We extend our previous work on a semi-Lagrangian adaptive rank (SLAR) integrator, in the finite difference framework for nonlinear Vlasov-Poisson systems, to the general high-order tensor setting. The proposed scheme retains the high-order accuracy of semi-Lagrangian methods, ensuring stability for large time steps and avoiding dimensional splitting errors. The primary contribution of this paper is the novel extension of the algorithm from the matrix to the high-dimensional tensor setting, which enables the simulation of Vlasov models in up to six dimensions. The key technical components include (1) a third-order high-dimensional polynomial reconstruction that scales as $O(d^2)$, providing a point-wise approximation of the solution at the foot of characteristics in a semi-Lagrangian scheme; (2) a recursive hierarchical adaptive cross approximation of high-order tensors in a hierarchical Tucker format, characterized by a tensor tree; (3) a low-complexity Poisson solver in the hierarchical Tucker format that leverages the FFT for efficiency. The computed adaptive rank kinetic solutions exhibit low-rank structures within branches of the tensor tree resulting in substantial computational savings in both storage and time. The resulting algorithm achieves a computational complexity of $O(d^4 N r^{3+\lceil\log_2d\rceil})$, where $N$ is the number of grid points per dimension, $d$ is the problem dimension, and $r$ is the maximum rank in the tensor tree, overcoming the curse of dimensionality. Through extensive numerical tests, we demonstrate the efficiency of the proposed algorithm and highlight its ability to capture complex solution structures while maintaining a computational complexity that scales linearly with $N$.

翻译：本文将我们先前在有限差分框架下针对非线性Vlasov-Poisson系统提出的半拉格朗日自适应秩（SLAR）积分器，推广至一般高阶张量设定。所提方案保持了半拉格朗日方法的高阶精度，确保了大时间步长下的稳定性，并避免了维度分裂误差。本文的主要贡献在于将算法从矩阵形式创新性地扩展至高维张量设定，从而能够模拟高达六维的Vlasov模型。关键技术组成部分包括：（1）一种计算复杂度为$O(d^2)$的三阶高维多变量多项式重构，在半拉格朗日格式中提供特征线起点处解的逐点近似；（2）基于张量树表征的层次Tucker格式中高阶张量的递归层次自适应交叉逼近；（3）利用快速傅里叶变换（FFT）实现高效计算的层次Tucker格式低复杂度Poisson求解器。计算得到的自适应秩动力学解在张量树的分支内呈现低秩结构，从而在存储和计算时间上实现显著节省。所得算法的计算复杂度为$O(d^4 N r^{3+\lceil\log_2d\rceil})$，其中$N$为每维度网格点数，$d$为问题维度，$r$为张量树中的最大秩，克服了维度灾难。通过大量数值实验，我们验证了所提算法的高效性，并突显了其在保持计算复杂度与$N$呈线性增长的同时捕捉复杂解结构的能力。