Multilevel approximation of Gaussian random fields: Covariance compression, estimation and spatial prediction

Centered Gaussian random fields (GRFs) indexed by compacta such as smooth, bounded domains in Euclidean space or smooth, compact and orientable manifolds are determined by their covariance operators. We consider centered GRFs given sample-wise as variational solutions to coloring operator equations driven by spatial white noise, with pseudodifferential coloring operator being elliptic, self-adjoint and positive from the Hörmander class. This includes the Matérn class of GRFs as a special case. Using microlocal tools and biorthogonal multiresolution analyses on the manifold, we prove that the precision and covariance operators, respectively, may be identified with bi-infinite matrices and finite sections may be diagonally preconditioned rendering the condition number independent of the dimension p of this section. We prove that a tapering strategy by thresholding as e.g. in [Bickel, P.J. and Levina, E. Covariance regularization by thresholding, Ann. Statist., 36 (2008), 2577–2604] applied on finite sections of the bi-infinite precision and covariance matrices results in optimally numerically sparse approximations. Numerical sparsity signifies that only asymptotically linearly many nonzero matrix entries are sufficient to approximate the original section of the bi-infinite covariance or precision matrix using this tapering strategy to arbitrary precision. This tapering strategy is non-adaptive and the locations of these nonzero matrix entries are known a priori. The tapered covariance or precision matrices may also be optimally diagonal preconditioned. Analysis of the relative size of the entries of the tapered covariance matrices motivates novel, multilevel Monte Carlo (MLMC) oracles for covariance estimation, in sample complexity that scales log-linearly with respect to the number p of parameters. This extends [Bickel, P.J. and Levina, E. Regularized Estimation of Large Covariance Matrices, Ann. Stat., 36 (2008), pp. 199–227] to estimation of (finite sections of) pseudodifferential covariances for GRFs by this fast MLMC method. Assuming at hand sections of the bi-infinite covariance matrix in wavelet coordinates, we propose and analyze a novel compressive algorithm for simulating and kriging of GRFs. The complexity (work and memory vs. accuracy) of these three algorithms scales near-optimally in terms of the number of parameters p of the sample-wise approximation of the GRF in Sobolev scales.