de/publikationen/prints.php?ID=1767
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  @inproceedings{brunnette2018greedy, abstract = {We present a novel acceleration method for the solution of parametric ODEs by single-step implicit solvers by means of greedy kernel-based surrogate models. In an offline phase, a set of trajectories is precomputed with a high-accuracy ODE solver for a selected set of parameter samples, and used to train a kernel model which predicts the next point in the trajectory as a function of the last one. This model is cheap to evaluate, and it is used in an online phase for new parameter samples to provide a good initialization point for the nonlinear solver of the implicit integrator. The accuracy of the surrogate reflects into a reduction of the number of iterations until convergence of the solver, thus providing an overall speedup of the full simulation. Interestingly, in addition to providing an acceleration, the accuracy of the solution is maintained, since the ODE solver is still used to guarantee the required precision. Although the method can be applied to a large variety of solvers and different ODEs, we will present in details its use with the Implicit Euler method for the solution of the Burgers equation, which results to be a meaningful test case to demonstrate the method's features.}, author = {Br{\"u}nnette, Tim and Santin, Gabriele and Haasdonk, Bernard}, institution = {University of Stuttgart}, owner = {santinge}, title = {Greedy kernel methods for accelerating implicit integrators for parametric ODEs}, url = {http://www.simtech.uni-stuttgart.de/publikationen/prints.php?ID=1767}, volume = {Proceedings of ENUMATH 2017}, year = 2018 }
  Link
  http://www.simtech.uni-stuttgart.de/publikationen/prints.php?ID=1767
2. S. De Marchi, A. Iske, and G. Santin, “Image reconstruction from scattered Radon data by weighted positive definite kernel functions,” Calcolo, vol. 55, no. 1, Art. no. 1, Feb. 2018, doi: 10.1007/s10092-018-0247-6.
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  @article{demarchi2018image, abstract = {We propose a novel kernel-based method for image reconstruction from scattered Radon data. To this end, we employ generalized Hermite--Birkhoff interpolation by positive definite kernel functions. For radial kernels, however, a straightforward application of the generalized Hermite--Birkhoff interpolation method fails to work, as we prove in this paper. To obtain a well-posed reconstruction scheme for scattered Radon data, we introduce a new class of weighted positive definite kernels, which are symmetric but not radially symmetric. By our construction, the resulting weighted kernels are combinations of radial positive definite kernels and positive weight functions. This yields very flexible image reconstruction methods, which work for arbitrary distributions of Radon lines. We develop suitable representations for the weighted basis functions and the symmetric positive definite kernel matrices that are resulting from the proposed reconstruction scheme. For the relevant special case, where Gaussian radial kernels are combined with Gaussian weights, explicit formulae for the weighted Gaussian basis functions and the kernel matrices are given. Supporting numerical examples are finally presented.}, author = {De Marchi, S. and Iske, A. and Santin, G.}, doi = {10.1007/s10092-018-0247-6}, file = {:http\://www.mathematik.uni-stuttgart.de/fak8/ians/publications/files/DeMarchi2017_www_kernel_radon_preprint.pdf:PDF}, issn = {1126-5434}, journal = {Calcolo}, month = {02}, number = 1, owner = {santinge}, pages = 2, title = {Image reconstruction from scattered Radon data by weighted positive definite kernel functions}, url = {https://doi.org/10.1007/s10092-018-0247-6}, volume = 55, year = 2018 }
  Link
  https://doi.org/10.1007/s10092-018-0247-6
3. J. Giesselmann, N. Kolbe, M. Lukacova-Medvidova, and N. Sfakianakis, “Existence and uniqueness of global classical solutions to a two species cancer invasion haptotaxis model,” Accepted for publication in Discrete Contin. Dyn. Syst. Ser. B., 2018, [Online]. Available: https://arxiv.org/abs/1704.08208
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  @article{giesselmann2018existence, author = {Giesselmann, Jan and Kolbe, Niklas and Lukacova-Medvidova, Maria and Sfakianakis, Nikolaos}, journal = {Accepted for publication in Discrete Contin. Dyn. Syst. Ser. B.}, owner = {seusdd}, title = {Existence and uniqueness of global classical solutions to a two species cancer invasion haptotaxis model}, url = {https://arxiv.org/abs/1704.08208}, year = 2018 }
  Link
  https://arxiv.org/abs/1704.08208
4. B. Haasdonk and G. Santin, “Greedy Kernel Approximation for Sparse Surrogate Modeling,” in Reduced-Order Modeling (ROM) for Simulation and Optimization: Powerful Algorithms as Key Enablers for Scientific Computing, W. Keiper, A. Milde, and S. Volkwein, Eds., in Reduced-Order Modeling (ROM) for Simulation and Optimization: Powerful Algorithms as Key Enablers for Scientific Computing. , Cham: Springer International Publishing, 2018, pp. 21--45. doi: 10.1007/978-3-319-75319-5_2.
