Gabriele Santin
Dr.
Gabriele
Santin


Phone 
0049 711 68565295

Room 
7.118 
Email address 
Link

Address

University of Stuttgart
Institute of Applied Analysis and Numerical Simulation
Pfaffenwaldring 57
D70569
Stuttgart
Germany

Research Interests
I work in the field of kernelbased approximation methods.
My current interest is in greedy algorithms for scalar and vectorial data, with applications
to surrogate models.
Preprints
Köppl, T.; Santin, G.; Haasdonk, B. & Helmig, R.:
Numerical modelling of a peripheral arterial stenosis using dimensionally reduced models and machine learning techniques,
University of Stuttgart,
2017.
@techreport
{KSHH2017,
author = {Köppl, Tobias and Santin, Gabriele and Haasdonk, Bernard and Helmig, Rainer}
,
title = {Numerical modelling of a peripheral arterial stenosis using dimensionally reduced models and machine learning techniques}
,
year = {2017}
,
url = {http://www.simtech.unistuttgart.de/publikationen/prints.php?ID=1743}
}
Abstract: In this work, we consider two kinds of model reduction techniques to simulate blood flow through the largest systemic arteries, where a stenosis is located in a peripheral artery i.e. in an artery that is located far away from the heart. For our simulations we place the stenosis in one of the tibial arteries belonging to the right lower leg (right post tibial artery). The model reduction techniques that are used are on the one hand side dimensionally reduced models (1D and 0D models, the socalled mixeddimension model) and on the other hand side surrogate models produced by kernel methods. Both methods are combined in such a way that the mixeddimension models yield training data for the surrogate model, where the surrogate model is parametrised by the degree of narrowing of the peripheral stenosis. By means of a welltrained surrogate model, we show that simulation data can be reproduced with a satisfactory accuracy and that parameter optimisation problems can be solved in a very efficient way. Furthermore it is demonstrated that a surrogate model enables us to present after a very short simulation time the impact of a varying degree of stenosis on blood flow.
Köppel, M.; Franzelin, F.; Kröker, I.; Oladyshkin, S.; Santin, G.; Wittwar, D.; Barth, A.; Haasdonk, B.; Nowak, W.; Pflüger, D. & Rohde, C.:
Comparison of datadriven uncertainty quantification methods for a carbon dioxide storage benchmark scenario,
2017.
@techreport
{UQcomparison2017,
author = {Köppel, M. and Franzelin, F. and Kröker, I. and Oladyshkin, S. and Santin, G. and Wittwar, D. and Barth, A. and Haasdonk, B. and Nowak, W. and Pflüger, D. and Rohde, C.}
,
title = {Comparison of datadriven uncertainty quantification methods for a carbon dioxide storage benchmark scenario}
,
year = {2017}
,
url = {http://www.simtech.unistuttgart.de/publikationen/prints.php?ID=1759}
}
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 datadriven 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 capillarityfree 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 nonintrusive and intrusive uncertainty quantification methods: arbitary polynomial chaos, spatially adaptive sparse grids, kernelbased 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.
Publications
Brünnette, T.; Santin, G. & Haasdonk, B.:
Greedy kernel methods for accelerating implicit integrators for parametric ODEs,
University of Stuttgart,
2018, Proceedings of ENUMATH 2017.
@inproceedings
{Bruennette_Enumath2017,
author = {Brünnette, Tim and Santin, Gabriele and Haasdonk, Bernard}
,
title = {Greedy kernel methods for accelerating implicit integrators for parametric ODEs}
,
year = {2018}
,
volume = {Proceedings of ENUMATH 2017}
,
url = {http://www.simtech.unistuttgart.de/publikationen/prints.php?ID=1767}
}
Abstract: We present a novel acceleration method for the solution of parametric ODEs by singlestep implicit solvers by means of greedy kernelbased surrogate models. In an offline phase, a set of trajectories is precomputed with a highaccuracy 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.
De Marchi, S.; Iske, A. & Santin, G.:
Image reconstruction from scattered Radon data by weighted positive definite kernel functions,
Calcolo,
2018, 55, 2.
