# Parallelization and scalability analysis of inverse factorization using the Chunks and Tasks programming model

## TL;DR

This paper introduces three parallel algorithms for inverse factorization of large sparse matrices, implemented using the Chunks and Tasks model, demonstrating linear scaling and improved efficiency in electronic structure calculations.

## Contribution

The paper develops and analyzes three scalable parallel inverse factorization methods using the Chunks and Tasks framework, tailored for large-scale electronic structure problems.

## Key findings

- All methods scale linearly with system size.
- Iterative refinement and localized inverse factorization outperform previous methods in weak scaling.
- Localized inverse factorization reduces communication compared to pure iterative refinement.

## Abstract

We present three methods for distributed memory parallel inverse factorization of block-sparse Hermitian positive definite matrices. The three methods are a recursive variant of the AINV inverse Cholesky algorithm, iterative refinement, and localized inverse factorization, respectively. All three methods are implemented using the Chunks and Tasks programming model, building on the distributed sparse quad-tree matrix representation and parallel matrix-matrix multiplication in the publicly available Chunks and Tasks Matrix Library (CHTML). Although the algorithms are generally applicable, this work was mainly motivated by the need for efficient and scalable inverse factorization of the basis set overlap matrix in large scale electronic structure calculations. We perform various computational tests on overlap matrices for quasi-linear Glutamic Acid-Alanine molecules and three-dimensional water clusters discretized using the standard Gaussian basis set STO-3G with up to more than 10 million basis functions. We show that for such matrices the computational cost increases only linearly with system size for all the three methods. We show both theoretically and in numerical experiments that the methods based on iterative refinement and localized inverse factorization outperform previous parallel implementations in weak scaling tests where the system size is increased in direct proportion to the number of processes. We show also that compared to the method based on pure iterative refinement the localized inverse factorization requires much less communication.