肖传福 讲师（湘潭大学）

黄记祖 副研究员

Provable low-rank tensor-train approximations in the inverse of large-scale structured matrices

11月24日（周一）9:30-10:30，思源楼817

This talk 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. A fundamental question arising in numerical linear algebra and numerical partial differential equations

(PDEs) is: Does the inverse of the large-scale structured matrix still admit the low-rank TT representation with guaranteed accuracy? In this talk, 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 and Boltzmann equations validate the correctness of our theory and the advantages of our methodology.