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  @inbook{haasdonk2018greedy, abstract = {Modern simulation scenarios frequently require multi-query or real-time responses of simulation models for statistical analysis, optimization, or process control. However, the underlying simulation models may be very time-consuming rendering the simulation task difficult or infeasible. This motivates the need for rapidly computable surrogate models. We address the case of surrogate modeling of functions from vectorial input to vectorial output spaces. These appear, for instance, in simulation of coupled models or in the case of approximating general input--output maps. We review some recent methods and theoretical results in the field of greedy kernel approximation schemes. In particular, we recall the vectorial kernel orthogonal greedy algorithm (VKOGA) for approximating vector-valued functions. We collect some recent convergence statements that provide sound foundation for these algorithms, in particular quasi-optimal convergence rates in case of kernels inducing Sobolev spaces. We provide some initial experiments that can be obtained with non-symmetric greedy kernel approximation schemes. The results indicate better stability and overall more accurate models in situations where the input data locations are not equally distributed.}, address = {Cham}, author = {Haasdonk, Bernard and Santin, Gabriele}, booktitle = {Reduced-Order Modeling (ROM) for Simulation and Optimization: Powerful Algorithms as Key Enablers for Scientific Computing}, doi = {10.1007/978-3-319-75319-5_2}, editor = {Keiper, Winfried and Milde, Anja and Volkwein, Stefan}, file = {:http\://www.mathematik.uni-stuttgart.de/fak8/ians/publications/files/HS2017_www_preprint.pdf:PDF}, isbn = {978-3-319-75319-5}, owner = {santinge}, pages = {21--45}, publisher = {Springer International Publishing}, title = {Greedy Kernel Approximation for Sparse Surrogate Modeling}, url = {https://doi.org/10.1007/978-3-319-75319-5_2}, year = 2018 }
  Link
  https://doi.org/10.1007/978-3-319-75319-5_2
5. D. Wittwar and B. Haasdonk, “Greedy Algorithms for Matrix-Valued Kernels,” University of Stuttgart, 2018. [Online]. Available: http://www.simtech.uni-stuttgart.de/publikationen/prints.php?ID=1773
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  @techreport{wittwar2018greedy, abstract = {We are interested in approximating vector-valued functions on a compact set $\Omega\subset \mathbb R^d$. We consider reproducing kernel Hilbert spaces of $\mathbb R^m$-valued functions which each admit a unique matrix-valued reproducing kernel $k$. These spaces seem promising, when modelling correlations between the target function components. The approximation of a function is a linear combination of matrix-valued kernel evaluations multiplied with coefficient vectors. To guarantee a fast evaluation of the approximant the expansion size, i.e. the number of centers $n$ is desired to be small. We thus present three different greedy algorithms by which a suitable set of centers is chosen in an incremental fashion: First, the $P$-Greedy which requires no function evaluations, second and third, the $f$-Greedy and $f/P$-Greedy which require function evaluations but produce centers tailored to the target function. The efficiency of the approaches is investigated on some data from an artificial model.}, author = {Wittwar, Dominik and Haasdonk, Bernard}, file = {:http\://www.mathematik.uni-stuttgart.de/fak8/ians/publications/files/WH18_www_greedy_matrix_valued.pdf:PDF}, institution = {University of Stuttgart}, owner = {santinge}, title = {Greedy Algorithms for Matrix-Valued Kernels}, url = {http://www.simtech.uni-stuttgart.de/publikationen/prints.php?ID=1773}, year = 2018 }
  Link
  http://www.simtech.uni-stuttgart.de/publikationen/prints.php?ID=1773
2017
1. S. De Marchi, A. Idda, and G. Santin, “A Rescaled Method for RBF Approximation,” in Approximation Theory XV: San Antonio 2016, G. E. Fasshauer and L. L. Schumaker, Eds., in Approximation Theory XV: San Antonio 2016. , Cham: Springer International Publishing, 2017, pp. 39--59. doi: 10.1007/978-3-319-59912-0_3.
