Tensor Networks and Hierarchical Tensors for the Solution of High-Dimensional Partial Differential Equations
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- 作者：Markus Bachmayr ; Reinhold Schneider…
- 关键词：Hierarchical tensors ; Low ; rank approximation ; High ; dimensional partial differential equations
- 刊名：Foundations of Computational Mathematics
- 出版年：2016
- 出版时间：December 2016
- 年：2016
- 卷：16
- 期：6
- 页码：1423-1472
- 全文大小：
- 刊物类别：Mathematics and Statistics
- 刊物主题：Numerical Analysis; Economics, general; Applications of Mathematics; Linear and Multilinear Algebras, Matrix Theory; Math Applications in Computer Science; Computer Science, general;
- 出版者：Springer US
- ISSN：1615-3383
- 卷排序：16
文摘
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.