Bayesian Calibration for Prediction in a Multi-Output Transposition Context

TL;DR

This work tackles the challenge of predicting multiple outputs when only a subset is observed (the transposition context). It develops a Bayesian calibration framework that augments the calibration vector with $q-p$ additional inputs to model cross-output discrepancies through a hierarchical prior on these augmented variables, enabling accurate prediction of unobserved outputs. Computation relies on Gaussian-process surrogates and an importance-sampling–based estimator for hyperparameters, with either a MAP plug-in or a full-Bayesian treatment for the hyperparameters $\mathbf{A}$. Applied to a three-output Taylor cylinder impact code with a small dataset, the hierarchical approaches substantially improve predictive accuracy and uncertainty quantification compared with no-error or embedded-discrepancy baselines, demonstrating the practical value of hierarchical model error in transposition settings.

Abstract

Numerical simulations are widely used to predict the behavior of physical systems, with Bayesian approaches being particularly well suited for this purpose. However, experimental observations are necessary to calibrate certain simulator parameters for the prediction. In this work, we use a multi-output simulator to predict all its outputs, including those that have never been experimentally observed. This situation is referred to as the transposition context. To accurately quantify the discrepancy between model outputs and real data in this context, conventional methods cannot be applied, and the Bayesian calibration must be augmented by incorporating a joint model error across all outputs. To achieve this, the proposed method is to consider additional numerical input parameters within a hierarchical Bayesian model, which includes hyperparameters for the prior distribution of the calibration variables. This approach is applied on a computer code with three outputs that models the Taylor cylinder impact test with a small number of observations. 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.

Figures (9)

• Figure 1: Description of the Taylor cylinder impact test on a Copper rod.
• Figure 2: Prediction of the three outputs from observations of $\ell_f - \ell_{0}.$
• Figure 3: Prediction of the three outputs from observations of $r_f$
• Figure 4: Prediction of the three outputs from observations of $\upepsilon_{\text{max}}.$
• Figure 5: Distribution of each marginal of the posterior samples $(\boldsymbol{\uplambda}_k)_{k=1}^M$ obtained with the Uniform error and the Hierarchical MAP strategies, and the samples $\left(\boldsymbol{\upalpha}^1_k + \boldsymbol{\upalpha}^2_k \boldsymbol{\upxi}_r \right)_{\underset{k=1, \ldots, M}{r=1, \ldots, R}}$ for the Embedded discrepancy, for the prediction at $\mathbf{x}_{10}$ from observations of $\ell_f - \ell_0$. The black cross is associated with $\boldsymbol{\uplambda}_0$, the parameters used for the measurements acquisition.