论文标题
树木张量网络的等级自适应时间整合
Rank-adaptive time integration of tree tensor networks
论文作者
论文摘要
提出和分析了通过树张量网络对高阶张量微分方程近似解的等级自适应积分器。在从叶到根部的递归中，积分器更新碱基，然后通过新碱和旧碱基跨越的增强子空间中的Galerkin方法更新连接张量。接下来是指定的误差公差内的等级截断。记忆要求按张量的顺序进行线性，并在最大模式维度下进行线性。积分器对连接张量的母亲的小奇异值具有鲁棒性。直到由给定的误差耐受性控制的等级截断误差，积分器保留了Schr Odinger方程的标准和能量，并且可以消除梯度系统中的能量。基本量子自旋系统的数值实验说明了所提出的算法的行为。
A rank-adaptive integrator for the approximate solution of high-order tensor differential equations by tree tensor networks is proposed and analyzed. In a recursion from the leaves to the root, the integrator updates bases and then evolves connection tensors by a Galerkin method in the augmented subspace spanned by the new and old bases. This is followed by rank truncation within a specified error tolerance. The memory requirements are linear in the order of the tensor and linear in the maximal mode dimension. The integrator is robust to small singular values of matricizations of the connection tensors. Up to the rank truncation error, which is controlled by the given error tolerance, the integrator preserves norm and energy for Schrodinger equations, and it dissipates the energy in gradient systems. Numerical experiments with a basic quantum spin system illustrate the behavior of the proposed algorithm.
