Multi-fidelity Bayesian Quadrature for adaptive uncertainty quantification in engineering

Propagation of stochasticity from inputs to outputs is a critical task in uncertainty quantification of simulation models. Due to the high cost of (high-fidelity) numerical simulators, the pursuit of higher computational efficiency, on the premise of reasonably quantifying and controlling computational errors, has kept receiving attention. Bayesian Quadrature has emerged to be a competitive method for addressing this issue due to its flexibility of balancing efficiency and accuracy, and theoretical guarantee of convergence for a specific class of model functions. On the other hand, developing multiple simulators with different levels of fidelity for a specific physical system has also been a promising scheme for saving computational cost. In this context, a multi-fidelity Bayesian Quadrature method has been developed to leverage the advantages of both to the fullest extent. The closed-form Bayesian quadrature rules based on multi-fidelity models are primarily derived, as a probabilistic description for both mean prediction and prediction uncertainty. A strategy, called Expected Variance Contribution, is then developed for automatic selection of fidelity levels to balance the cost and expected improvement in prediction accuracy. Further, a data acquisition strategy based on the Generalized Uncertainty Sampling function is introduced for adaptive design of training points. All these components are integrated to form the developed multi-fidelity Bayesian Quadrature method, of which the effectiveness is demonstrated with both numerical examples and real-world simulators.