SIAM Journal on Numerical Analysis, Volume 59, Issue 1, Page 289-313, January 2021.
Dynamical low-rank approximation by tree tensor networks is studied for the data-sparse approximation of large time-dependent data tensors and unknown solutions to tensor differential equations. A time integration method for tree tensor networks of prescribed tree rank is presented and analyzed. It extends the known projector-splitting integrators for dynamical low-rank approximation by matrices and rank-constrained Tucker tensors and is shown to inherit their favorable properties. The integrator is based on recursively applying the low-rank Tucker tensor integrator. In every time step, the integrator climbs up and down the tree: it uses a recursion that passes from the root to the leaves of the tree for the construction of initial value problems on subtree tensor networks using appropriate restrictions and prolongations, and another recursion that passes from the leaves to the root for the update of the factors in the tree tensor network. The integrator reproduces given time-dependent tree tensor networks of the specified tree rank exactly and is robust to the typical presence of small singular values in matricizations of the connection tensors, in contrast to standard integrators applied to the differential equations for the factors in the dynamical low-rank approximation by tree tensor networks.

中文翻译：

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树张量网络的时间积分

SIAM数值分析学报，第59卷，第1期，第289-313页，2021年1月。
针对大的时间相关数据张量的数据稀疏近似和张量微分方程的未知解，研究了基于树张量网络的动态低秩逼近。提出并分析了指定树等级的树张量网络的时间积分方法。它扩展了已知的投影仪分解积分器，可通过矩阵和受等级约束的Tucker张量进行动态低秩逼近，并显示出继承的良好特性。积分器基于递归地应用低阶Tucker张量积分器。在每个时间步中，积分器都会在树上上下爬：它使用从树的根到叶的递归，通过适当的限制和延长，在子树张量网络上构造初始值问题，另一个递归从叶传递到根，以更新树张量网络中的因子。积分器精确地重现了指定树等级的给定时间相关的树张量网络，并且对于连接张量矩阵中的小奇异值的典型存在具有鲁棒性，这与应用于动态因素的微分方程的标准积分器不同树张量网络的低秩逼近