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  @inbook{demarchi2017rescaled, abstract = {In the recent paper [1], a new method to compute stable kernel-based interpolants has been presented. This rescaled interpolation method combines the standard kernel interpolation with a properly defined rescaling operation, which smooths the oscillations of the interpolant. Although promising, this procedure lacks a systematic theoretical investigation. Through our analysis, this novel method can be understood as standard kernel interpolation by means of a properly rescaled kernel. This point of view allows us to consider its error and stability properties.}, address = {Cham}, author = {De Marchi, Stefano and Idda, Andrea and Santin, Gabriele}, booktitle = {Approximation Theory XV: San Antonio 2016}, doi = {10.1007/978-3-319-59912-0_3}, editor = {Fasshauer, Gregory E. and Schumaker, Larry L.}, isbn = {978-3-319-59912-0}, owner = {santinge}, pages = {39--59}, publisher = {Springer International Publishing}, title = {A Rescaled Method for {RBF} Approximation}, url = {https://doi.org/10.1007/978-3-319-59912-0_3}, year = 2017 }
  Link
  https://doi.org/10.1007/978-3-319-59912-0_3
2. S. De Marchi, A. Iske, and G. Santin, “Image Reconstruction from Scattered Radon Data by Weighted Positive Definite Kernel Functions,” 2017.
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  @techreport{demarchi2017image, author = {De Marchi, Stefano and Iske, Armin and Santin, Gabriele}, file = {:http\://www.mathematik.uni-stuttgart.de/fak8/ians/publications/files/DeMarchi2017_www_kernel_radon_preprint.pdf:PDF}, owner = {santinge}, title = {Image Reconstruction from Scattered Radon Data by Weighted Positive Definite Kernel Functions}, year = 2017 }
3. M. Kutter, C. Rohde, and A.-M. Sändig, “Well-posedness of a two scale model for liquid phase epitaxy with elasticity,” Contin. Mech. Thermodyn., vol. 29, no. 4, Art. no. 4, 2017, doi: 10.1007/s00161-015-0462-1.
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  @article{kutter2017wellposedness, author = {Kutter, Michael and Rohde, Christian and S\"andig, Anna-Margarete}, doi = {10.1007/s00161-015-0462-1}, issn = {0935-1175}, journal = {Contin. Mech. Thermodyn.}, number = 4, owner = {seusdd}, pages = {989-1016}, title = {Well-posedness of a two scale model for liquid phase epitaxy with elasticity}, url = {http://dx.doi.org/10.1007/s00161-015-0462-1}, volume = 29, year = 2017 }
  Link
  http://dx.doi.org/10.1007/s00161-015-0462-1
4. G. Santin and B. Haasdonk, “Convergence rate of the data-independent P-greedy algorithm in kernel-based approximation,” Dolomites Research Notes on Approximation, vol. 10, pp. 68--78, 2017, [Online]. Available: http://www.emis.de/journals/DRNA/9-2.html
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  @article{santin2017convergence, abstract = {Kernel-based methods provide flexible and accurate algorithms for the reconstruction of functions from meshless samples. A major question in the use of such methods is the influence of the samples� locations on the behavior of the approximation, and feasible optimal strategies are not known for general problems. Nevertheless, efficient and greedy point-selection strategies are known. This paper gives a proof of the convergence rate of the data-independent $P$-greedy algorithm, based on the application of the convergence theory for greedy algorithms in reduced basis methods. The resulting rate of convergence is shown to be quasi-optimal in the case of kernels generating Sobolev spaces. As a consequence, this convergence rate proves that, for kernels of Sobolev spaces, the points selected by the algorithm are asymptotically uniformly distributed, as conjectured in the paper where the algorithm has been introduced}, author = {Santin, G. and Haasdonk, B.}, file = {:http\://www.mathematik.uni-stuttgart.de/fak8/ians/publications/files/SH16b_www_p-greedy.pdf:PDF}, journal = {Dolomites Research Notes on Approximation}, owner = {haasdonk}, pages = {68--78}, title = {Convergence rate of the data-independent {P}-greedy algorithm in kernel-based approximation}, url = {http://www.emis.de/journals/DRNA/9-2.html}, volume = 10, year = 2017 }
  Link
  http://www.emis.de/journals/DRNA/9-2.html
2016
1. D. Amsallem and B. Haasdonk, “PEBL-ROM: Projection-Error Based Local Reduced-Order Models,” AMSES, Advanced Modeling and Simulation in Engineering Sciences, vol. 3, no. 6, Art. no. 6, 2016, doi: 10.1186/s40323-016-0059-7.
