Regularized kernel based reconstruction in generalized Besov spaces
Regularized kernel based reconstruction in generalized Besov spaces
View/Date of Publication
2015Publisher
Institut für Numerische Simulation (INS)Publisher Location
BonnMetadata
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Citable Link (Handle URI) 
https://hdl.handle.net/20.500.11811/11898
Abstract
We present a theoretical framework for reproducing kernel based reconstruction methods in certain generalized Besov spaces based on positive, essentially self-adjoint operators. An explicit representation of the reproducing kernel is given in terms of an infinite series. We provide stability estimates for the kernel, including inverse Bernstein-type estimates for kernel-based trial spaces, and we give condition estimates for the interpolation matrix. Then, a deterministic error analysis for regularized reconstruction schemes is presented by means of sampling inequalities. In particular, we provide error bounds for a regularized reconstruction scheme based on a numerically feasible approximation of the kernel. This allows us to derive explicit coupling relations between the series truncation, the regularization parameters and the data set.
Subjects
Reproducing kernels, A priori error analysis, Generalized Besov spaces, Feasible reconstruction schemes, Spline smoothing
Classification (DDC)
510 Mathematik 518 Numerische AnalysisFirst Publication
https://ins.uni-bonn.de/publication/preprintsRelated Publications
https://doi.org/10.1007/s10208-017-9346-zPart of Series
INS Preprints ; 1517Griebel, Michael; Rieger, Christian; Zwicknagl, Barbara: Regularized kernel based reconstruction in generalized Besov spaces. In: INS Preprints, 1517.
Online-Ausgabe in bonndoc: https://hdl.handle.net/20.500.11811/11898
Online-Ausgabe in bonndoc: https://hdl.handle.net/20.500.11811/11898
@unpublished{handle:20.500.11811/11898,
author = {{Michael Griebel} and {Christian Rieger} and {Barbara Zwicknagl}},
title = {Regularized kernel based reconstruction in generalized Besov spaces},
publisher = {Institut für Numerische Simulation (INS)},
year = 2015,
INS Preprints},
volume = 1517,
note = {We present a theoretical framework for reproducing kernel based reconstruction methods in certain generalized Besov spaces based on positive, essentially self-adjoint operators. An explicit representation of the reproducing kernel is given in terms of an infinite series. We provide stability estimates for the kernel, including inverse Bernstein-type estimates for kernel-based trial spaces, and we give condition estimates for the interpolation matrix. Then, a deterministic error analysis for regularized reconstruction schemes is presented by means of sampling inequalities. In particular, we provide error bounds for a regularized reconstruction scheme based on a numerically feasible approximation of the kernel. This allows us to derive explicit coupling relations between the series truncation, the regularization parameters and the data set.},
url = {https://hdl.handle.net/20.500.11811/11898}
}
author = {{Michael Griebel} and {Christian Rieger} and {Barbara Zwicknagl}},
title = {Regularized kernel based reconstruction in generalized Besov spaces},
publisher = {Institut für Numerische Simulation (INS)},
year = 2015,
INS Preprints},
volume = 1517,
note = {We present a theoretical framework for reproducing kernel based reconstruction methods in certain generalized Besov spaces based on positive, essentially self-adjoint operators. An explicit representation of the reproducing kernel is given in terms of an infinite series. We provide stability estimates for the kernel, including inverse Bernstein-type estimates for kernel-based trial spaces, and we give condition estimates for the interpolation matrix. Then, a deterministic error analysis for regularized reconstruction schemes is presented by means of sampling inequalities. In particular, we provide error bounds for a regularized reconstruction scheme based on a numerically feasible approximation of the kernel. This allows us to derive explicit coupling relations between the series truncation, the regularization parameters and the data set.},
url = {https://hdl.handle.net/20.500.11811/11898}
}
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