Skip to Main Content

Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Low-rank approximation of continuous functions in Sobolev spaces with dominating mixed smoothness
HTML articles powered by AMS MathViewer

by Michael Griebel, Helmut Harbrecht and Reinhold Schneider HTML | PDF
Math. Comp. 92 (2023), 1729-1746 Request permission


Let $\Omega _i\subset \mathbb {R}^{n_i}$, $i=1,\ldots ,m$, be given domains. In this article, we study the low-rank approximation with respect to $L^2(\Omega _1\times \dots \times \Omega _m)$ of functions from Sobolev spaces with dominating mixed smoothness. To this end, we first estimate the rank of a bivariate approximation, i.e., the rank of the continuous singular value decomposition. In comparison to the case of functions from Sobolev spaces with isotropic smoothness, compare Griebel and Harbrecht [IMA J. Numer. Anal. 34 (2014), pp. 28–54] and Griebel and Harbrecht [IMA J. Numer. Anal. 39 (2019), pp. 1652–1671], we obtain improved results due to the additional mixed smoothness. This convergence result is then used to study the tensor train decomposition as a method to construct multivariate low-rank approximations of functions from Sobolev spaces with dominating mixed smoothness. We show that this approach is able to beat the curse of dimension.
Similar Articles
Additional Information
  • Michael Griebel
  • Affiliation: Institut für Numerische Simulation, Universität Bonn, Friedrich-Hirzebruch-Allee 7, 53115 Bonn, Germany; and Fraunhofer Institute for Algorithms and Scientific Computing (SCAI), Schloss Birlinghoven, 53754 Sankt Augustin, Germany
  • MR Author ID: 270664
  • Email:
  • Helmut Harbrecht
  • Affiliation: Departement Mathematik und Informatik, Universität Basel, Spiegelgasse 1, 4051 Basel, Switzerland
  • MR Author ID: 682387
  • Email:
  • Reinhold Schneider
  • Affiliation: Institute of Mathematics, Technical University of Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany
  • MR Author ID: 224723
  • Email:
  • Received by editor(s): March 7, 2022
  • Received by editor(s) in revised form: August 12, 2022, November 24, 2022, and February 20, 2023
  • Published electronically: March 7, 2023
  • Additional Notes: The first author was supported by the Hausdorff Center for Mathematics in Bonn, funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy -EXC-2047/1-390685813 and the Sonderforschungsbereich 1060 The Mathematics of Emergent Effects.
  • © Copyright 2023 American Mathematical Society
  • Journal: Math. Comp. 92 (2023), 1729-1746
  • MSC (2020): Primary 41A17, 41A25, 41A30, 41A65
  • DOI:
  • MathSciNet review: 4570339