We consider the statistical linear inverse problem of making inference on an unknown source function in an elliptic partial differential equation from noisy observations of its solution. We employ nonparametric Bayesian procedures based on Gaussian priors, leading to convenient conjugate formulae for posterior inference. We review recent results providing theoretical guarantees on the quality of the resulting posterior-based estimation and uncertainty quantification, and we discuss the application of the theory to the important classes of Gaussian series priors defined on the Dirichlet-Laplacian eigenbasis and Mat\'ern process priors. We provide an implementation of posterior inference for both classes of priors, and investigate its performance in a numerical simulation study.

翻译：我们考虑从带噪声的观测解中推断椭圆型偏微分方程未知源函数的统计线性反问题。采用基于高斯先验的非参数贝叶斯方法，得到便于后验推断的共轭计算公式。本文综述了关于后验估计与不确定性量化质量的理论保证的最新研究成果，并讨论了该理论在两类重要先验——基于狄利克雷-拉普拉斯特征基的高斯级数先验与Matérn过程先验——中的应用。我们实现了针对这两类先验的后验推断算法，并通过数值模拟研究评估其性能。