Bayesian Calibration for Prediction in a Multi-Output Transposition Context

TL;DR

This paper introduces a hierarchical Bayesian calibration method for multi-output simulators, enabling accurate predictions even for outputs without experimental data, demonstrated on a Taylor impact test model.

Contribution

It proposes a novel hierarchical Bayesian approach that incorporates joint model error and additional parameters for multi-output transposition calibration.

Findings

Effective calibration of all outputs, including unobserved ones.

Hierarchical model outperforms traditional methods with embedded errors.

Demonstrated on a three-output Taylor impact simulation.

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.