Bayesian calibration is an effective approach for ensuring that numerical simulations accurately reflect the behavior of physical systems. However, because numerical models are never perfect, a discrepancy known as model error exists between the model outputs and the observed data, and must be quantified. Conventional methods can not be implemented in transposition situations, such as when a model has multiple outputs but only one is experimentally observed. To account for the model error in this context, we propose augmenting the calibration process by introducing additional input numerical parameters through a hierarchical Bayesian model, which includes hyperparameters for the prior distribution of the calibration variables. Importance sampling estimators are used to avoid increasing computational costs. Performance metrics are introduced to assess the proposed probabilistic model and the accuracy of its predictions. The method is applied on a computer code with three outputs that models the Taylor cylinder impact test. The outputs are considered as the observed variables one at a time, to work with three different transposition situations. The proposed method is compared with other approaches that embed model errors to demonstrate the significance of the hierarchical formulation.

翻译：贝叶斯校准是确保数值模拟准确反映物理系统行为的有效方法。然而，由于数值模型永远无法做到完美，模型输出与观测数据之间存在称为模型误差的差异，必须对此进行量化。传统方法无法在转置情境中实施，例如当模型具有多个输出但仅有一个输出被实验观测时。为在此情境下考虑模型误差，我们提出通过引入额外输入数值参数来增强校准过程，这些参数通过包含校准变量先验分布超参数的层次贝叶斯模型实现。采用重要性采样估计器以避免增加计算成本。引入性能指标以评估所提出的概率模型及其预测精度。该方法应用于模拟泰勒圆柱冲击试验的具有三个输出的计算机程序。通过每次将其中一个输出视为观测变量，构建了三种不同的转置情境。将所提方法与其它包含模型误差的方法进行比较，以证明层次化建模框架的重要性。