# Assessment of localized and randomized algorithms for electronic structure

## TL;DR

This paper evaluates the performance of localized, randomized, and hybrid algorithms for electronic structure calculations, highlighting their advantages and limitations across different temperature regimes and proposing advanced hybrid methods with promising initial results.

## Contribution

It provides a comparative analysis of various linear-scaling algorithms and introduces sophisticated hybrid algorithms that improve efficiency in electronic structure simulations.

## Key findings

- Linear-scaling algorithms are effective at high temperatures.
- Cost prefactors limit their efficiency near ambient temperatures.
- Structured linear algebra-based hybrid algorithms show promising initial performance.

## Abstract

As electronic structure simulations continue to grow in size, the system-size scaling of computational costs increases in importance relative to cost prefactors. Presently, linear-scaling costs for three-dimensional systems are only attained by localized or randomized algorithms that have large cost prefactors in the difficult regime of low-temperature metals. Using large copper clusters in a minimal-basis semiempirical model as our reference system, we study the costs of these algorithms relative to a conventional cubic-scaling algorithm using matrix diagonalization and a recent quadratic-scaling algorithm using sparse matrix factorization and rational function approximation. The linear-scaling algorithms are competitive at the high temperatures relevant for warm dense matter, but their cost prefactors are prohibitive near ambient temperatures. To further reduce costs, we consider hybridized algorithms that combine localized and randomized algorithms. While simple hybridized algorithms do not improve performance, more sophisticated algorithms using recent concepts from structured linear algebra show promising initial performance results on a simple-cubic orthogonal tight-binding model.