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

• Published in: Foundations of computational mathematics Vol. 16; no. 6; pp. 1423 - 1472
• Main Authors: Bachmayr, Markus, Schneider, Reinhold, Uschmajew, André
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2016; Springer; Springer Nature B.V
• Subjects: Analysis; Applications of Mathematics; Approximation; Computation; Computer Science; Decomposition; Differential equations, Partial; Economics; Iterative methods; Linear and Multilinear Algebras; Manifolds (mathematics); Math Applications in Computer Science; Mathematical analysis; Mathematical models; Mathematics; Mathematics and Statistics; Matrix Theory; Metastable state; Molecular chains; Molecular dynamics; Numerical Analysis; Partial differential equations; Riemann manifold; Schrodinger equation; Singular value decomposition; Subspaces; Tensors; Tensors (Mathematics); United States; 65-02; 35C; High-dimensional partial differential equations; Low-rank approximation; 65F99; 49M; Hierarchical tensors; 65J
• ISSN: 1615-3375; 1615-3383
• DOI: 10.1007/s10208-016-9317-9

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.