We consider the $\mathcal{H}^2$-formatted compression and computational estimation of covariance functions on a compact set in $\mathbb{R}^d$. The classical sample covariance or Monte Carlo estimator is prohibitively expensive for many practically relevant problems, where often approximation spaces with many degrees of freedom and many samples for the estimator are needed. In this article, we propose and analyze a data sparse multilevel sample covariance estimator, i.e., a multilevel Monte Carlo estimator. For this purpose, we generalize the notion of asymptotically smooth kernel functions to a Gevrey type class of kernels for which we derive new variable-order $\mathcal{H}^2$-approximation rates. These variable-order $\mathcal{H}^2$-approximations can be considered as a variant of $hp$-approximations. Our multilevel sample covariance estimator then uses an approximate multilevel hierarchy of variable-order $\mathcal{H}^2$-approximations to compress the sample covariances on each level. The non-nestedness of the different levels makes the reduction to the final estimator nontrivial and we present a suitable algorithm which can handle this task in linear complexity. This allows for a data sparse multilevel estimator of Gevrey covariance kernel functions in the best possible complexity for Monte Carlo type multilevel estimators, which is quadratic. Numerical examples which estimate covariance matrices with tens of billions of entries are presented.

翻译：我们认为 $\ mathcal{ H\\\ 2 $ 2 格式化压缩和计算估算 以 $\ mathbb{ R\ = d$ 。 对于许多实际相关的问题, 古典样样变换或 Monte Carlo 估测器非常昂贵, 对于很多相关的问题, 通常需要将空格近似多个自由度的空格和校验器的许多样本。 在此文章中, 我们提议并分析一个数据稀疏的多级样样变调调调调调控值, 即多级的 Monte Carlo 估测器。 为此, 我们将无气调调制平滑内核功能的概念推广到 G\ calcal 类的Gevrey 样变换型调控器类别。 在调制式调控器的每个变换调调调调调调器级别中, 我们的多级变调调调调调调调调制调制调序值将使用一个近多级的多级级级的调和调调调调调调调调调调调调调调调调调, 调调调调的调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调的调调调调调调调调调调调调调调调调调调调调调调的调调调调调调调调调调调调调调调调调调的调调调调调调调调调调调调调调调的调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调的调调调调调调调调调调调调调调的调调调调调调调调调调调调调调调调的调调调调调调调调调调调调调调的调调调调调调的调调调调调调调调调调调调调调调的调调调调调调调调调调调调调调调调调调调调调调调调的调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调