Polynomial and Rational Approximation for Electronic Structure Theory
Date/Time:14 Aug 2019 16:00
Venue: S17 #05-11 SR5
Speaker: Simon Etter, National University of Singapore
Polynomial and Rational Approximation for Electronic Structure Theory
Density Functional Theory (DFT) and related electronic structure models are some of the largest consumers of computing power in supercomputing centres worldwide, and the bulk of this time is spent on evaluating functions of large, sparse matrices. Traditionally, such matrix functions have been computed by means of the eigenvalue decomposition, but the cubic scaling of standard eigensolvers with respect to the matrix size renders this ansatz prohibitively expensive in many applications. Several alternative algorithms have been developed in recent decades, among which the Pole Expansion and Selected Inversion (PEXSI) algorithm by Lin Lin and collaborators stands out for its ease of use and excellent parallel scaling. However, the runtime of the PEXSI scheme grows with $N^{3/2}$ for two-dimensional systems of $N$ atoms and with $N^2$ for three-dimensional systems, which is an important drawback of this method compared to most of its competitors.
The first part of this talk will present a modification of the PEXSI algorithm which reduces its cost to linear in the system size regardless of the dimension. This modification is based on the observation that the LU factorisation of a sparse, well-conditioned matrix is localised, i.e. the magnitudes of the fill-in entries decay exponentially as we move away from the nonzero entries of the original matrix. The second part of this talk will discuss conductivity calculations, which introduce a new twist to the theory of electronic structure algorithms in that the matrix function to evaluate in this case is a bivariate one. The runtime of such calculations is proportional to the number of significant Chebyshev coefficients of the function $f(x,y) := 1/(x – y + s)$ with small imaginary $s$, and we will see that this number scales with only $|s|^{3/2}$ rather than $|s|^2$ as one might expect based on one-dimensional arguments.
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