Tensor Networks and Hierarchical Tensors for the Solution of High-Dimensional Partial Differential Equations
Tensor Networks and Hierarchical Tensors for the Solution of High-Dimensional Partial Differential Equations
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DOI：
10.1007/s10208-016-9317-9
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发表时间：
2016-04
影响因子：
3
通讯作者：
M. Bachmayr;R. Schneider;André Uschmajew
中科院分区：
文献类型：
--
作者：
M. Bachmayr;R. Schneider;André Uschmajew
分层张量可以被看作是高阶张量的奇异值分解的推广，保留了许多重要的特征。对于给定的张量积空间，将坐标集递归分解为维树给出了嵌套子空间和相应的嵌套基的层次结构。这些子空间的维数产生了多线性秩的概念。这个秩元组，以及准最佳的低秩近似秩截断，可以通过分层奇异值分解。对于固定的多线性秩，这些分层表示的存储和操作复杂性仅以张量的顺序线性缩放。在矩阵的情况下，一个给定的多线性秩的层次张量的集合不是一个凸集，但形成一个开放的光滑流形。一些技术的计算分层低秩近似已经开发，包括局部优化技术黎曼流形以及截断迭代法，可以应用于解决高维偏微分方程。本文对这些发展作一综述。我们还讨论了应用程序的问题，不确定性量化，解决电子薛定谔方程的强相关制度，并计算亚稳态分子动力学。
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.