This paper studies the low-rank property of the inverse of a class of large-scale structured matrices in the tensor-train (TT) format, which is typically discretized from differential operators. An interesting question that we are concerned with is: Does the inverse of the large-scale structured matrix still admit the low-rank TT representation with guaranteed accuracy? In this paper, we provide a computationally verifiable sufficient condition such that the inverse matrix can be well approximated in a low-rank TT format. It not only answers what kind of structured matrix whose inverse has the low-rank TT representation but also motivates us to develop an efficient TT-based method to compute the inverse matrix. Furthermore, we prove that the inverse matrix indeed has the low-rank tensor format for a class of large-scale structured matrices induced by differential operators involved in several PDEs, such as the Poisson, Boltzmann, and Fokker-Planck equations. Thus, the proposed algorithm is suitable for solving these PDEs with massive degrees of freedom. Numerical results on the Poisson, Boltzmann, and Fokker-Planck equations validate the correctness of our theory and the advantages of our methodology.

翻译：本文研究了以张量链格式表示的一类大规模结构化矩阵的逆的低秩性质，这类矩阵通常由微分算子离散化得到。我们关注的一个有趣问题是：大规模结构化矩阵的逆是否仍能以有精度保证的低秩张量链格式表示？本文提出了一个可通过计算验证的充分条件，使得逆矩阵能够以低秩张量链格式良好逼近。这不仅回答了何种结构化矩阵的逆具有低秩张量链表示，还促使我们开发了一种基于张量链的高效计算逆矩阵方法。进一步地，我们证明了对于由若干偏微分方程（如泊松方程、玻尔兹曼方程和福克-普朗克方程）中微分算子诱导的一类大规模结构化矩阵，其逆矩阵确实具有低秩张量格式。因此，所提算法适用于求解具有海量自由度的这类偏微分方程。在泊松方程、玻尔兹曼方程和福克-普朗克方程上的数值结果验证了我们理论的正确性及方法的优越性。