Linear systems with a tensor product structure arise naturally when considering the discretization of Laplace type differential equations or, more generally, multidimensional operators with separable coefficients. In this work, we focus on the numerical solution of linear systems of the form $$ \left(I\otimes \dots\otimes I \otimes A_1+\dots + A_d\otimes I \otimes\dots \otimes I\right)x=b,$$ where the matrices $A_t\in\mathbb R^{n\times n}$ are symmetric positive definite and belong to the class of hierarchically semiseparable matrices. We propose and analyze a nested divide-and-conquer scheme, based on the technology of low-rank updates, that attains the quasi-optimal computational cost $\mathcal O(n^d (\log(n) + \log(\kappa)^2 + \log(\kappa) \log(\epsilon^{-1})))$ where $\kappa$ is the condition number of the linear system, and $\epsilon$ the target accuracy. Our theoretical analysis highlights the role of inexactness in the nested calls of our algorithm and provides worst case estimates for the amplification of the residual norm. The performances are validated on 2D and 3D case studies.

翻译：当考虑拉普拉斯型微分方程的离散化，或更一般地，具有可分离系数的多维算子时，具有张量积结构的线性系统自然出现。本文聚焦于如下形式的线性系统的数值求解：$$ \left(I\otimes \dots\otimes I \otimes A_1+\dots + A_d\otimes I \otimes\dots \otimes I\right)x=b,$$ 其中矩阵$A_t\in\mathbb R^{n\times n}$为对称正定矩阵且属于层级半可分离矩阵类。我们提出并分析了一种基于低秩更新技术的嵌套分治方案，该方案达到了拟最优计算复杂度$\mathcal O(n^d (\log(n) + \log(\kappa)^2 + \log(\kappa) \log(\epsilon^{-1})))$，其中$\kappa$为线性系统的条件数，$\epsilon$为目标精度。我们的理论分析揭示了算法嵌套调用中非精确性的作用，并给出了残差范数放大的最坏情况估计。通过在二维和三维案例研究上的数值实验验证了算法性能。