This paper analyzes a popular computational framework to solve infinite-dimensional Bayesian inverse problems, discretizing the prior and the forward model in a finite-dimensional weighted inner product space. We demonstrate the benefit of working on a weighted space by establishing operator-norm bounds for finite element and graph-based discretizations of Mat\'ern-type priors and deconvolution forward models. For linear-Gaussian inverse problems, we develop a general theory to characterize the error in the approximation to the posterior. We also embed the computational framework into ensemble Kalman methods and MAP estimators for nonlinear inverse problems. Our operator-norm bounds for prior discretizations guarantee the scalability and accuracy of these algorithms under mesh refinement.

翻译：本文分析了一种求解无穷维贝叶斯逆问题的流行计算框架，该框架在有限维加权内积空间中对先验和正演模型进行离散化。通过建立Matérn型先验和去卷积正演模型的有限元及图离散化的算子范数界，我们展示了在加权空间上工作的优势。针对线性高斯逆问题，我们发展了一般理论来刻画后验近似的误差。同时，我们将该计算框架嵌入到非线性逆问题的集合卡尔曼方法和MAP估计器中。我们得到的先验离散化算子范数界保证了这些算法在网格细化下的可扩展性和准确性。