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.

翻译：计算高维张量微分方程的数值解可能导致计算成本与内存需求过高。为降低内存与计算开销，动态低秩近似已被证明是一种有前景的方法。该方法将解表示为低秩张量分解形式，并随时间演化所得的低秩因子。动态低秩近似的核心挑战在于寻找对出现的小奇异值具有鲁棒性的时间积分格式。近期针对矩阵微分方程已推导出一种鲁棒的并行基更新与伽辽金积分器，可同时演化所有低秩因子。本工作将并行低秩矩阵积分器扩展至Tucker张量与一般树张量网络，提出一种在时间步长内并行演化所有基矩阵与连接张量的算法。我们阐述了算法框架，给出了鲁棒误差界，并通过量子多体物理、不确定性量化及辐射传输等领域的问题验证了新积分器的高效性。