Tile Low Rank Cholesky Factorization for Climate/Weather Modeling Applications on Manycore Architectures

Abstract

Covariance matrices are ubiquitous in computational science and engineering. In particular, large covariance matrices arise from multivariate spatial data sets, for instance, in climate/weather modeling applications to improve prediction using statistical methods and spatial data. One of the most time-consuming computational steps consists in calculating the Cholesky factorization of the symmetric, positive-definite covariance matrix problem. The structure of such covariance matrices is also often data-sparse, in other words, effectively of low rank, though formally dense. While not typically globally of low rank, covariance matrices in which correlation decays with distance are nearly always hierarchically of low rank. While symmetry and positive definiteness should be, and nearly always are, exploited for performance purposes, exploiting low rank character in this context is very recent, and will be a key to solving these challenging problems at large-scale dimensions. The authors design a new and flexible tile row rank Cholesky factorization and propose a high performance implementation using OpenMP task-based programming model on various leading-edge manycore architectures. Performance comparisons and memory footprint saving on up to $200K\times 200K$ covariance matrix size show a gain of more than an order of magnitude for both metrics, against state-of-the-art open-source and vendor optimized numerical libraries, while preserving the numerical accuracy fidelity of the original model. This research represents an important milestone in enabling large-scale simulations for covariance-based scientific applications.