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), we introduced some preliminary notions in the theory of tensor product spaces. We analyzed low-rank approximations in H1 and the error of the SVD performed in the ambient L2 space. In this work (part II), we continue by considering variants of the SVD in norms stronger than the L2-norm. Overall and, perhaps surprisingly, this leads to a more difficult control of the H1-error. We briefly consider an isometric embedding of H1 that allows direct application of the SVD to H1-functions. Finally, we provide a few numerical examples that support our theoretical findings.
Cite
@article{arxiv.1912.11293,
title = {Singular Value Decomposition in Sobolev Spaces: Part II},
author = {Mazen Ali and Anthony Nouy},
journal= {arXiv preprint arXiv:1912.11293},
year = {2022}
}
Comments
17 pages, 2 figures. Portions of this work previously appeared as arXiv:1809.11001 which was split during the refereeing for publication