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Zeitschrift für Analysis und ihre Anwendungen


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Volume 39, Issue 4, 2020, pp. 371–394
DOI: 10.4171/ZAA/1664

Published online: 2020-10-22

Singular Value Decomposition in Sobolev Spaces: Part II

Mazen Ali[1] and Anthony Nouy[2]

(1) Centrale Nantes, France
(2) Centrale Nantes, France

Under certain conditions, an element of a tensor product space can be identified with a compact operator and the singular value decomposition (SVD) applies to the latter. These conditions are not fulfilled in Sobolev spaces. In the previous part of this work (part I) [Z. Anal. Anwend. 39 (2020), 349–369], we introduced some preliminary notions in the theory of tensor product spaces. We analyzed low-rank approximations in $H^1$ and the error of the SVD performed in the ambient $L^2$ space. In this work (part II), we continue by considering variants of the SVD in norms stronger than the $L^2norm. Overall and, perhaps surprisingly, this leads to a more difficult control of the $H^1$-error. We briefly consider an isometric embedding of $H^1$ that allows direct application of the SVD to $H^1$-functions. Finally, we provide a few numerical examples that support our theoretical findings.

Keywords: Singular value decomposition (SVD), higher-order singular value decomposition (HOSVD), low-rank approximation, tensor intersection spaces, Sobolev spaces, minimal subspaces

Ali Mazen, Nouy Anthony: Singular Value Decomposition in Sobolev Spaces: Part II. Z. Anal. Anwend. 39 (2020), 371-394. doi: 10.4171/ZAA/1664