Estimation of the Robin coefficient field in a Poisson problem with uncertain conductivity field

We consider the reconstruction of a heterogeneous coefficient field in a Robin boundary condition on an inaccessible part of the boundary in a Poisson problem with an uncertain (or unknown) inhomogeneous conductivity field in the interior of the domain. To account for model errors that stem from the uncertainty in the conductivity coefficient, we treat the unknown conductivity as a nuisance parameter and carry out approximative premarginalization over it, and invert for the Robin coefficient field only. We approximate the related modelling errors via the Bayesian approximation error (BAE) approach. The uncertainty analysis presented here relies on a local linearization of the parameter-to-observable map at the maximum a posteriori (MAP) estimates, which leads to a normal (Gaussian) approximation of the parameter posterior density. To compute the MAP point we apply an inexact Newton conjugate gradient approach based on the adjoint methodology. The construction of the covariance is made tractable by invoking a low-rank approximation of the data misfit component of the Hessian. Two numerical experiments are considered: one where the prior covariance on the conductivity is isotropic, and one where the prior covariance on the conductivity is anisotropic. Results are compared to those based on standard error models, with particular emphasis on the feasibility of the posterior uncertainty estimates. We show that the BAE approach is a feasible one in the sense that the predicted posterior uncertainty is consistent with the actual estimation errors, while neglecting the related modelling error yields infeasible estimates for the Robin coefficient. In addition, we demonstrate that the BAE approach is approximately as computationally expensive (measured in the number of PDE solves) as the conventional error approach.

我们考虑在一个泊松问题中，在区域内部具有不确定（或未知）的非均匀电导率场的情况下，对边界不可达部分的罗宾边界条件中的非均匀系数场进行重构。为了解决由电导率系数的不确定性导致的模型误差，我们将未知电导率视为一个干扰参数，并对其进行近似的先边际化，仅对罗宾系数场进行反演。我们通过贝叶斯近似误差（BAE）方法来近似相关的建模误差。这里呈现的不确定性分析依赖于在最大后验（MAP）估计处参数到可观测量映射的局部线性化，这导致了参数后验密度的正态（高斯）近似。为了计算MAP点，我们应用一种基于伴随方法的不精确牛顿共轭梯度方法。通过对黑塞矩阵的数据不匹配分量进行低秩近似，使协方差的构建变得可行。考虑了两个数值实验：一个是电导率的先验协方差是各向同性的，另一个是电导率的先验协方差是各向异性的。将结果与基于标准误差模型的结果进行比较，特别强调后验不确定性估计的可行性。我们表明，BAE方法是一种可行的方法，因为预测的后验不确定性与实际估计误差是一致的，而忽略相关的建模误差会导致对罗宾系数的不可行估计。此外，我们证明BAE方法在计算上（以偏微分方程求解的次数来衡量）与常规误差方法大致相同。