Polynomial and rational approximation for electronic structure calculations

Abstract

Atomic-scale simulation of matter has become an important research tool in physics, chemistry, material science and biology as it allows for insights which neither theoretical nor experimental investigation can provide. The most accurate of these simulations are based on the laws of quantum mechanics, in which case the main computational bottleneck becomes the evaluation of functions f(H) of a sparse matrix H (the Hamiltonian). One way to evaluate such matrix functions is through polynomial and rational approximation, the theory of which is reviewed in Chapter 2 of this thesis. It is well known that rational functions can approximate the relevant functions with much lower degrees than polynomials, but they are more challenging to use in practice since they require fast algorithms for evaluating rational functions r(H) of a matrix argument H. Such an algorithm has recently been proposed in the form of the Pole Expansion and Selected Inversion (PEXSI) scheme, which evaluates r(H) by writing r(x) = P k ck x−zk in partial-fraction-decomposed form and then employing advanced sparse factorisation techniques to evaluate only a small subset of the entries of the resolvents (H − z) −1 . This scheme scales better than cubically in the matrix dimension, but it is not a linear scaling algorithm in general. We overcome this limitation in Chapter 3 by devising a modified, linear-scaling PEXSI algorithm which exploits that most of the fill-in entries in the triangular factorisations computed by the PEXSI algorithm are negligibly small. Finally, Chapter 4 presents a novel algorithm for computing electric conductivities which requires evaluating a bivariate matrix function f(H, H). We show that the Chebyshev coefficients ck1k2 of the relevant function f(x1, x2) concentrate along the diagonal k1 ∼ k2 and that this allows us to approximate f(x1, x2) much more efficiently than one would expect based on a straightforward tensor-product extension of the one-dimensional arguments