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Linear systems with a tensor product structure arise naturally whenconsidering the discretization of Laplace type differential equations or, moregenerally, multidimensional operators with separable coefficients. In thiswork, 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 $A_t\in\mathbb R^{n\times n}$ aresymmetric positive definite and belong to the class of hierarchicallysemiseparable matrices.  We propose and analyze a nested divide-and-conquer scheme, based on thetechnology of low-rank updates, that attains the quasi-optimal computationalcost $\mathcal O(n^d (\log(n) + \log(\kappa)^2 + \log(\kappa)\log(\epsilon^{-1})))$ where $\kappa$ is the condition number of the linearsystem, and $\epsilon$ the target accuracy. Our theoretical analysis highlightsthe role of inexactness in the nested calls of our algorithm and provides worstcase estimates for the amplification of the residual norm. The performances arevalidated on 2D and 3D case studies.

A nested divide-and-conquer method for tensor Sylvester equations with positive definite hierarchically semiseparable coefficients