×

zbMATH — the first resource for mathematics

Singular value decomposition in Sobolev spaces: part I. (English) Zbl 07242731
Summary: A well known result from functional analysis states that any compact operator between Hilbert spaces admits a singular value decomposition (SVD). This decomposition is a powerful tool that is the workhorse of many methods both in mathematics and applied fields. A prominent application in recent years is the approximation of high-dimensional functions in a low-rank format. This is based on the fact that, under certain conditions, a tensor can be identified with a compact operator and SVD applies to the latter. One key assumption for this application is that the tensor product norm is not weaker than the injective norm. This assumption is not fulfilled in Sobolev spaces, which are widely used in the theory and numerics of partial differential equations. Our goal is the analysis of the SVD in Sobolev spaces. This work consists of two parts. In this manuscript (part I), we address low-rank approximations and minimal subspaces in \(H^1\). We analyze the \(H^1\)-error of the SVD performed in the ambient \(L^2\)-space. In part II, we will address variants of the SVD in norms stronger than the \(L^2\)-norm. We will provide a few numerical examples that support our theoretical findings.
MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46N40 Applications of functional analysis in numerical analysis
65J99 Numerical analysis in abstract spaces
Software:
redbKIT
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Benner, P., Cohen, A., Ohlberger, M. and Willcox, K. (eds.),Model Reduction and Approximation. Theory and Algorithms. Comput. Sci. Eng. 15. Philadelphia (PA): SIAM 2017. · Zbl 1378.65010
[2] Dahmen, W.,Some remarks on multiscale transformations, stability, and biorthogonality.In:Wavelets, Images, and Surface Fitting(Proceedings Chamonix-Mont-Blanc (France) 1993; eds.: P.-J. Laurent et al.). Wellesley (MA): A K Peters 1994, pp. 157 - 188.
[3] Diestel, J., Fourie, J. H. and Swart, J.,The Metric Theory of Tensor Products. Providence (RI): Amer. Math. Soc. 2008. · Zbl 1186.46004
[4] Grasedyck, L., Hierarchical singular value decomposition of tensors.SIAM J. Matrix Anal. Appl.31 (2009/10)(4), 2029 - 2054. · Zbl 1210.65090
[5] Hackbusch, W.,Tensor Spaces and Numerical Tensor Calculus. Springer Ser. Comput. Math. 42. Heidelberg: Springer 2012. · Zbl 1244.65061
[6] Hackbusch, W.,L∞estimation of tensor truncations.Numer. Math.125 (2013)(3), 419 - 440. · Zbl 1282.65046
[7] Hackbusch, W., Truncation of tensors in the hierarchical format.SeMA(2019).
[8] Hackbusch, W. and Khoromskij, B. N. Low-rank Kronecker-product approximation to multi-dimensional nonlocal operators. II. HKT representation of certain operators.Computing76 (2006)(3-4), 203 - 225. · Zbl 1087.65050
[9] Khoromskaia, V.Tensor Numerical Methods in Quantum Chemistry. Oldenbourg: de Gruyter 2018. · Zbl 06562816
[10] Khoromskij, B. N.,Tensor Numerical Methods in Scientific Computing. Oldenbourg: de Gruyter 2018.
[11] Nouy, A., Low-rank methods for high-dimensional approximation and model order reduction. In:Model Reduction and Approximation. Theory and Algorithms(eds.: P. Benner et al.). Philadelphia (PA): SIAM 2017, Chapter 4.
[12] Quarteroni, A., Manzoni, A. and Negri, F.,Reduced Basis Methods for Partial Differential Equations. Unitext 92. Cham: Springer 2016. · Zbl 1337.65113
[13] Urban, K.,Wavelet Methods for Elliptic Partial Differential Equations. Numer. Math. Sci. Comput. Oxford: Oxford Univ. Press 2009. · Zbl 1158.65002
[14] Uschmajew, A.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.