SIAM/ASA Journal on Uncertainty Quantification, Volume 11, Issue 4, Page 1278-1307, December 2023.
Abstract. The statistical finite element method (StatFEM) is an emerging probabilistic method that allows observations of a physical system to be synthesized with the numerical solution of a PDE intended to describe it in a coherent statistical framework, to compensate for model error. This work presents a new theoretical analysis of the StatFEM demonstrating that it has similar convergence properties to the finite element method on which it is based. Our results constitute a bound on the 2-Wasserstein distance between the ideal prior and posterior and the StatFEM approximation thereof, and show that this distance converges at the same mesh-dependent rate as finite element solutions converge to the true solution. Several numerical examples are presented to demonstrate our theory, including an example which tests the robustness of StatFEM when extended to nonlinear quantities of interest.

中文翻译：

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统计有限元法的理论保证

SIAM/ASA 不确定性量化杂志，第 11 卷，第 4 期，第 1278-1307 页，2023 年 12 月。
摘要。统计有限元方法 (StatFEM) 是一种新兴的概率方法，它允许将物理系统的观测结果与偏微分方程的数值解相结合，旨在在连贯的统计框架中描述它，以补偿模型误差。这项工作提出了 StatFEM 的新理论分析，证明它与其所基于的有限元方法具有相似的收敛特性。我们的结果构成了理想先验和后验及其 StatFEM 近似之间的 2-Wasserstein 距离的界限，并表明该距离以与有限元解收敛到真实解相同的网格相关速率收敛。提供了几个数值示例来证明我们的理论，其中包括一个测试 StatFEM 在扩展到感兴趣的非线性量时的鲁棒性的示例。