Communications in Mathematical Sciences

Fast algorithm for extracting the diagonal of the inverse matrix with application to the electronic structure analysis of metallic systems

Lin Lin, Jianfeng Lu, Lexing Ying, Roberto Car, and Weinan E

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Abstract

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.

Article information

Source
Commun. Math. Sci., Volume 7, Number 3 (2009), 755-777.

Dates
First available in Project Euclid: 26 October 2009

Permanent link to this document
https://projecteuclid.org/euclid.cms/1256562822

Mathematical Reviews number (MathSciNet)
MR2569032

Zentralblatt MATH identifier
1182.65072

Subjects
Primary: 65F30: Other matrix algorithms
Secondary: 65Z05: Applications to physics

Keywords
Diagonal extraction hierarchical Schur complement electronic structure calculation

Citation

Lin, Lin; Lu, Jianfeng; Ying, Lexing; Car, Roberto; E, Weinan. Fast algorithm for extracting the diagonal of the inverse matrix with application to the electronic structure analysis of metallic systems. Commun. Math. Sci. 7 (2009), no. 3, 755--777. https://projecteuclid.org/euclid.cms/1256562822


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