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  @article{amsallem2016peblrom, author = {Amsallem, D. and Haasdonk, B.}, doi = {10.1186/s40323-016-0059-7}, journal = {AMSES, Advanced Modeling and Simulation in Engineering Sciences}, number = 6, owner = {santinge}, title = {{PEBL-ROM}: Projection-Error Based Local Reduced-Order Models}, url = {https://amses-journal.springeropen.com/articles/10.1186/s40323-016-0059-7}, volume = 3, year = 2016 }
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  https://amses-journal.springeropen.com/articles/10.1186/s40323-016-0059-7
2. R. Cavoretto, S. De Marchi, A. De Rossi, E. Perracchione, and G. Santin, “Approximating basins of attraction for dynamical systems via stable radial bases,” in AIP Conf. Proc., in AIP Conf. Proc. 2016. doi: 10.1063/1.4952177.
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  @inproceedings{cavoretto2016approximating, abstract = {In applied sciences it is often required to model and supervise temporal evolution of populations via dynamical systems. In this paper, we focus on the problem of approximating the basins of attraction of such models for each stable equilibrium point. We propose to reconstruct the basins via an implicit interpolant using stable radial bases, obtaining the surfaces by partitioning the phase space into disjoint regions. An application to a competition model presenting jointly three stable equilibria is considered.}, author = {Cavoretto, Roberto and De Marchi, Stefano and De Rossi, Alessandra and Perracchione, Emma and Santin, Gabriele}, booktitle = {AIP Conf. Proc.}, doi = {10.1063/1.4952177}, file = {:http\://www.mathematik.uni-stuttgart.de/fak8/ians/publications/files/Cavoretto2016b_www_Approx_basins_attraction_RBF.pdf:PDF}, owner = {santinge}, title = {Approximating basins of attraction for dynamical systems via stable radial bases}, url = {http://dx.doi.org/10.1063/1.4952177}, year = 2016 }
  Link
  http://dx.doi.org/10.1063/1.4952177
3. R. Cavoretto, S. De Marchi, A. De Rossi, E. Perracchione, and G. Santin, “Partition of unity interpolation using stable kernel-based techniques,” Applied Numerical Mathematics, 2016, doi: 10.1016/j.apnum.2016.07.005.
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  @article{cavoretto2016partition, abstract = {In this paper we propose a new stable and accurate approximation technique which is extremely effective for interpolating large scattered data sets. The Partition of Unity (PU) method is performed considering Radial Basis Functions (RBFs) as local approximants and using locally supported weights. In particular, the approach consists in computing, for each \{PU\} subdomain, a stable basis. Such technique, taking advantage of the local scheme, leads to a significant benefit in terms of stability, especially for flat kernels. Furthermore, an optimized searching procedure is applied to build the local stable bases, thus rendering the method more efficient.}, author = {Cavoretto, Roberto and De Marchi, Stefano and De Rossi, Alessandra and Perracchione, Emma and Santin, Gabriele}, doi = {10.1016/j.apnum.2016.07.005}, journal = {Applied Numerical Mathematics}, owner = {santinge}, title = {Partition of unity interpolation using stable kernel-based techniques}, url = {http://dx.doi.org/10.1016/j.apnum.2016.07.005}, year = 2016 }
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  http://dx.doi.org/10.1016/j.apnum.2016.07.005
4. D. Garmatter, B. Haasdonk, and B. Harrach, “A reduced Landweber Method for Nonlinear Inverse Problems,” Inverse Problems, vol. 32, no. 3, Art. no. 3, 2016, doi: http://dx.doi.org/10.1088/0266-5611/32/3/035001.
