在具有不确定导电率场的泊松问题中估计鲁金系数场
Estimation of the Robin coefficient field in a Poisson problem with uncertain conductivity field
AI总结：
本文研究了在不可达边界部分的鲁金边界条件下的异质系数场重建问题，通过贝叶斯近似误差方法处理导电率不确定性，并利用局部线性化和低秩近似来分析不确定性，展示了鲁金系数场估计的可行性。
摘要：
我们考虑在具有不确定（或未知）非均匀导电率场的域内情况下，在不可达边界部分的鲁金边界条件下的异质系数场重建。为处理源于导电率系数不确定性的模型误差，我们将未知导电率视为噪声参数，并对其进行近似预边缘化，仅反演鲁金系数场。我们通过贝叶斯近似误差（BAE）方法近似相关建模误差。此处呈现的不确定性分析依赖于在最大后验（MAP）估计处对参数到观测映射的局部线性化，这导致参数后验密度的正态（高斯）近似。为计算MAP点，我们应用基于伴随方法的不精确牛顿共轭梯度方法。通过调用数据不匹配部分的Hessian的低秩近似，使协方差构造变得可行。考虑了两个数值实验：一个中导电率的先验协方差是各向同性的，另一个是各向异性的。结果与基于标准误差模型的结果进行了比较，特别强调后验不确定性估计的可行性。我们展示了BAE方法在意义上是可行的，因为预测的后验不确定性与实际估计误差一致，而忽略相关建模误差会导致鲁金系数的不可行估计。此外，我们还证明BAE方法在计算上大约与传统误差方法一样昂贵（以PDE求解次数衡量）
英文摘要：
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.
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