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Preprint 28-2015

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

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Author(s) : Markus Bachmayr , Reinhold Schneider , André Uschmajew

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 28-2015

MSC 2000

65-02 Research exposition

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 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. In a number of important cases, quasi-optimality of approximation ranks and computational complexity have been analysed. 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

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