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  @article{garmatter2016reduced, author = {Garmatter, D. and Haasdonk, B. and Harrach, B.}, doi = {http://dx.doi.org/10.1088/0266-5611/32/3/035001}, journal = {Inverse Problems}, number = 3, owner = {santinge}, pages = {1--21}, title = {A reduced {L}andweber Method for Nonlinear Inverse Problems}, volume = 32, year = 2016 }
5. M. Redeker and B. Haasdonk, “A POD-EIM reduced two-scale model for precipitation in porous media,” MCMDS, Mathematical and Computer Modelling of Dynamical Systems, 2016, doi: 10.1080/13873954.2016.1198384.
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  @article{redeker2016podeim, author = {Redeker, M. and Haasdonk, B.}, doi = {10.1080/13873954.2016.1198384}, journal = {MCMDS, Mathematical and Computer Modelling of Dynamical Systems}, owner = {santinge}, title = {A {POD-EIM} reduced two-scale model for precipitation in porous media}, url = {http://www.tandfonline.com/doi/full/10.1080/13873954.2016.1198384}, year = 2016 }
  Link
  http://www.tandfonline.com/doi/full/10.1080/13873954.2016.1198384
6. G. Santin, “Approximation in kernel-based spaces, optimal subspaces and approximation of eigenfunction,” Doctoral School in Mathematical Sciences, University of Padova, 2016. [Online]. Available: http://paduaresearch.cab.unipd.it/9186/
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  @phdthesis{santin2016approximation, author = {Santin, Gabriele}, owner = {santinge}, school = {Doctoral School in Mathematical Sciences, University of Padova}, title = {Approximation in kernel-based spaces, optimal subspaces and approximation of eigenfunction}, url = {http://paduaresearch.cab.unipd.it/9186/}, year = 2016 }
  Link
  http://paduaresearch.cab.unipd.it/9186/
7. G. Santin and R. Schaback, “Approximation of eigenfunctions in kernel-based spaces,” Adv. Comput. Math., vol. 42, no. 4, Art. no. 4, 2016, doi: 10.1007/s10444-015-9449-5.
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  @article{santin2016approximation, abstract = {Kernel-based methods in Numerical Analysis have the advantage of yielding optimal recovery processes in the ``native'' Hilbert space H {\$}{\backslash}mathcal {\{}H{\}}{\$} in which they are reproducing. Continuous kernels on compact domains have an expansion into eigenfunctions that are both L 2-orthonormal and orthogonal in H {\$}{\backslash}mathcal {\{}H{\}}{\$} (Mercer expansion). This paper examines the corresponding eigenspaces and proves that they have optimality properties among all other subspaces of H {\$}{\backslash}mathcal {\{}H{\}}{\$} . These results have strong connections to n-widths in Approximation Theory, and they establish that errors of optimal approximations are closely related to the decay of the eigenvalues. Though the eigenspaces and eigenvalues are not readily available, they can be well approximated using the standard n-dimensional subspaces spanned by translates of the kernel with respect to n nodes or centers. We give error bounds for the numerical approximation of the eigensystem via such subspaces. A series of examples shows that our numerical technique via a greedy point selection strategy allows to calculate the eigensystems with good accuracy.}, author = {Santin, Gabriele and Schaback, Robert}, doi = {10.1007/s10444-015-9449-5}, file = {:http\://www.mathematik.uni-stuttgart.de/fak8/ians/publications/files/SaSc2015_www_Approx_eigen_kernel_based_spaces.pdf:PDF}, fjournal = {Advances in Computational Mathematics}, issn = {1572-9044}, journal = {Adv. Comput. Math.}, number = 4, owner = {santinge}, pages = {973--993}, title = {Approximation of eigenfunctions in kernel-based spaces}, url = {http://dx.doi.org/10.1007/s10444-015-9449-5}, volume = 42, year = 2016 }
  Link
  http://dx.doi.org/10.1007/s10444-015-9449-5
2015
1. R. Cavoretto, S. De Marchi, A. De Rossi, E. Perracchione, and G. Santin, “RBF approximation of large datasets by partition of unity and local stabilization,” in CMMSE 2015 : Proceedings of the 15th International Conference on Mathematical Methods in Science and Engineering, J. Vigo-Aguiar, Ed., in CMMSE 2015 : Proceedings of the 15th International Conference on Mathematical Methods in Science and Engineering. 2015, pp. 317--326.
