标题： 树张量网络的并行基更新和伽辽金积分器 显示英文标题

标题： A parallel Basis Update and Galerkin Integrator for Tree Tensor Networks

摘要： 计算高维张量微分方程的数值解可能导致高昂的计算成本和内存需求。 为了减少内存和计算开销，动态低秩逼近（DLRA）已被证明是一种有前景的方法。 DLRA将解表示为低秩张量分解，并随时间演化得到的低秩因子。 DLRA中的一个核心挑战是找到对出现的小奇异值具有鲁棒性的时间积分方案。 最近，一种鲁棒的并行基更新与Galerkin积分器被推导出来，用于矩阵微分方程，该方法同时演化所有低秩因子。 这项工作将并行低秩矩阵积分器扩展到Tucker张量和一般的树状张量网络，产生了一种算法，在一个时间步内并行演化所有基和连接张量。 我们提出了该算法，提供了鲁棒的误差界，并展示了新积分器在量子多体物理、不确定性量化和辐射传输问题中的效率。 显示英文摘要

摘要： Computing the numerical solution to high-dimensional tensor differential equations can lead to prohibitive computational costs and memory requirements. To reduce the memory and computational footprint, dynamical low-rank approximation (DLRA) has proven to be a promising approach. DLRA represents the solution as a low-rank tensor factorization and evolves the resulting low-rank factors in time. A central challenge in DLRA is to find time integration schemes that are robust to the arising small singular values. A robust parallel basis update & Galerkin integrator, which simultaneously evolves all low-rank factors, has recently been derived for matrix differential equations. This work extends the parallel low-rank matrix integrator to Tucker tensors and general tree tensor networks, yielding an algorithm in which all bases and connecting tensors are evolved in parallel over a time step. We formulate the algorithm, provide a robust error bound, and demonstrate the efficiency of the new integrators for problems in quantum many-body physics, uncertainty quantification, and radiative transfer.