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：我们考虑统计线性反问题，即从含噪声的解观测数据中推断椭圆偏微分方程中的未知源函数。我们采用基于高斯先验的非参数贝叶斯方法，通过共轭公式实现便捷的后验推断。综述了近期为后验估计质量与不确定性量化提供理论保证的研究成果，并讨论了该理论在两类重要先验——定义于Dirichlet-Laplacian特征基上的高斯级数先验与Matérn过程先验——中的应用。我们提供了这两类先验的后验推断实现方法，并通过数值模拟研究了其表现性能。