  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 \emph{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 ${\mathcal O}(n^2)$ storage and ${\mathcal O}(n^3)$
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 framework (i.e.,
ExaGeoStat) to support the Tile Low-Rank (TLR) approximation technique, 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. Performance
results of TLR-based computations on shared and distributed-memory systems
attain up to 13X and 5X speedups, respectively, compared to full accuracy
simulations using synthetic and real datasets (up to 2M), while ensuring
adequate prediction accuracy.
