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：本文考虑受偏微分方程控制的贝叶斯非线性反问题在模型不确定性下的最优实验设计。具体而言，我们研究除反演参数外控制方程还包含辅助不确定参数的反问题。重点关注具有无穷维反演参数与辅助参数的问题，并提出适用于此类问题最优设计的可扩展计算框架。所提方法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。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，并结合无穷维贝叶斯非线性反问题的A-最优设计方法。具体采用最大后验概率点处后验的高斯近似来定义考虑不确定性的最优实验设计目标函数，该目标函数便于评估与优化。特别地，该目标函数的计算成本（以偏微分方程求解次数衡量）不随离散化反演参数与辅助参数的维度增长而增长。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，并采用贪心算法这一实用方法寻找最优设计。通过三维区域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或推理阶段忽略建模不确定性的潜在风险。