@article
{DeMarchi2018,
author = {De Marchi, S. and Iske, A. and Santin, G.}
,
title = {Image reconstruction from scattered Radon data by weighted positive definite kernel functions}
,
journal = {Calcolo}
,
year = {2018}
,
volume = {55}
,
number = {1}
,
pages = {2}
,
url = {https://doi.org/10.1007/s1009201802476}
,
doi = {10.1007/s1009201802476}
}
Abstract: We propose a novel kernelbased method for image reconstruction from scattered Radon data. To this end, we employ generalized HermiteBirkhoff interpolation by positive definite kernel functions. For radial kernels, however, a straightforward application of the generalized HermiteBirkhoff interpolation method fails to work, as we prove in this paper. To obtain a wellposed 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.
De Marchi, S.; Idda, A. & Santin, G.:
Fasshauer, Gregory E. and Schumaker, Larry L. (Eds.),
A Rescaled Method for RBF Approximation,
Approximation Theory XV: San Antonio 2016,
Springer International Publishing,
2017, 3959.
@inbook
{DeMarchi2017b,
author = {De Marchi, Stefano and Idda, Andrea and Santin, Gabriele}
,
editor = {Fasshauer, Gregory E. and Schumaker, Larry L.}
,
title = {A Rescaled Method for RBF Approximation}
,
booktitle = {Approximation Theory XV: San Antonio 2016}
,
publisher = {Springer International Publishing}
,
year = {2017}
,
pages = {3959}
,
url = {https://doi.org/10.1007/9783319599120_3}
,
doi = {10.1007/9783319599120_3}
}
Abstract: In the recent paper [1], a new method to compute stable kernelbased 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.
Haasdonk, B. & Santin, G.:
Greedy Kernel Approximation for Sparse Surrogate Modelling,
Proceedings of the KoMSO Challenge Workshop on ReducedOrder Modeling for Simulation and Optimization,
2017.
@inproceedings
{HS2017a,
author = {B. Haasdonk and G. Santin}
,
title = {Greedy Kernel Approximation for Sparse Surrogate Modelling}
,
booktitle = {Proceedings of the KoMSO Challenge Workshop on ReducedOrder Modeling for Simulation and Optimization}
,
year = {2017}
,
url = {http://www.simtech.unistuttgart.de/publikationen/prints.php?ID=1613}
}
Abstract: Modern simulation scenarios frequently require multiquery or realtime responses of simulation models for statistical analysis, optimization or process control. However, the underlying simulation models may be very timeconsuming rendering the simulation task difficult or infeasible. This motivates the need for rapidly computable surrogate models. We address the case of surrogate modelling 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 inputoutput maps. We review some recent methods and theoretical results in the field of greedy kernel approximation schemes. In particular we recall the VKOGA procedure for approximating vector valued functions. We collect some recent con vergence statements that provide sound foundation for these algorithms, in particular quasioptimal convergence rates in case of kernels inducing Sobolev spaces. We provide some initial experiments that can be obtained with nonsymmetric greedy kernel approximation schemes. The results indicate better stability and hence overall more accurate models.
Santin, G. & Haasdonk, B.:
Convergence rate of the dataindependent Pgreedy algorithm in kernelbased approximation,
Dolomites Research Notes on Approximation,
2017, 10, 6878.
@article
{SH16b,
author = {G. Santin and B. Haasdonk}
,
title = {Convergence rate of the dataindependent Pgreedy algorithm in kernelbased approximation}
,
journal = {Dolomites Research Notes on Approximation}
,
year = {2017}
,
volume = {10}
,
pages = {6878}
,
url = {http://www.emis.de/journals/DRNA/92.html}
}
Abstract: Kernelbased 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 pointselection strategies are known. This paper gives a proof of the convergence rate of the dataindependent $P$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 quasioptimal 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
Cavoretto, R.; De Marchi, S.; De Rossi, A.; Perracchione, E. & Santin, G.:
Approximating basins of attraction for dynamical systems via stable radial bases,
AIP Conf. Proc.,
2016.
@inproceedings
{Cavoretto2016a,
author = {Cavoretto, Roberto and De Marchi, Stefano and De Rossi, Alessandra and Perracchione, Emma and Santin, Gabriele}
,
title = {Approximating basins of attraction for dynamical systems via stable radial bases}
,
booktitle = {AIP Conf. Proc.}
,
year = {2016}
,
url = {http://dx.doi.org/10.1063/1.4952177}
,
doi = {10.1063/1.4952177}
}
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.
