Multilevel communication optimal LU and QR factorizations for hierarchical platforms

TL;DR

This paper introduces a new hierarchical platform model and develops multilevel LU and QR algorithms optimized for reducing communication costs on such platforms, supported by theoretical analysis and numerical experiments.

Contribution

It presents a novel Hierarchical Cluster Platform model and tailored multilevel LU and QR algorithms that minimize communication across hierarchy levels.

Findings

Lower bounds on communication extended for hierarchical platforms

Multilevel algorithms outperform traditional methods in communication efficiency

Numerical experiments confirm the effectiveness of the proposed algorithms

Abstract

This study focuses on the performance of two classical dense linear algebra algorithms, the LU and the QR factorizations, on multilevel hierarchical platforms. We first introduce a new model called Hierarchical Cluster Platform (HCP), encapsulating the characteristics of such platforms. The focus is set on reducing the communication requirements of studied algorithms at each level of the hierarchy. Lower bounds on communications are therefore extended with respect to the HCP model. We then introduce multilevel LU and QR algorithms tailored for those platforms, and provide a detailed performance analysis. We also provide a set of numerical experiments and performance predictions demonstrating the need for such algorithms on large platforms.