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.

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