Cavoretto, R.; De Marchi, S.; De Rossi, A.; Perracchione, E. & Santin, G.:
Partition of unity interpolation using stable kernelbased techniques,
Applied Numerical Mathematics,
2016.
@article
{Cavoretto2016,
author = {Cavoretto, Roberto and De Marchi, Stefano and De Rossi, Alessandra and Perracchione, Emma and Santin, Gabriele}
,
title = {Partition of unity interpolation using stable kernelbased techniques}
,
journal = {Applied Numerical Mathematics}
,
year = {2016}
,
url = {http://dx.doi.org/10.1016/j.apnum.2016.07.005}
,
doi = {10.1016/j.apnum.2016.07.005}
}
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.
Santin, G. & Schaback, R.:
Approximation of eigenfunctions in kernelbased spaces,
Adv. Comput. Math.,
2016, 42, 973993.
@article
{Santin2016a,
author = {Santin, Gabriele and Schaback, Robert}
,
title = {Approximation of eigenfunctions in kernelbased spaces}
,
journal = {Adv. Comput. Math.}
,
year = {2016}
,
volume = {42}
,
number = {4}
,
pages = {973993}
,
url = {http://dx.doi.org/10.1007/s1044401594495}
,
doi = {10.1007/s1044401594495}
}
Abstract: Kernelbased methods in Numerical Analysis have the advantage of yielding optimal recovery processes in the ``native'' Hilbert space ℋ backslashmathcal H in which they are reproducing. Continuous kernels on compact domains have an expansion into eigenfunctions that are both L 2orthonormal and orthogonal in ℋ backslashmathcal H (Mercer expansion). This paper examines the corresponding eigenspaces and proves that they have optimality properties among all other subspaces of ℋ backslashmathcal H . These results have strong connections to nwidths 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 ndimensional 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.
Cavoretto, R.; De Marchi, S.; De Rossi, A.; Perracchione, E. & Santin, G.:
VigoAguiar, J. (Eds.),
RBF approximation of large datasets by partition of unity and local stabilization,
CMMSE 2015 : Proceedings of the 15th International Conference on Mathematical Methods in Science and Engineering,
2015, 317326.
@inproceedings
{Cavoretto2015,
author = {Cavoretto, Roberto and De Marchi, Stefano and De Rossi, Alessandra and Perracchione, Emma and Santin, Gabriele}
,
editor = {VigoAguiar, J.}
,
title = {RBF approximation of large datasets by partition of unity and local stabilization}
,
booktitle = {CMMSE 2015 : Proceedings of the 15th International Conference on Mathematical Methods in Science and Engineering}
,
year = {2015}
,
pages = {317326}
}
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 eciently deal with large problems and it is robust with respect to the typical instability of kernel methods.
De Marchi, S. & Santin, G.:
Fast computation of orthonormal basis for RBF spaces through Krylov space methods,
BIT Numerical Mathematics,
Springer Netherlands,
2015, 55, 949966.
@article
{DeMarchi2015a,
author = {De Marchi, Stefano and Santin, Gabriele}
,
title = {Fast computation of orthonormal basis for RBF spaces through Krylov space methods}
,
journal = {BIT Numerical Mathematics}
,
publisher = {Springer Netherlands}
,
year = {2015}
,
volume = {55}
,
number = {4}
,
pages = {949966}
,
url = {http://dx.doi.org/10.1007/s1054301405376}
,
doi = {10.1007/s1054301405376}
}
Abstract: In recent years, in the setting of radial basis function, the study of approximation algorithms has particularly focused on the construction of (stable) bases for the associated Hilbert spaces. One of the ways of describing such spaces and their properties is the study of a particular integral operator and its spectrum. We proposed in a recent work the socalled WSVD basis, which is strictly connected to the eigendecomposition of this operator and allows to overcome some problems related to the stability of the computation of the approximant for a wide class of radial kernels. Although effective, this basis is computationally expensive to compute. In this paper we discuss a method to improve and compute in a fast way the basis using methods related to Krylov subspaces. After reviewing the connections between the two bases, we concentrate on the properties of the new one, describing its behavior by numerical tests.
