JournalszaaVol. 39, No. 4pp. 371–394

Singular Value Decomposition in Sobolev Spaces: Part II

  • Mazen Ali

    Centrale Nantes, France
  • Anthony Nouy

    Centrale Nantes, France
Singular Value Decomposition in Sobolev Spaces: Part II cover
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Abstract

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 H1H^1 and the error of the SVD performed in the ambient L2L^2 space. In this work (part II), we continue by considering variants of the SVD in norms stronger than the L2norm.Overalland,perhapssurprisingly,thisleadstoamoredifficultcontroloftheL^2norm. Overall and, perhaps surprisingly, this leads to a more difficult control of the H^1error.Webrieflyconsideranisometricembeddingof-error. We briefly consider an isometric embedding of H^1 thatallowsdirectapplicationoftheSVDtothat allows direct application of the SVD to H^1$-functions. Finally, we provide a few numerical examples that support our theoretical findings.

Cite this article

Mazen Ali, Anthony Nouy, Singular Value Decomposition in Sobolev Spaces: Part II. Z. Anal. Anwend. 39 (2020), no. 4, pp. 371–394

DOI 10.4171/ZAA/1664