发表机构
Universität Paderborn; Technical University of Munich（帕德博恩大学; 慕尼黑工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对Tucker张量和树张量网络的动态低秩近似，提出两种二阶BUG时间积分方法，证明其对连接张量小奇异值的鲁棒性，数值实验验证了理论结果。
AI 中文摘要
我们针对Tucker张量和树张量网络上的动态低秩近似问题，提出并分析了两种二阶基更新与Galerkin（BUG）时间积分方法。两种方法均基于顺序预计算扫描构建，之后通过Galerkin方法演化所有基矩阵和连接张量的微分方程。此外，所提方法天生具有秩自适应性。第一种是二阶并行BUG积分器，可完全并行求解所有这些微分方程，适用于并行架构，随后执行顺序增广与截断步骤；第二种是二阶增广BUG积分器，放弃完全并行性，但对薛定谔方程可保持范数和能量，对梯度流可在截断容差范围内耗散能量。对于两种积分器，我们证明了二阶误差界，该界对连接张量矩阵化的小奇异值具有鲁棒性。辐射输运和量子自旋系统的数值实验验证了理论结果。
英文摘要
We propose and analyze two second-order basis-update \& Galerkin (BUG) time integration methods for dynamical low-rank approximation on Tucker tensors and tree tensor networks. Both are built from a sequential pre-computation sweep, after which the differential equations for all basis matrices and connecting tensors are evolved by a Galerkin method. Further, the proposed methods are rank-adaptive by construction. The first, the second-order parallel BUG integrator, solves all these differential equations fully in parallel, which is favorable on parallel architectures, followed by a sequential augmentation and truncation step. The second, the second-order augmented BUG integrator, gives up full parallelism but conserves norm and energy for Schrödinger equations and dissipates energy for gradient flows up to the truncation tolerance. For both integrators, we prove a second-order error bound that is robust with respect to small singular values of the matricization of connecting tensors. Numerical experiments for radiative transfer and quantum spin systems validate the theoretical findings.
Comments34 pages, 3 figures
