### A Machine Learning Framework for Large-Scale Weather and Climate Prediction using Exact and Approximate Linear Algebra Computation

Maximum likelihood estimation is an important statistical technique for estimating missing data, for example in climate and environmental applications, which are usually large and feature data points that are irregularly spaced. In particular, the Gaussian log-likelihood function is the de facto model, which operates on the resulting sizable dense covariance matrix. The advent of high-performance systems with advanced computing power and memory capacity have enabled full simulations only for rather small dimensional climate problems, solved at the machine precision accuracy. The challenge for high dimensional problems lies in the computation requirements of the log-likelihood function, which necessitates O(n^2) storage and O(n^3) operations, where n represents the number of given spatial locations. This prohibitive computational cost may be reduced by using approximation techniques that not only enable large-scale simulations otherwise intractable but also maintain the accuracy and the fidelity of the spatial statistics model. In this paper, we extend the Exascale GeoStatistics software (i.e., ExaGeoStat) to support the Tile Low-Rank (TLR) approximation, which exploits the data sparsity of the dense covariance matrix by compressing the off-diagonal tiles up to a user-defined accuracy threshold. The underlying linear algebra operations may then be carried out on this data compression format, which may ultimately reduce the arithmetic complexity of the maximum likelihood estimation and the corresponding memory footprint. We analyze the performance of the proposed framework across most recent Intel architectures. We achieve up to 4.48X speedup using TLR approximation compared to a full-time solution on Intel 56-core salable Skylake chip, 9.16X speedup on Intel 28-core Broadwell chip, 6.21X speedup on Intel 36-Haswell chip, and 3X speedup on Knights Landing (KNL) chip. With a distributed system, performance results of TLR-based computations on Shaheen-II attain up to 5X speedup, compared to full accuracy simulations using synthetic and real datasets (up to 2M), while ensuring adequate prediction accuracy.

### Event Name

IXPUG Annual Fall Conference 2018

### Keywords

maximum likelihood estimation,optimization