We study the asymptotic frequentist coverage and Gaussian approximation of Bayes posterior credible sets in nonlinear inverse problems when a Gaussian prior is placed on the parameter of the PDE. The aim is to ensure valid frequentist coverage of Bayes credible intervals when estimating continuous linear functionals of the parameter. Our results show that Bayes credible intervals have conservative coverage under certain smoothness assumptions on the parameter and a compatibility condition between the likelihood and the prior, regardless of whether an efficient limit exists and/or Bernstein von-Mises theorem holds. In the latter case, our results yield a corollary with more relaxed sufficient conditions than previous works. We illustrate practical utility of the results through the example of estimating the conductivity coefficient of a second order elliptic PDE, where a near-$N^{-1/2}$ contraction rate and conservative coverage results are obtained for linear functionals that were shown not to be estimable efficiently.

翻译：本文研究当偏微分方程参数采用高斯先验时，非线性反问题中贝叶斯后验可信集的渐近频域覆盖性与高斯逼近特性。研究目标在于确保估计参数连续线性泛函时，贝叶斯可信区间具有有效的频域覆盖性。结果表明：在参数满足特定光滑性假设、似然函数与先验分布满足相容性条件的前提下，无论有效极限是否存在和/或Bernstein von-Mises定理是否成立，贝叶斯可信区间均具有保守覆盖性。在后一种情形下，我们的结论可推导出比前人研究条件更宽松的充分条件推论。通过估计二阶椭圆型偏微分方程传导系数的实例，我们展示了该结论的实际应用价值：对于已被证明无法有效估计的线性泛函，我们获得了接近$N^{-1/2}$的收缩率与保守覆盖性结果。