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

**
****
Christoph Schwab**

Seminar of Applied Mathematics (SAM)

ETH Zürich

Switzerland

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 [1]
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 [2]
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)

**References**

[1] P.J. Bickel and E. Levina:
Covariance
regularization by thresholding,
Ann. Statist., 36 (2008), 2577-2604

[2] P.J. Bickel and E. Levina: Regularized
Estimation
of Large Covariance Matrices, Ann. Stat., 36 (2008), pp. 199-227