Low-rank approximation of continuous functions in Sobolev spaces with dominating mixed smoothness
Low-rank approximation of continuous functions in Sobolev spaces with dominating mixed smoothness
Dokument öffnen (583.1KB)Datum der Veröffentlichung
03.2022Verlag / herausgebende Einrichtung
Institut für Numerische SimulationVerlagsort
BonnMetadaten
Zur Langanzeige
Zitierbarer Link (Handle URI) 
https://hdl.handle.net/20.500.11811/11786
Inhalt
Let Ωi ⊂ Rni , i = 1, . . . , m, be given domains. In this article, we study the low-rank approximation with respect to L2(Ω1 × · · · × Ωm) of functions from Sobolev spaces with dominating mixed smoothness. To this end, we first estimate the rank of a bivariate approximation, i.e., the rank of the continuous singular value decomposition. In comparison to the case of functions from Sobolev spaces with isotropic smoothness, compare [13, 14], we obtain improved results due to the additional mixed smoothness. This convergence result is then used to study the tensor train decomposition as a method to construct multivariate low-rank approximations of functions from Sobolev spaces with dominating mixed smoothness. We show that this approach is able to beat the curse of dimension.
Schlagwörter
Low-rank approximation, Sobolev spaces with dominating mixed smoothness, approximation error, rank complexity
Klassifikation (DDC)
510 Mathematik 518 Numerische AnalysisErstpublikation
https://ins.uni-bonn.de/publication/preprintsZugehörige Publikation(en)
https://doi.org/10.48550/arXiv.2203.04100Reihe, Nummer
INS Preprints ; 2203Griebel, Michael; Harbrecht, Helmut; Schneider, Reinhold: Low-rank approximation of continuous functions in Sobolev spaces with dominating mixed smoothness. In: INS Preprints, 2203.
Online-Ausgabe in bonndoc: https://hdl.handle.net/20.500.11811/11786
Online-Ausgabe in bonndoc: https://hdl.handle.net/20.500.11811/11786
@unpublished{handle:20.500.11811/11786,
author = {{Michael Griebel} and {Helmut Harbrecht} and {Reinhold Schneider}},
title = {Low-rank approximation of continuous functions in Sobolev spaces with dominating mixed smoothness},
publisher = {Institut für Numerische Simulation},
year = 2022,
month = mar,
INS Preprints},
volume = 2203,
note = {Let Ωi ⊂ Rni , i = 1, . . . , m, be given domains. In this article, we study the low-rank approximation with respect to L2(Ω1 × · · · × Ωm) of functions from Sobolev spaces with dominating mixed smoothness. To this end, we first estimate the rank of a bivariate approximation, i.e., the rank of the continuous singular value decomposition. In comparison to the case of functions from Sobolev spaces with isotropic smoothness, compare [13, 14], we obtain improved results due to the additional mixed smoothness. This convergence result is then used to study the tensor train decomposition as a method to construct multivariate low-rank approximations of functions from Sobolev spaces with dominating mixed smoothness. We show that this approach is able to beat the curse of dimension.},
url = {https://hdl.handle.net/20.500.11811/11786}
}
author = {{Michael Griebel} and {Helmut Harbrecht} and {Reinhold Schneider}},
title = {Low-rank approximation of continuous functions in Sobolev spaces with dominating mixed smoothness},
publisher = {Institut für Numerische Simulation},
year = 2022,
month = mar,
INS Preprints},
volume = 2203,
note = {Let Ωi ⊂ Rni , i = 1, . . . , m, be given domains. In this article, we study the low-rank approximation with respect to L2(Ω1 × · · · × Ωm) of functions from Sobolev spaces with dominating mixed smoothness. To this end, we first estimate the rank of a bivariate approximation, i.e., the rank of the continuous singular value decomposition. In comparison to the case of functions from Sobolev spaces with isotropic smoothness, compare [13, 14], we obtain improved results due to the additional mixed smoothness. This convergence result is then used to study the tensor train decomposition as a method to construct multivariate low-rank approximations of functions from Sobolev spaces with dominating mixed smoothness. We show that this approach is able to beat the curse of dimension.},
url = {https://hdl.handle.net/20.500.11811/11786}
}
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