We consider robust optimal experimental design (ROED) for nonlinear Bayesian inverse problems governed by partial differential equations (PDEs). An optimal design is one that maximizes some utility quantifying the quality of the solution of an inverse problem. However, the optimal design is dependent on elements of the inverse problem such as the simulation model, the prior, or the measurement error model. ROED aims to produce an optimal design that is aware of the additional uncertainties encoded in the inverse problem and remains optimal even after variations in them. We follow a worst-case scenario approach to develop a new framework for robust optimal design of nonlinear Bayesian inverse problems. The proposed framework a) is scalable and designed for infinite-dimensional Bayesian nonlinear inverse problems constrained by PDEs; b) develops efficient approximations of the utility, namely, the expected information gain; c) employs eigenvalue sensitivity techniques to develop analytical forms and efficient evaluation methods of the gradient of the utility with respect to the uncertainties we wish to be robust against; and d) employs a probabilistic optimization paradigm that properly defines and efficiently solves the resulting combinatorial max-min optimization problem. The effectiveness of the proposed approach is illustrated for optimal sensor placement problem in an inverse problem governed by an elliptic PDE.

翻译：本文研究偏微分方程约束下非线性贝叶斯反问题的鲁棒最优实验设计。最优设计旨在最大化量化反问题解质量的某种效用函数，但该设计受反问题中仿真模型、先验分布或测量误差模型等因素的影响。鲁棒最优实验设计旨在生成能识别反问题中隐含附加不确定性、并在这些因素变化时仍保持最优性的设计方案。本研究采用最坏情形场景方法，构建了非线性贝叶斯反问题鲁棒最优设计的新框架。该框架具有以下特征：a) 可扩展性强，专为偏微分方程约束的无限维贝叶斯非线性反问题设计；b) 建立了效用函数（即期望信息增益）的高效近似方法；c) 运用特征值灵敏度技术，推导出效用函数相对于目标不确定性的梯度解析形式及高效计算方法；d) 采用概率优化范式，明确定义并高效求解由此产生的组合式最大最小优化问题。通过椭圆型偏微分方程反问题中的最优传感器布置实例，验证了所提方法的有效性。