Abstract
Geostatistical modeling, one of the prime motivating applications for exascale computing, is a technique for predicting desired quantities from geographically distributed data, based on statistical models and optimization of parameters. Spatial data are assumed to possess properties of stationarity or non-stationarity via a kernel fitted to a covariance matrix. A primary workhorse of stationary spatial statistics is Gaussian maximum log-likelihood estimation (MLE), whose central data structure is a dense, symmetric positive definite covariance matrix of the dimension of the number of correlated observations. Two essential operations in MLE are the application of the inverse and evaluation of the determinant of the covariance matrix. These can be rendered through the Cholesky decomposition and triangular solution. In this contribution, we reduce the precision of weakly correlated locations to single- or half- precision based on distance. We thus exploit mathematical structure to migrate MLE to a three-precision approximation that takes advantage of contemporary architectures offering BLAS3-like operations in a single instruction that are extremely fast for reduced precision. We illustrate application-expected accuracy worthy of double-precision from a majority half-precision computation, in a context where uniform single-precision is by itself insufficient. In tackling the complexity and imbalance caused by the mixing of three precisions, we deploy the PaRSEC runtime system. PaRSEC delivers on-demand casting of precisions while orchestrating tasks and data movement in a multi-GPU distributed-memory environment within a tile-based Cholesky factorization. Application-expected accuracy is maintained while achieving up to <inline-formula><tex-math notation="LaTeX">1.59X</tex-math> <mml:math><mml:mrow><mml:mn>1</mml:mn><mml:mo>.</mml:mo><mml:mn>59</mml:mn><mml:mi>X</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href="abdulah-ieq1-3084071.gif"/> </inline-formula> by mixing FP64/FP32 operations on 1536 nodes of HAWK or 4096 nodes of Shaheen II , and up to <inline-formula><tex-math notation="LaTeX">2.64X</tex-math> <mml:math><mml:mrow><mml:mn>2</mml:mn><mml:mo>.</mml:mo><mml:mn>64</mml:mn><mml:mi>X</mml:mi></mml:mrow></mml:math><inline-graphic xlink:href="abdulah-ieq2-3084071.gif"/> </inline-formula> by mixing FP64/FP32/FP16 operations on 128 nodes of Summit , relative to FP64-only operations. This translates into up to 4.5, 4.7, and 9.1 (mixed) PFlop/s sustained performance, respectively, demonstrating a synergistic combination of exascale architecture, dynamic runtime software, and algorithmic adaptation applied to challenging environmental problems.