Estimation of the Robin coefficient field in a Poisson problem with uncertain conductivity field
Estimation of the Robin coefficient field in a Poisson problem with uncertain conductivity field
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DOI：
10.1088/1361-6420/aad91e
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发表时间：
2018-01
期刊：
影响因子：
2.1
通讯作者：
R. Nicholson;N. Petra;J. Kaipio
中科院分区：
文献类型：
--
作者：
R. Nicholson;N. Petra;J. Kaipio
我们考虑重建的非均匀系数场在罗宾边界条件上的一个不可访问的部分边界的泊松问题与不确定的（或未知的）非均匀的电导率场的内部的域。考虑到模型的误差，源于电导率系数的不确定性，我们把未知的电导率作为一个讨厌的参数，并进行近似premarginalizing超过它，并只反演罗宾系数字段。我们通过贝叶斯近似误差（BAE）方法近似相关的建模误差。这里提出的不确定性分析依赖于最大后验概率（MAP）估计的参数-可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。为了计算MAP点，我们应用基于伴随方法的不精确牛顿共轭梯度法。通过调用Hessian的数据失配分量的低秩近似，协方差的构造变得易于处理。两个数值实验被认为是：一个在导电性的先验协方差是各向同性的，和一个在导电性的先验协方差是各向异性的。结果进行了比较，标准误差模型的基础上，特别强调后验不确定性估计的可行性。我们表明，BAE方法是一个可行的意义上，预测的后验不确定性是一致的实际估计误差，而忽略相关的建模误差产生不可行的估计罗宾系数。此外，我们证明了BAE方法是近似的计算昂贵的PDE解决的数量（测量）作为传统的错误的方法。
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.