We consider goal-oriented optimal design of experiments for infinite-dimensional Bayesian linear inverse problems governed by partial differential equations (PDEs). Specifically, we seek sensor placements that minimize the posterior variance of a prediction or goal quantity of interest. The goal quantity is assumed to be a nonlinear functional of the inversion parameter. We propose a goal-oriented optimal experimental design (OED) approach that uses a quadratic approximation of the goal-functional to define a goal-oriented design criterion. The proposed criterion, which we call the Gq-optimality criterion, is obtained by integrating the posterior variance of the quadratic approximation over the set of likely data. Under the assumption of Gaussian prior and noise models, we derive a closed-form expression for this criterion. To guide development of discretization invariant computational methods, the derivations are performed in an infinite-dimensional Hilbert space setting. Subsequently, we propose efficient and accurate computational methods for computing the Gq-optimality criterion. A greedy approach is used to obtain Gq-optimal sensor placements. We illustrate the proposed approach for two model inverse problems governed by PDEs. Our numerical results demonstrate the effectiveness of the proposed strategy. In particular, the proposed approach outperforms non-goal-oriented (A-optimal) and linearization-based (c-optimal) approaches.

翻译：本文研究偏微分方程约束下无限维贝叶斯线性反问题的目标导向实验最优设计。具体而言，我们寻求最小化预测量或目标关注量的后验方差的传感器布设方案。目标量被假定为反演参数的非线性泛函。我们提出一种目标导向最优实验设计方法，该方法利用目标泛函的二次近似来定义目标导向设计准则。所提出的准则（称为Gq最优准则）通过对可能数据集上二次近似的后验方差积分获得。在高斯先验与噪声模型的假设下，我们推导了该准则的闭式表达式。为建立离散化无关的计算方法，推导过程在无限维希尔伯特空间框架下进行。随后，我们提出了计算Gq最优准则的高效精确数值方法，并采用贪婪算法获取Gq最优传感器布设。通过两个偏微分方程约束的模型反问题验证了所提方法。数值结果表明该策略具有显著优势，特别是在性能上超越了非目标导向的A最优方法和基于线性化的c最优方法。