Seminar of Applied Mathematics (SAM)
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 analyze the numerical approximation of centered GRFs given sample-wise as variational solutions to coloring operator equations driven by spatial white noise, with coloring operator being elliptic, self-adjoint and positive from the Hörmander class. This includes the Matérn class of GRFs as a special case.
Precision and covariance operators can be represented as bi-infinite matrices. Finite sections of these may be diagonally preconditioned rendering the condition number independent of the dimension p of this section. and compressed (“tapered” in the language of graphical statistical models). We prove that tapering by thresholding as e.g. in  applied on finite sections of the bi-infinite precision and covariance matrices results in optimally numerically sparse approximations.
“Numerical sparsity” signifies that, asymptotically, a number of nonzero matrix entries that grows linearly with the number of GRF parameters. The tapering strategy is non-adaptive and the locations of these nonzero matrix entries are known a priori.
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  to estimation of (finite sections of) pseudodifferential covariances for GRFs by this fast MLMC method.
This is joint work with
H. Harbrecht (University of Basel, Switzerland)
L. Herrmann (RICAM, Linz, Austria)
K. Kirchner (TU Delft, The Netherlands)
Preprint (SAM Report 2021-09)
 P.J. Bickel and E. Levina: Covariance regularization by thresholding, Ann. Statist., 36 (2008), 2577-2604
 P.J. Bickel and E. Levina: Regularized Estimation of Large Covariance Matrices, Ann. Stat., 36 (2008), pp. 199-227