De Marchi, S. & Santin, G.:
A new stable basis for radial basis function interpolation,
J. Comput. Appl. Math.,
2013, 253, 113.
@article
{DeMarchi2013,
author = {De Marchi, Stefano and Santin, Gabriele}
,
title = {A new stable basis for radial basis function interpolation}
,
journal = {J. Comput. Appl. Math.}
,
year = {2013}
,
volume = {253}
,
pages = {113}
,
url = {http://dx.doi.org/10.1016/j.cam.2013.03.048}
,
doi = {10.1016/j.cam.2013.03.048}
}
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 Φ ( Ω ) associated to a kernel Φ on a domain Ω ⊂ 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 Φ ( Ω ) and in ℓ 2 , w ( X ) , where X is a set of data sites of the domain Ω . 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 Φ : N Φ ( Ω ) → N Φ ( Ω ) , T Φ [ f ] ( x ) = ∫ Ω Φ ( x , y ) f ( y ) d y ∀ x ∈ Ω , and provides a connection with the continuous basis that arises from an eigendecomposition of T Φ . Finally, using the eigenvalues of this operator, we provide convergence estimates and stability bounds for interpolation and discrete leastsquares approximation.
Santin, G.; Sommariva, A. & Vianello, M.:
An algebraic cubature formula on curvilinear polygons,
Applied Mathematics and Computation,
2011, 217, 1000310015.
@article
{Santin2011,
author = {Santin, Gabriele and Sommariva, Alvise and Vianello, Marco}
,
title = {An algebraic cubature formula on curvilinear polygons}
,
journal = {Applied Mathematics and Computation}
,
year = {2011}
,
volume = {217}
,
number = {24}
,
pages = {1000310015}
,
url = {http://dx.doi.org/10.1016/j.amc.2011.04.071}
,
doi = {10.1016/j.amc.2011.04.071}
}
Abstract: We implement in Matlab a Gausslike 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.
Thesis
Santin, G.:
Approximation in kernelbased spaces, optimal subspaces and approximation of eigenfunction,
Doctoral School in Mathematical Sciences, University of Padova,
2016.
@phdthesis
{Santin2016Th,
author = {Santin, Gabriele}
,
title = {Approximation in kernelbased spaces, optimal subspaces and approximation of eigenfunction}
,
school = {Doctoral School in Mathematical Sciences, University of Padova}
,
year = {2016}
,
url = {http://paduaresearch.cab.unipd.it/9186/}
}
Teaching
Cosupervised Theses:
 Inverse Radon Transformation mit MultiskalenKernen, MSc in Mathematics.
 Kernel Methods for Accelerating Implicit Integrators, BSc in Simulation Technology.
 Interpolation mit MultiskalenKernen, BSc in Mathematics.
 A comparison of some RBF interpolation methods: theory and numerics, MSc in Mathematics (at
University of Padova).
 Kernelbased medical image reconstruction from Radon data, MSc in Mathematics (at University of
Padova).
Talks
26.9.2017
Greedy kernel methods for accelerating implicit integrators for parametric ODEs,
ENUMATH2017
13.9.2016
Nonsymmetric kernelbased approximation,
DWCAA 2016
30.3.2016
Greedy Kernel Interpolation Surrogate Modeling (Poster),
MORML 2016
610.7.2015
RBF approximation of large datasets by partition of unity and local stabilization,
CMMSE2015
25.9.2014
Approximation in kernel based spaces,
SPAN
813.9.2013
WSVD basis for RBF and Krylov subspaces (Poster),
DRWA13
59.8.2013
A orthonormal basis for Radial Basis Function approximation ,
Isaac 9th Congress
915.6.2013
A fast algorithm for computing a truncated orthonormal basis for RBF native spaces ,
CTF2013
914.9.2012
A new stable basis for RBF approximation (Poster),
DWCAA2012
Further information
My profiles on
ORCID,
Google Scholar,
Research Gate.
Software
2016
Pgreedy:implementation
of the Pgreedy algorithm (MATLAB).
2015
EigenApprox:Approximation
of eigenfunctions in kernel based spaces (MATLAB).
2015
KBMIR:Kernel
based medical image reconstruction (MATLAB).
2014
WSVD
and FCoOB:RBF Approximation with WSVDBasis and Fast WSVDBasis (MATLAB).