  Physics-based covariance models provide a systematic way to construct
covariance models that are consistent with the underlying physical laws in
Gaussian process analysis. The unknown parameters in the covariance models can
be estimated using maximum likelihood estimation, but direct construction of
the covariance matrix and classical strategies of computing with it requires
$n$ physical model runs, $n^2$ storage complexity, and $n^3$ computational
complexity. To address such challenges, we propose to approximate the
discretized covariance function using hierarchical matrices. By utilizing
randomized range sketching for individual off-diagonal blocks, the construction
process of the hierarchical covariance approximation requires $O(\log{n})$
physical model applications and the maximum likelihood computations require
$O(n\log^2{n})$ effort per iteration. We propose a new approach to compute
exactly the trace of products of hierarchical matrices which results in the
expected Fischer information matrix being computable in $O(n\log^2{n})$ as
well. The construction is totally matrix-free and the derivatives of the
covariance matrix can then be approximated in the same hierarchical structure
by differentiating the whole process. Numerical results are provided to
demonstrate the effectiveness, accuracy, and efficiency of the proposed method
for parameter estimations and uncertainty quantification.
