This work describes a computational approach for rapidly estimating and characterizing the covariance structure of a Gaussian random field from a modest number of i.i.d. replicates by formulating the problem in terms of a sparse inverse Cholesky factor of the data's covariance matrix. Borrowing from the extensive literature of using directed acyclic graphs to impose sparsity on inverse Cholesky factors (for example, Vecchia's approximation or nearest-neighbor Gaussian process models), a probabilistically parsimonious a priori sparsity pattern is imposed, and with careful formulation the estimation problem is given as an extremely well-structured convex problem that can be solved for millions of unknowns in seconds on a single processor core. After describing the problem and specific formulation of the optimization, we demonstrate the estimator's speed and effectiveness at capturing a variety of nonstationary structures in spatial or spatio-temporal data, and in ensemble data assimilation.
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