We study the consistency of the estimator in spatial regression with partial differential equation (PDE) regularization. This new smoothing technique allows to accurately estimate spatial fields over complex two-dimensional domains, starting from noisy observations; the regularizing term involves a PDE that formalizes problem specific information about the phenomenon at hand. Differently from classical smoothing methods, the solution of the infinite-dimensional estimation problem cannot be computed analytically. An approximation is obtained via the finite element method, considering a suitable triangulation of the spatial domain. We first consider the consistency of the estimator in the infinite-dimensional setting. We then study the consistency of the finite element estimator, resulting from the approximated PDE. We study the bias and variance of the estimators, with respect to the sample size and to the value of the smoothing parameter. Some final simulation studies provide numerical evidence of the rates derived for the bias, variance and mean square error.

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关于空间回归与偏微分方程正则化的一致性的一些初步结果

我们研究了空间回归中估计量与偏微分方程 (PDE) 正则化的一致性。这种新的平滑技术允许从嘈杂的观察开始准确估计复杂二维域上的空间场；正则化项涉及一个偏微分方程，该偏微分方程将关于手头现象的问题特定信息形式化。与经典的平滑方法不同，无限维估计问题的解不能通过解析计算。考虑到空间域的合适三角剖分，通过有限元方法获得近似值。我们首先考虑估计器在无限维设置中的一致性。然后我们研究由近似 PDE 产生的有限元估计器的一致性。我们研究估计量的偏差和方差，关于样本大小和平滑参数的值。一些最终的模拟研究提供了偏差、方差和均方误差的比率的数值证据。