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  @inproceedings{cavoretto2015approximation, abstract = {We present an algorithm to approximate large dataset by Radial Basis Function (RBF) techniques. The method couples a fast domain decomposition procedure with a localized stabilization method. The resulting algorithm can e�ciently deal with large problems and it is robust with respect to the typical instability of kernel methods.}, author = {Cavoretto, Roberto and De Marchi, Stefano and De Rossi, Alessandra and Perracchione, Emma and Santin, Gabriele}, booktitle = {CMMSE 2015 : Proceedings of the 15th International Conference on Mathematical Methods in Science and Engineering}, editor = {Vigo-Aguiar, J.}, file = {:http\://www.mathematik.uni-stuttgart.de/fak8/ians/publications/files/CaDeRoPeSa2015b_www_RBF_PU_WSVD.pdf:PDF}, isbn = {978-84-617-2230-3}, issn = {2312-0177}, owner = {santinge}, pages = {317--326}, title = {RBF approximation of large datasets by partition of unity and local stabilization}, year = 2015 }
2014
1. W. L. Wendland, “Martin Costabel’s version of the trace theorem revisited,” Math. Methods Appl. Sci., vol. 37 (13), pp. 1924–1955, 2014.
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  @article{wendland2014martin, author = {Wendland, Wolfgang L.}, journal = {Math. Methods Appl. Sci.}, owner = {ingrid}, pages = {1924-1955}, title = {Martin Costabel's version of the trace theorem revisited}, volume = {37 (13)}, year = 2014 }
2013
1. S. De Marchi and G. Santin, “A new stable basis for radial basis function interpolation,” J. Comput. Appl. Math., vol. 253, pp. 1--13, 2013, doi: 10.1016/j.cam.2013.03.048.
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  @article{demarchi2013stable, abstract = {Abstract It is well known that radial basis function interpolants suffer from bad conditioning if the basis of translates is used. In the recent work by Pazouki and Schaback (2011), [5], the authors gave a quite general way to build stable and orthonormal bases for the native space N F ( O ) associated to a kernel F on a domain O ? R s . The method is simply based on the factorization of the corresponding kernel matrix. Starting from that setting, we describe a particular basis which turns out to be orthonormal in N F ( O ) and in l 2 , w ( X ) , where X is a set of data sites of the domain O . The basis arises from a weighted singular value decomposition of the kernel matrix. This basis is also related to a discretization of the compact operator T F : N F ( O ) ? N F ( O ) , T F [ f ] ( x ) = ? O F ( x , y ) f ( y ) d y ? x ? O , and provides a connection with the continuous basis that arises from an eigendecomposition of T F . Finally, using the eigenvalues of this operator, we provide convergence estimates and stability bounds for interpolation and discrete least-squares approximation.}, author = {De Marchi, Stefano and Santin, Gabriele}, doi = {10.1016/j.cam.2013.03.048}, file = {:http\://www.mathematik.uni-stuttgart.de/fak8/ians/publications/files/DeMSa2013_www_WSVD_basis.pdf:PDF}, fjournal = {Journal of Computational and Applied Mathematics}, issn = {0377-0427}, journal = {J. Comput. Appl. Math.}, mrclass = {65D05 (41A05 41A30)}, mrnumber = {3056591}, mrreviewer = {Tae-wan Kim}, owner = {santinge}, pages = {1--13}, title = {A new stable basis for radial basis function interpolation}, url = {http://dx.doi.org/10.1016/j.cam.2013.03.048}, volume = 253, year = 2013 }
  Link
  http://dx.doi.org/10.1016/j.cam.2013.03.048
2011
1. M. Kohr, C. Pintea, and W. L. Wendland, “Dirichlet-transmission problems for general Brinkman operators on Lipschitz and $C^1$ domains in Riemannian manifolds,” Discrete Contin. Dyn. Syst. Ser. B, vol. 15, no. 4, Art. no. 4, 2011, doi: 10.3934/dcdsb.2011.15.999.
