Singular Value Decomposition in Sobolev Spaces: Part II

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 $H^1$ 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 $L^{2}norm.Overalland,perhapssurprisingly,thisleadstoamoredifficultcontrolofthe$ H^1$-error.Webrieflyconsideranisometricembeddingof$ H^1 $thatallowsdirectapplicationoftheSVDto$ H^1$-functions. Finally, we provide a few numerical examples that support our theoretical findings.