This work considers Bayesian experimental design for the inverse boundary value problem of linear elasticity in a two-dimensional setting. The aim is to optimize the positions of compactly supported pressure activations on the boundary of the examined body in order to maximize the value of the resulting boundary deformations as data for the inverse problem of reconstructing the Lam\'e parameters inside the object. We resort to a linearized measurement model and adopt the framework of Bayesian experimental design, under the assumption that the prior and measurement noise distributions are mutually independent Gaussians. This enables the use of the standard Bayesian A-optimality criterion for deducing optimal positions for the pressure activations. The (second) derivatives of the boundary measurements with respect to the Lam\'e parameters and the positions of the boundary pressure activations are deduced to allow minimizing the corresponding objective function, i.e., the trace of the covariance matrix of the posterior distribution, by a gradient-based optimization algorithm. Two-dimensional numerical experiments are performed to demonstrate the functionality of our approach.

翻译：本文考虑二维环境下线性弹性力学逆边界值问题的贝叶斯实验设计。研究旨在优化受检物体边界上紧支撑压力激励的位置，以最大化由此产生的边界形变作为数据对反演物体内部拉梅参数问题的价值。我们采用线性化测量模型，并在先验分布与测量噪声分布相互独立且均服从高斯分布的假设下，引入贝叶斯实验设计框架。这使得我们能够使用标准贝叶斯A最优性准则来推导压力激励的最优位置。通过推导边界测量值关于拉梅参数及边界压力激励位置的二阶导数，可借助梯度优化算法最小化相应目标函数（即后验分布协方差矩阵的迹）。最后通过二维数值实验验证了本方法的有效性。