Singular Value Decomposition in Sobolev Spaces: Part II

TL;DR

This paper extends the analysis of singular value decomposition in Sobolev spaces, exploring stronger norms than L2, and discusses the challenges and potential solutions with numerical illustrations.

Contribution

It introduces variants of SVD in stronger Sobolev norms and examines their theoretical properties and practical implications.

Findings

Stronger norms complicate H1-error control.

Embedding H1 allows direct SVD application.

Numerical examples support theoretical results.

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.