On the decay rate of the singular values of bivariate functions
On the decay rate of the singular values of bivariate functions
Dokument öffnen (426.3KB)Datum der Veröffentlichung
11.2017Verlag / herausgebende Einrichtung
Institut für Numerische Simulation (INS)Verlagsort
BonnMetadaten
Zur Langanzeige
Zitierbarer Link (Handle URI) 
https://hdl.handle.net/20.500.11811/11839
Inhalt
In this work, we establish a new truncation error estimate of the singular value decomposition (SVD) for a class of Sobolev smooth bivariate functions κ ∈ L2(Ω, Hs(D)), s ≥ 0 and κ ∈ L2(Ω, Ḣs(D)) with D ⊂ ℝ d, where Hs(D) := Ws,2(D) and Ḣs(D) := {ν ∈ L2(D) : (−∆)s/2ν ∈ L2(D)} with −∆ being the negative Laplacian on D coupled with specific boundary conditions. To be precise, we show the order O(M −s/d) for the truncation error of the SVD series expansion after the M -th term. This is achieved by deriving the sharp decay rate O(n−1−2s⁄d) for the square of the n-th largest singular value of the associated integral operator, which improves on known results in the literature. We then use this error estimate to analyze an algorithm for solving a class of elliptic PDEs with random coefficient in the multi-query context, which employs the Karhunen-Loève approximation of the stochastic diffusion coefficient to truncate the model.
Schlagwörter
eigenvalue decay, approximation of bivariate functions, Karhunen-Loève approximation, PDEs with random coefficient
Klassifikation (DDC)
510 Mathematik 518 Numerische AnalysisErstpublikation
https://ins.uni-bonn.de/publication/preprintsZugehörige Publikation(en)
https://doi.org/10.1137/17M1117550Reihe, Nummer
INS Preprints ; 1702Griebel, Michael; Li, Guanglian: On the decay rate of the singular values of bivariate functions. In: INS Preprints, 1702.
Online-Ausgabe in bonndoc: https://hdl.handle.net/20.500.11811/11839
Online-Ausgabe in bonndoc: https://hdl.handle.net/20.500.11811/11839
@unpublished{handle:20.500.11811/11839,
author = {{Michael Griebel} and {Guanglian Li}},
title = {On the decay rate of the singular values of bivariate functions},
publisher = {Institut für Numerische Simulation (INS)},
year = 2017,
month = nov,
INS Preprints},
volume = 1702,
note = {In this work, we establish a new truncation error estimate of the singular value decomposition (SVD) for a class of Sobolev smooth bivariate functions κ ∈ L2(Ω, Hs(D)), s ≥ 0 and κ ∈ L2(Ω, Ḣs(D)) with D ⊂ ℝ d, where Hs(D) := Ws,2(D) and Ḣs(D) := {ν ∈ L2(D) : (−∆)s/2ν ∈ L2(D)} with −∆ being the negative Laplacian on D coupled with specific boundary conditions. To be precise, we show the order O(M −s/d) for the truncation error of the SVD series expansion after the M -th term. This is achieved by deriving the sharp decay rate O(n−1−2s⁄d) for the square of the n-th largest singular value of the associated integral operator, which improves on known results in the literature. We then use this error estimate to analyze an algorithm for solving a class of elliptic PDEs with random coefficient in the multi-query context, which employs the Karhunen-Loève approximation of the stochastic diffusion coefficient to truncate the model.},
url = {https://hdl.handle.net/20.500.11811/11839}
}
author = {{Michael Griebel} and {Guanglian Li}},
title = {On the decay rate of the singular values of bivariate functions},
publisher = {Institut für Numerische Simulation (INS)},
year = 2017,
month = nov,
INS Preprints},
volume = 1702,
note = {In this work, we establish a new truncation error estimate of the singular value decomposition (SVD) for a class of Sobolev smooth bivariate functions κ ∈ L2(Ω, Hs(D)), s ≥ 0 and κ ∈ L2(Ω, Ḣs(D)) with D ⊂ ℝ d, where Hs(D) := Ws,2(D) and Ḣs(D) := {ν ∈ L2(D) : (−∆)s/2ν ∈ L2(D)} with −∆ being the negative Laplacian on D coupled with specific boundary conditions. To be precise, we show the order O(M −s/d) for the truncation error of the SVD series expansion after the M -th term. This is achieved by deriving the sharp decay rate O(n−1−2s⁄d) for the square of the n-th largest singular value of the associated integral operator, which improves on known results in the literature. We then use this error estimate to analyze an algorithm for solving a class of elliptic PDEs with random coefficient in the multi-query context, which employs the Karhunen-Loève approximation of the stochastic diffusion coefficient to truncate the model.},
url = {https://hdl.handle.net/20.500.11811/11839}
}
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