  - BibTeX
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  BibTeX
  @article{kohr2011dirichlettransmission, author = {Kohr, Mirela and Pintea, Cornel and Wendland, Wolfgang L.}, doi = {10.3934/dcdsb.2011.15.999}, fjournal = {Discrete and Continuous Dynamical Systems. Series B. A Journal Bridging Mathematics and Sciences}, issn = {1531-3492}, journal = {Discrete Contin. Dyn. Syst. Ser. B}, mrclass = {35R01 (31C12 35J25 42B20 58J32 76D03)}, mrnumber = {2786365 (2012d:35387)}, number = 4, pages = {999--1018}, title = {Dirichlet-transmission problems for general {B}rinkman operators on {L}ipschitz and {$C^1$} domains in {R}iemannian manifolds}, url = {http://dx.doi.org/10.3934/dcdsb.2011.15.999}, volume = 15, year = 2011 }
  Link
  http://dx.doi.org/10.3934/dcdsb.2011.15.999
2. G. Santin, A. Sommariva, and M. Vianello, “An algebraic cubature formula on curvilinear polygons,” Applied Mathematics and Computation, vol. 217, no. 24, Art. no. 24, 2011, doi: 10.1016/j.amc.2011.04.071.
  - BibTeX
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  BibTeX
  @article{santin2011algebraic, abstract = {We implement in Matlab a Gauss-like cubature formula on bivariate domains whose boundary is a piecewise smooth Jordan curve (curvilinear polygons). The key tools are Green�s integral formula, together with the recent software package Chebfun to approximate the boundary curve close to machine precision by piecewise Chebyshev interpolation. Several tests are presented, including some comparisons of this new routine ChebfunGauss with the recent SplineGauss that approximates the boundary by splines.}, author = {Santin, Gabriele and Sommariva, Alvise and Vianello, Marco}, coden = {AMHCBQ}, doi = {10.1016/j.amc.2011.04.071}, file = {:http\://www.mathematik.uni-stuttgart.de/fak8/ians/publications/files/SaSoVi2011_www_An_algebraic_cubature_formula.pdf:PDF}, fjournal = {Appl. Math. Comput.}, issn = {0096-3003}, journal = {Applied Mathematics and Computation}, mrclass = {65D30 (65D32)}, mrnumber = {2806387}, number = 24, owner = {santinge}, pages = {10003--10015}, title = {An algebraic cubature formula on curvilinear polygons}, url = {http://dx.doi.org/10.1016/j.amc.2011.04.071}, volume = 217, year = 2011 }
  Link
  http://dx.doi.org/10.1016/j.amc.2011.04.071
2008
1. B. Haasdonk and M. Ohlberger, “Reduced basis method for finite volume approximations of parametrized linear evolution equations,” ESAIM: M2AN, vol. 42, no. 2, Art. no. 2, Mar. 2008, doi: 10.1051/m2an:2008001.
  - BibTeX
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  BibTeX
  @article{haasdonk2008reduced, author = {Haasdonk, B. and Ohlberger, M.}, doi = {10.1051/m2an:2008001}, journal = {ESAIM: M2AN}, month = {03}, number = 2, owner = {santinge}, pages = {277--302}, title = {Reduced basis method for finite volume approximations of parametrized linear evolution equations}, url = {http://dx.doi.org/10.1051/m2an:2008001}, volume = 42, year = 2008 }
  Link
  http://dx.doi.org/10.1051/m2an:2008001
2. G. C. Hsiao and W. L. Wendland, Boundary integral equations, vol. 164. in Applied Mathematical Sciences, vol. 164. Berlin: Springer-Verlag, 2008, p. xx+618. doi: 10.1007/978-3-540-68545-6.
  - BibTeX
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  BibTeX
  @book{hsiao2008boundary, address = {Berlin}, author = {Hsiao, George C. and Wendland, Wolfgang L.}, doi = {10.1007/978-3-540-68545-6}, isbn = {978-3-540-15284-2}, mrclass = {45-02 (35A08 35J05 35S05 47G10 65-02)}, mrnumber = {2441884 (2009i:45001)}, mrreviewer = {Paul Andrew Martin}, pages = {xx+618}, publisher = {Springer-Verlag}, series = {Applied Mathematical Sciences}, title = {Boundary integral equations}, url = {http://dx.doi.org/10.1007/978-3-540-68545-6}, volume = 164, year = 2008 }
  Link
  http://dx.doi.org/10.1007/978-3-540-68545-6
3. A. Lalegname, A. Sändig, and G. Sewell, “Analytical and numerical treatment of a dynamic crack model,” International Journal of Fracture, vol. 152, no. 2, Art. no. 2, 2008, doi: 10.1007/s10704-008-9274-7.
