In this paper we investigate the Bayesian approach to inverse Robin problems. These are problems for certain elliptic boundary value problems of determining a Robin coefficient on a hidden part of the boundary from Cauchy data on the observable part. Such a nonlinear inverse problem arises naturally in the initialisation of large-scale ice sheet models that are crucial in climate and sea-level predictions. We motivate the Bayesian approach for a prototypical Robin inverse problem by showing that the posterior mean converges in probability to the data-generating ground truth as the number of observations increases. Related to the stability theory for inverse Robin problems, we establish a logarithmic convergence rate for Sobolev-regular Robin coefficients, whereas for analytic coefficients we can attain an algebraic rate. The use of rescaled analytic Gaussian priors in posterior consistency for nonlinear inverse problems is new and may be of separate interest in other inverse problems. Our numerical results illustrate the convergence property in two observation settings.

翻译：本文研究了贝叶斯方法在逆Robin问题中的应用。这类问题涉及特定椭圆边值问题，旨在通过可观测边界上的柯西数据，确定隐藏边界上的Robin系数。此类非线性逆问题在大型冰盖模型的初始化中自然出现，而冰盖模型对气候和海平面预测至关重要。我们通过证明后验均值随观测次数增加而依概率收敛于生成数据的真实值，为典型Robin逆问题的贝叶斯方法提供了理论依据。结合逆Robin问题的稳定性理论，我们建立了Sobolev正则Robin系数的对数收敛速率，而对于解析系数则可达到代数收敛速率。将重新缩放的解析高斯先验应用于非线性逆问题的后验一致性属于创新方法，可能对其它逆问题具有独立参考价值。数值结果在两种观测设置下验证了该收敛性质。