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A Fast Parallel Algorithm for Selected Inversion of Structured Sparse Matrices with Application to 2D Electronic Structure Calculations

Abstract

We present an efficient parallel algorithm and its implementation for computing the diagonal of $H^-1$ where $H$ is a 2D Kohn-Sham Hamiltonian discretized on a rectangular domain using a standard second order finite difference scheme. This type of calculation can be used to obtain an accurate approximation to the diagonal of a Fermi-Dirac function of $H$ through a recently developed pole-expansion technique \cite{LinLuYingE2009}. The diagonal elements are needed in electronic structure calculations for quantum mechanical systems \citeHohenbergKohn1964, KohnSham 1965,DreizlerGross1990. We show how elimination tree is used to organize the parallel computation and how synchronization overhead is reduced by passing data level by level along this tree using the technique of local buffers and relative indices. We analyze the performance of our implementation by examining its load balance and communication overhead. We show that our implementation exhibits an excellent weak scaling on a large-scale high performance distributed parallel machine. When compared with standard approach for evaluating the diagonal a Fermi-Dirac function of a Kohn-Sham Hamiltonian associated a 2D electron quantum dot, the new pole-expansion technique that uses our algorithm to compute the diagonal of $(H-z_i I)^-1$ for a small number of poles $z_i$ is much faster, especially when the quantum dot contains many electrons.

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