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  BibTeX
  @article{lalegname2008analytical, abstract = {We discuss the propagation of a running crack in a bounded linear elastic body under shear waves for a simplified 2D-model. This model is described by two coupled equations in the actual configuration: a two-dimensional scalar wave equation in a cracked, bounded domain and an ordinary differential equation derived from an energy balance law. The unknowns are the displacement fields u = u ( y , t ) and the one-dimensional crack tip trajectory h = h ( t ). We assume that the crack grows straight. Based on a paper of Nicaise-S{\"a}ndig, we derive an improved formula for the ordinary differential equation of motion for the crack tip, where the dynamical stress intensity factor occurs. The numerical simulation is an iterative procedure starting from the wave field at time t = t i . The dynamic stress intensity factor will be extracted at t = t i . Its knowledge allows us to compute the crack-tip motion h ( t i +1 ) with corresponding nonuniform crack speed assuming ( t i +1 - t i ) is small. Now, we start from the cracked configuration at time t = t i +1 and repeat the steps. The wave displacements are computed with the FEM-package PDE2D. Some numerical examples demonstrate the proposed method. The influence of finite length of the crack and finite size of the sample on the dynamic stress intensity factor will be discussed in detail.}, affiliation = {Institute of Applied Analysis and Numerical Simulation, University of Stuttgart Pfaffenwaldring 57 70569 Stuttgart Germany}, author = {Lalegname, A. and S{\"a}ndig, A. and Sewell, G.}, doi = {10.1007/s10704-008-9274-7}, issn = {0376-9429}, journal = {International Journal of Fracture}, keyword = {Chemie und Materialwissenschaften}, note = {10.1007/s10704-008-9274-7}, number = 2, pages = {97--125}, publisher = {Springer Netherlands}, title = {Analytical and numerical treatment of a dynamic crack model}, url = {http://dx.doi.org/10.1007/s10704-008-9274-7}, volume = 152, year = 2008 }
  Link
  http://dx.doi.org/10.1007/s10704-008-9274-7
2005
1. B. Haasdonk, “Feature Space Interpretation of SVMs with Indefinite Kernels,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 27, no. 4, Art. no. 4, 2005, doi: http://doi.ieeecomputersociety.org/10.1109/TPAMI.2005.78.
  - BibTeX
  BibTeX
  @article{haasdonk2005feature, address = {Los Alamitos, CA, USA}, author = {Haasdonk, B.}, doi = {http://doi.ieeecomputersociety.org/10.1109/TPAMI.2005.78}, issn = {0162-8828}, journal = {IEEE Transactions on Pattern Analysis and Machine Intelligence}, number = 4, owner = {santinge}, pages = {482-492}, publisher = {IEEE Computer Society}, title = {Feature Space Interpretation of {SVMs} with Indefinite Kernels}, volume = 27, year = 2005 }
2004
1. B. Haasdonk, A. Halawani, and H. Burkhardt, “Adjustable invariant features by partial Haar-integration,” in Proceedings of the 17th International Conference on Pattern Recognition, in Proceedings of the 17th International Conference on Pattern Recognition, vol. 2. 2004, pp. 769–774. doi: http://dx.doi.org/10.1109/ICPR.2004.1334372.
  - BibTeX
  BibTeX
  @inproceedings{haasdonk2004adjustable, author = {Haasdonk, B. and Halawani, A. and Burkhardt, H.}, booktitle = {Proceedings of the 17th International Conference on Pattern Recognition}, doi = {http://dx.doi.org/10.1109/ICPR.2004.1334372}, number = 2, owner = {santinge}, pages = {769- 774}, title = {Adjustable invariant features by partial {H}aar-integration}, volume = 2, year = 2004 }

Teaching

Co-supervised Theses and Projects:

VKOGA validation and selection by log-marginal likelihood, SimTech Projektarbeit
Inverse Radon Transformation mit Multiskalen-Kernen, MSc in Mathematics.
Kernel Methods for Accelerating Implicit Integrators, BSc in Simulation Technology.
Interpolation mit Multiskalen-Kernen, BSc in Mathematics.
A comparison of some RBF interpolation methods: theory and numerics, MSc in Mathematics (at University of Padova).
Kernel-based medical image reconstruction from Radon data, MSc in Mathematics (at University of Padova).