| 2019Bastian Bohn, Michael Griebel and Jens Oettershagen
Optimally rotated coordinate systems for adaptive least-squares regression on sparse grids
2019
https://ins.uni-bonn.de/media/public/publication-media/INSPreprint_RotRegr.pdf
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| 2018D. Dũng, M. Griebel, V. N. Huy and C. Rieger
$\varepsilon$-dimension in infinite dimensional hyperbolic cross approximation and application to parametric elliptic PDEs
Journal of Complexity, 46: 66--89 2018
10.1016/j.jco.2017.12.001
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| M. Griebel, C. Rieger and B. Zwicknagl
Regularized Kernel-Based Reconstruction in Generalized Besov Spaces
Foundations of Computational Mathematics, 18(2): 459--508 2018
10.1007/s10208-017-9346-z
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| Michael Griebel, Christian Rieger and Peter Zaspel
Kernel-based stochastic collocation for the random two-phase Navier-Stokes equations
2018
https://ins.uni-bonn.de/media/public/publication-media/INSPreprint1813.pdf
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| 2017B. Bohn and M. Griebel
Error estimates for multivariate regression on discretized function spaces
SIAM Journal on Numerical Analysis, 55(4): 1843--1866 2017
10.1137/15M1013973
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| B. Bohn, M. Griebel and C. Rieger
A representer theorem for deep kernel learning
2017
http://wissrech.ins.uni-bonn.de/research/pub/bohn/INSPreprint_concatRegr.pdf
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| M. Griebel and C. Rieger
Reproducing kernel Hilbert spaces for parametric partial differential equations
SIAM/ASA J. Uncertainty Quantification, 5: 111-137 2017
10.1137/15M1026870
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| 2016B. Bohn, J. Garcke and M. Griebel
A sparse grid based method for generative dimensionality reduction of high-dimensional data
Journal of Computational Physics, 309: 1--17 2016
https://www.sciencedirect.com/science/article/pii/S0021999115008529
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| M. Griebel and J. Oettershagen
On tensor product approximation of analytic functions
Journal of Approximation Theory, 207: 348--379 2016
http://wissrech.ins.uni-bonn.de/research/pub/oettershagen/INSPreprint1512.pdf
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| M. Griebel and P. Oswald
Schwarz Iterative Methods: Infinite Space Splittings
Constructive Approximation, 44(1): 121--139 2016
http://wissrech.ins.uni-bonn.de/research/pub/griebel/GreedyRandomSchwarzInf.pdf
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| 2015Dinh Dũng and Michael Griebel
Hyperbolic cross approximation in infinite dimensions
Journal of Complexity 2015
http://arxiv.org/pdf/1501.01119v1
Abstract: We give tight upper and lower bounds of the cardinality of
the index sets of certain hyperbolic crosses which reflect mixed
Sobolev–Korobov-type smoothness and mixed Sobolev-analytictype
smoothness in the infinite-dimensional case where specific
summability properties of the smoothness indices are fulfilled.
These estimates are then applied to the linear approximation of
functions from the associated spaces in terms of the ε-dimension
of their unit balls. Here, the approximation is based on linear
information. Such function spaces appear for example for the
solution of parametric and stochastic PDEs. The obtained upper
and lower bounds of the approximation error as well as of the
associated ε-complexities are completely independent of any parametric
or stochastic dimension. Moreover, the rates are independent
of the parameters which define the smoothness properties
of the infinite-variate parametric or stochastic part of the solution.
These parameters are only contained in the order constants.
This way, linear approximation theory becomes possible in the
infinite-dimensional case and corresponding infinite-dimensional
problems get tractable. |
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| Michael Griebel, Christian Rieger and Barbara Zwicknagl
Multiscale approximation and reproducing kernel Hilbert space methods
SIAM Journal on Numerical Analysis, 53(2): 852-873 2015
http://dx.doi.org/10.1137/130932144
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| Michael Griebel, Alexander Hullmann and Oeter Oswald
Optimal scaling parameters for sparse grid discretizations
Numerical Linear Algebra with Applications, 22(1): 76-100 2015
http://dx.doi.org/10.1002/nla.1939
Abstract: We apply iterative subspace correction methods to elliptic PDE problems discretized by generalized sparse grid systems. The involved subspace solvers are based on the combination of all anisotropic full grid spaces that are contained in the sparse grid space. Their relative scaling is at our disposal and has significant influence on the performance of the iterative solver. In this paper, we follow three approaches to obtain close-to-optimal or even optimal scaling parameters of the subspace solvers and thus of the overall subspace correction method. We employ a Linear Program that we derive from the theory of additive subspace splittings, an algebraic transformation that produces partially negative scaling parameters which result in improved asymptotic convergence properties, and finally we use the OptiCom method as a variable non-linear preconditioner. |
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| 2014Michael Griebel, Jan Hamaekers and Frederik Heber
A bond order dissection ANOVA approach for efficient electronic structure calculations
In Extraction of Quantifiable Information from Complex Systems, Volume 102 of Lecture Notes in Computational Science and Engineering
Chapter 11, page 211-235.
Publisher: Springer International
2014
http://dx.doi.org/10.1007/978-3-319-08159-5
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| Michael Griebel and Alexander Hullmann
Dimensionality Reduction of High-Dimensional Data with a NonLinear Principal Component Aligned Generative Topographic Mapping
SIAM J. Sci. Comput., 36(3): A1027-A1047 2014
http://dx.doi.org/10.1137/130931382
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| Michael Griebel and Helmut Harbrecht
On the convergence of the combination technique
In Sparse grids and Applications, Volume 97 of Lecture Notes in Computational Science and Engineering
page 55-74.
2014
http://dx.doi.org/10.1007/978-3-319-04537-5_3
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| Michael Griebel and Alexander Hullmann
A Sparse Grid Based Generative Topographic Mapping for the Dimensionality Reduction of High-Dimensional Data
In Modeling, Simulation and Optimization of Complex Processes - HPSC 2012
page 51-62.
2014
http://dx.doi.org/10.1007/978-3-319-09063-4_5
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| Michael Griebel and Jens Oettershagen
Dimension-adaptive sparse grid quadrature for integrals with boundary singularities
In Sparse grids and Applications, Volume 97 of Lecture Notes in Computational Science and Engineering
page 109-136.
2014
http://dx.doi.org/10.1007/978-3-319-04537-5_5
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| Michael Griebel and Jan Hamaekers
Fast Discrete Fourier Transform on Generalized Sparse Grids
In Sparse grids and Applications, Lecture Notes in Computational Science and Engineering Vol. 97, Springer, Volume 97 of Lecture Notes in Computational Science and Engineering
page 75-108.
2014
http://dx.doi.org/10.1007/978-3-319-04537-5_4
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| 2013Michael Griebel and Helmut Harbrecht
A note on the construction of L-fold sparse tensor product spaces
Constructive Approximation, 38(2): 235-251 2013
http://dx.doi.org/10.1007/s00365-012-9178-7
Abstract: In the present paper, we consider the construction of general sparse tensor product spaces in arbitrary space dimensions when the single subdomains are of different dimensionality and the associated ansatz spaces possess different approximation properties. Our theory extends the results from Griebel and Harbrecht (Math. Comput., 2013) for the construction of two-fold sparse tensor product space to arbitrary L-fold sparse tensor product spaces. |
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