论文标题
树张量网络的时间集成
Time integration of tree tensor networks
论文作者
论文摘要
研究了通过树张量网络的动态低级别近似值，以进行数据 - 帕斯斯对大的时间依赖性数据张量和张量微分方程的未知溶液的近似。提出和分析了开处方树等级的树张量网络的时间集成方法。它扩展了已知的投影仪拆分积分器，用于通过矩阵和塔克张量进行动态低级别近似，并显示出继承其有利的性质。集成器基于递归应用Tucker Tensor Integrator。在每个时间步骤中，积分器都会上下爬上树：它使用从根部到树叶的递归，用于在子树张量网络上使用适当的限制和延长构建初始价值问题，而另一个递归则从叶子传递到叶子到根部的根部，以更新树木张镜网络中的因子。集成剂可重现指定树等的时间相关的树张量网络，并且与连接张量的典型奇异值的典型存在相反，相反，与标准集成仪相反，该集成量适用于标准的集成仪，用于在树上通过树量张力网络进行动态低量近似值的因子。
Dynamical low-rank approximation by tree tensor networks is studied for the data-sparse approximation to large time-dependent data tensors and unknown solutions of 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 Tucker tensors and is shown to inherit their favorable properties. The integrator is based on recursively applying the 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.
