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

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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.

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Notes

  1. A contraction variable \(k_\eta \) is called inactive in \({\mathbf {C}}^\nu \) if \({\mathbf {C}}^\nu \) does not depend on this index. The other variables are called active. The notation will be adjusted to reflect the dependence on active variables only later for special cases.

  2. If \(\mathbf {u} = \sum _{k=1}^r \mathbf {u}^1_k \otimes \mathbf {u}^2_k\), then \({\mathbf {u}} \in {{\mathrm{span}}}\{\mathbf {u}^1_1,\dots ,\mathbf {u}^1_r \} \otimes {{\mathrm{span}}}\{ \mathbf {u}^2_1,\dots ,\mathbf {u}^2_r \}\). Conversely, if \(\mathbf {u}\) is in such a subspace, then there exist \(a_{ij}\) such that \(\mathbf {u} = \sum _{i=1}^r \sum _{j=1}^r a_{ij} \mathbf {u}^1_i \otimes \mathbf {u}^2_j = \sum _{i=1}^r \mathbf {u}^1_i \otimes \left( \sum _{j=1}^r a_{ij} \mathbf {u}^2_j \right) \).

  3. One can think of \(\pi _{{\mathbb {T}}}(\mathbf {u})\) as a reshape of the tensor \(\mathbf {u}\) which relabels the physical variables according to the permutation \(\varPi \) induced by the order of the tree vertices. Note that in the pointwise formula (3.9) it is not needed.

  4. Even without assuming our knowledge that \({\mathscr {H}}_{\mathfrak {r}}\) is an embedded submanifold, these considerations show that \(\tau _{\text {TT}}\) is a smooth map of constant co-rank \(r_1^2 + \dots + r_{d-1}^2\) on \({\mathscr {W}}^*\). This already implies that the image is a locally embedded submanifold [79].

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Acknowledgments

M.B. was supported by ERC AdG BREAD; R.S. was supported through Matheon by the Einstein Foundation Berlin and DFG project ERA Chemistry.

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Authors and Affiliations

  1. Sorbonne Universités, UPMC Univ Paris 06, CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 4 place Jussieu, 75005, Paris, France

    Markus Bachmayr

  2. Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623, Berlin, Germany

    Reinhold Schneider

  3. Hausdorff Center for Mathematics & Institute for Numerical Simulation, University of Bonn, 53115, Bonn, Germany

    André Uschmajew

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  1. Markus Bachmayr
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Correspondence to Reinhold Schneider.

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Communicated by W. Dahmen.

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Bachmayr, M., Schneider, R. & Uschmajew, A. Tensor Networks and Hierarchical Tensors for the Solution of High-Dimensional Partial Differential Equations. Found Comput Math 16, 1423–1472 (2016). https://doi.org/10.1007/s10208-016-9317-9

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