We propose an algorithm for extracting the diagonal of the inverse matrices arising from electronic structure calculation. The proposed algorithm uses a hierarchical decomposition of the computational domain. It first constructs hierarchical Schur complements of the interior points for the blocks of the domain in a bottom-up pass and then extracts the diagonal entries efficiently in a top-down pass by exploiting the hierarchical local dependence of the inverse matrices. The overall cost of our algorithm is $O(N^3/2)$ for a two dimensional problem with $N$ degrees of freedom. Numerical results in electronic structure calculation illustrate the efficiency and accuracy of the proposed algorithm.
Lin Lin. Jianfeng Lu. Lexing Ying. Roberto Car. Weinan E. "Fast algorithm for extracting the diagonal of the inverse matrix with application to the electronic structure analysis of metallic systems." Commun. Math. Sci. 7 (3) 755 - 777, September 2009.