Abstract
Tensor truncation techniques are based on singular value decompositions. Therefore, the direct error control is restricted to \(\ell ^{2}\) or \(L^{2}\) norms. On the other hand, one wants to approximate multivariate (grid) functions in appropriate tensor formats in order to perform cheap pointwise evaluations, which require \(\ell ^{\infty }\) or \(L^{\infty }\) error estimates. Due to the huge dimensions of the tensor spaces, a direct estimate of \(\left\| \cdot \right\| _{\infty }\) by \(\left\| \cdot \right\| _{2}\) is hopeless. In the paper we prove that, nevertheless, in cases where the function to be approximated is smooth, reasonable error estimates with respect to \(\left\| \cdot \right\| _{\infty }\) can be derived from the Gagliardo–Nirenberg inequality because of the special nature of the singular value decomposition truncation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
If one is interested in estimates with respect to the \(L^{p}\) norm (\(2<p<\infty \)), one can derive the corresponding Gagliardo–Nirenberg inequality \(\left\| \cdot \right\| _{L^{p}}\lesssim \left| \cdot \right| _{m}^{\frac{d}{2m}(1-\frac{2}{p} )}\left\| \cdot \right\| _{L^{2}}^{1-\frac{d}{2m}(1-\frac{2}{p})}.\) Alternatively, one uses the \(L^{\infty }\) estimate from above and the interpolation inequality \(\left\| \cdot \right\| _{L^{p}}\le \left\| \cdot \right\| _{L^{2}}^{2/p}\left\| \cdot \right\| _{\infty }^{1-2/p} .\)
The norm \(\left| \cdot \right| _{m}\) is defined in [8] by all derivatives of order \(m,\) whereas here we use only the non-mixed derivatives.
The induced scalar product in V \(\times \mathbf{V}\) is completely defined by\(\left\langle \mathbf{v},\mathbf{w}\right\rangle =\prod _{j=1}^{d}\left\langle v^{(j)} ,w^{(j)}\right\rangle _{V_{j}}\) for elementary tensors v = \(\bigotimes _{j=1}^{d}v^{(j)}\) and w = \(\bigotimes _{j=1}^{d}w^{(j)}.\)
Truncation reduces the indices \(i_{j}\) to \(\{1,\ldots ,s_{j}\}\) corresponding to the largest singular eigenvalues. The following statements are true for the reduction to any index subset of \(\{1,\ldots ,r_{j} \}.\)
\(\mathcal L (X,Y)\) denotes the space of bounded linear mappings from \(X\) into \(Y.\)
\(A^{(j)}\) may be a mapping from \(domain(A^{(j)})\subset V_{j}\) into \(V_{j}\) or into another Hilbert space \(W_{j}.\) In the latter case, the operator norm has to be changed accordingly.
By assumption v belongs to the domain of A \(_{1}.\) Reading the previous equality from right to left, we conclude from \(\Pi _{[1]}\mathbf{A}_{1}\mathbf{v=A}_{1}\Pi _{[1]}\mathbf{v,}\) that also \(\Pi _{[1]}\mathbf{v}\) belongs to the domain, etc. Hence, \(\Pi \mathbf{v}=\mathbf{w}\) belongs to the domain.
The sum \(\sum _{\alpha }\) is taken over all \(\alpha \in T_{D}\) except \(\alpha =D\) and one of the sons of \(D.\) Hence, the sum contains \(2d-3\) terms.
\(\Pi _{\alpha _{0}}=\Pi _{D}\) can be omitted, since \(\Pi _{D}=I\) because of \(s_{D}=r_{D}=1\).
Note that \(Q_{1}\) is not only the identity on U \(_{\alpha _{1}^{c}}^{\prime }\), but also on \(\varvec{\hat{V}}_{\alpha _{1}^{c}}.\)
For simplicity, the field \(\mathbb R \) is assumed.
In general, \(\Omega =\Omega _{1}\times \cdots \times \Omega _{d}\) must be assumed to be a cone (with origin at \(0\)), i.e., \(x\in \Omega \) implies \(\lambda x\in \Omega \) for all \(\lambda \ge 0.\)
The estimate is not optimised concerning the choice of \(\alpha \) and \(\beta .\) The present choice corresponds to \(\cos (\delta )=\sin (\delta )=1/\sqrt{2}\) in the previous proof. Hence, \(a_{m}=\sqrt{2}\) may be improved.
The sequence \(f=(f_{\nu })\) may be identified with the piecewise constant function \(f=f_{\nu }\) on \([(\nu -1/2)h,(\nu +1/2)h).\)
References
Ballani, J.: Fast evaluation of singular BEM integrals based on tensor approximations. Numer. Math. 121, 433–460 (2012)
Caffarelli, L., Kohn, R., Nirenberg, L.: First order interpolation inequalities with weights. Compos. Math. 53, 259–275 (1984)
De Lathauwer, L., De Moor, B., Vandewalle, J.: A multilinear singular value decomposition. SIAM J. Matrix Anal. Appl. 21, 1253–1278 (2000)
Grasedyck, L.: Hierarchical singular value decomposition of tensors. SIAM J. Matrix Anal. Appl. 31, 2029–2054 (2010)
Hackbusch, W.: Elliptic differential equations. In: Theory and numerical treatment, 2nd edn, SCM No. 18. Springer, Berlin (2003)
Hackbusch, W.: Tensor spaces and numerical tensor calculus, SCM No. 42. Springer, Berlin (2012)
Hitchcock, F.L.: Multiple invariants and generalized rank of a p-way matrix or tensor. J. Math. Phys. 7, 40–79 (1927)
Maz’ja, V.G.: Sobolev spaces, 2nd edn. Springer, Berlin (2011)
Nirenberg, L.: On elliptic partial differential equations. Ann. Sc. Norm. Sup. Pisa Cl. Sci. 3 Ser. 13, 115–162 (1959)
Oberhettinger, F.: Tabellen zur fourier transformation. Springer, Berlin (1957)
Uschmajew, A.: Regularity of tensor product approximations to square integrable functions. Constr. Approx. 34, 371–391 (2011)
Zeidler, E. (ed.): Oxford users’ guide to mathematics. Oxford University Press, Oxford (2004)
Acknowledgments
We thank Prof. Dr. H. Triebel for bibliographical hints.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hackbusch, W. \(L^{\infty }\) estimation of tensor truncations. Numer. Math. 125, 419–440 (2013). https://doi.org/10.1007/s00211-013-0544-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-013-0544-6