In this article, we study Bayesian inverse problems with multi-layered Gaussian priors. We first describe the conditionally Gaussian layers in terms of a system of stochastic partial differential equations. We build the computational inference method using a finite-dimensional Galerkin method. We show that the proposed approximation has a convergence-in-probability property to the solution of the original multi-layered model. We then carry out Bayesian inference using the preconditioned Crank--Nicolson algorithm which is modified to work with multi-layered Gaussian fields. We show via numerical experiments in signal deconvolution and computerized X-ray tomography problems that the proposed method can offer both smoothing and edge preservation at the same time.

中文翻译：

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贝叶斯反演的非平稳多层高斯先验

在本文中，我们研究具有多层高斯先验的贝叶斯逆问题。我们首先根据随机偏微分方程组来描述条件高斯层。我们使用有限维伽辽金方法构建计算推理方法。我们表明，所提出的近似对原始多层模型的解具有概率收敛性。然后，我们使用预处理的 Crank--Nicolson 算法进行贝叶斯推理，该算法经过修改以适用于多层高斯场。我们通过信号反卷积和计算机 X 射线断层扫描问题的数值实验表明，所提出的方法可以同时提供平滑和边缘保留。