Centered Gaussian random fields (GRFs) indexed by compacta such as smooth, bounded Euclidean domains or smooth, compact and orientable manifolds are determined by their covariance operators. We consider centered GRFs given as variational solutions to coloring operator equations driven by spatial white noise, with an elliptic self-adjoint pseudodifferential coloring operator from the H\"ormander class. This includes the Mat\'ern class of GRFs as a special case. Using 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 applied on finite sections of the bi-infinite precision and covariance matrices results in optimally numerically sparse approximations. That is, asymptotically only 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. The locations of these nonzero matrix entries are known a priori. The tapered covariance or precision matrices may also be optimally diagonally 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. In addition, 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.

翻译：中央高尔斯随机字段( GRF ), 由光滑、 约束的 Euclidean 域或平滑、 紧凑和可调整的元件由共变运算符决定。 我们认为, 中心GRF 是由空间白色噪音驱动的调色操作方程式的变异解决方案, 由 H\" ormander 类的 椭圆形自相交配的假相色运算符。 包括 Mat\' ernal 类 GRF, 作为特例 。 使用 平流多分辨率多分辨率的多分辨率分析, 我们证明, 精密和共变异性操作的操作者, 精密的精密度和易变异性运算符的精度, 精确度的精度和精确度, 精确度的直径直度, 精确度和精确度的直径直的G- 。 我们证明, 这些精度的直径直的直径直径直的直径比值, 和精确度的直径直径直径直的直的基的直径直径直径直径直径直的G- 。