Foundations of Computational Mathematics

, Volume 16, Issue 6, pp 1423–1472 | Cite as

Tensor Networks and Hierarchical Tensors for the Solution of High-Dimensional Partial Differential Equations

  • Markus Bachmayr
  • Reinhold Schneider
  • André Uschmajew
Article

Abstract

Hierarchical tensors can be regarded as a generalisation, preserving many crucial features, of the singular value decomposition to higher-order tensors. For a given tensor product space, a recursive decomposition of the set of coordinates into a dimension tree gives a hierarchy of nested subspaces and corresponding nested bases. The dimensions of these subspaces yield a notion of multilinear rank. This rank tuple, as well as quasi-optimal low-rank approximations by rank truncation, can be obtained by a hierarchical singular value decomposition. For fixed multilinear ranks, the storage and operation complexity of these hierarchical representations scale only linearly in the order of the tensor. As in the matrix case, the set of hierarchical tensors of a given multilinear rank is not a convex set, but forms an open smooth manifold. A number of techniques for the computation of hierarchical low-rank approximations have been developed, including local optimisation techniques on Riemannian manifolds as well as truncated iteration methods, which can be applied for solving high-dimensional partial differential equations. This article gives a survey of these developments. We also discuss applications to problems in uncertainty quantification, to the solution of the electronic Schrödinger equation in the strongly correlated regime, and to the computation of metastable states in molecular dynamics.

Keywords

Hierarchical tensors Low-rank approximation High-dimensional partial differential equations 

Mathematics Subject Classification

65-02 65F99 65J 49M 35C 

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Copyright information

© SFoCM 2016

Authors and Affiliations

  • Markus Bachmayr
    • 1
  • Reinhold Schneider
    • 2
  • André Uschmajew
    • 3
  1. 1.Sorbonne Universités, UPMC Univ Paris 06, CNRS, UMR 7598Laboratoire Jacques-Louis LionsParisFrance
  2. 2.Institut für MathematikTechnische Universität BerlinBerlinGermany
  3. 3.Hausdorff Center for Mathematics & Institute for Numerical SimulationUniversity of BonnBonnGermany

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