We consider optimal experimental design (OED) for Bayesian nonlinear inverse problems governed by partial differential equations (PDEs) under model uncertainty. Specifically, we consider inverse problems in which, in addition to the inversion parameters, the governing PDEs include secondary uncertain parameters. We focus on problems with infinite-dimensional inversion and secondary parameters and present a scalable computational framework for optimal design of such problems. The proposed approach enables Bayesian inversion and OED under uncertainty within a unified framework. We build on the Bayesian approximation error (BAE) approach, to incorporate modeling uncertainties in the Bayesian inverse problem, and methods for A-optimal design of infinite-dimensional Bayesian nonlinear inverse problems. Specifically, a Gaussian approximation to the posterior at the maximum a posteriori probability point is used to define an uncertainty aware OED objective that is tractable to evaluate and optimize. In particular, the OED objective can be computed at a cost, in the number of PDE solves, that does not grow with the dimension of the discretized inversion and secondary parameters. The OED problem is formulated as a binary bilevel PDE constrained optimization problem and a greedy algorithm, which provides a pragmatic approach, is used to find optimal designs. We demonstrate the effectiveness of the proposed approach for a model inverse problem governed by an elliptic PDE on a three-dimensional domain. Our computational results also highlight the pitfalls of ignoring modeling uncertainties in the OED and/or inference stages.

翻译：我们研究偏微分方程控制下受模型不确定性影响的贝叶斯非线性反问题的最优实验设计。具体而言，我们考虑的反问题中，除反演参数外，控制偏微分方程还包含次要不确定参数。我们聚焦于具有无限维反演参数和次要参数的问题，并提出可扩展的计算框架以实现此类问题的最优设计。所提方法能够在统一框架内实现不确定性条件下的贝叶斯反演与最优实验设计。本研究基于贝叶斯近似误差方法，将建模不确定性纳入贝叶斯反问题，并结合无限维贝叶斯非线性反问题的A-最优设计方法。具体而言，通过最大后验概率点处的高斯后验近似，定义了可评估和优化的不确定性感知最优实验设计目标函数。特别地，该目标函数的计算成本（以偏微分方程求解次数计）不随离散化反演参数和次要参数的维度增长。最优实验设计问题被表述为二元双层偏微分方程约束优化问题，并采用贪婪算法这一实用方法寻找最优设计方案。我们通过三维域上椭圆型偏微分方程控制的模型反问题验证了所提方法的有效性。计算结果同时揭示了在最优实验设计和/或推断阶段忽略建模不确定性可能带来的缺陷。