Bayesian calibration is a powerful framework for identifying parameters in complex and large-scale computational models. However, when there is insufficient or poorly suited data for the calibration process, significant uncertainty about the identified parameters remains. This uncertainty can hinder effective decision-making and understanding of the system. In many applications, the experiment that generates the data can be influenced by design variables, such as sensor placements, loading conditions, or test configurations. Often, however, these design variables can not be chosen arbitrarily but only from a finite set of possible experimental designs. We propose an adaptive algorithm for Bayesian optimal experimental design over a finite design set, specifically tailored for applications involving expensive computational models. The method integrates an accelerated nested Monte Carlo estimator that reuses parameter samples to reduce model evaluations. Additionally, it employs common random numbers and Rao--Blackwellization to reduce variance in pairwise expected information gain comparisons. To support decision-making within the algorithm, we estimate the probability that one design outperforms another using bootstrap sampling. Starting with small sample sizes, the algorithm iteratively eliminates inferior designs based on these probabilistic comparisons, allocating additional computational effort only to promising candidates until a single design remains. The resulting approach achieves high reliability at substantially reduced computational cost, making it well-suited for large-scale engineering applications.

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