Uncertainty quantification analyses often employ surrogate models as computationally efficient approximations of computer codes simulating the physical phenomena. The accuracy and economy in the construction of surrogate models depends on the quality and quantity of data collected from the computationally expensive system models. Computationally efficient methods for accurate surrogate model training are thus required. This paper develops a novel approach to surrogate model construction based on the hierarchical decomposition of the approximation error. The proposed algorithm employs sparse Gaussian processes on a hierarchical grid to achieve a sparse nonlinear approximation of the underlying function. In contrast to existing methods, which are based on minimizing prediction variance, the proposed approach focuses on model bias and aims to improve the quality of reconstruction represented by the model. The performance of the algorithm is compared to existing methods using several numerical examples. In the examples considered, the proposed method demonstrates significant improvement in the quality of reconstruction for the same sample size.
One of the most fundamental challenges in predictive modeling and simulation involving materials is quantifying and minimizing the errors that originate from the use of approximate constitutive laws (with uncertain parameters and/or model form). We propose to use functional derivatives of the quantity of interest (QoI) with respect to the input constitutive laws to quantify how the QoI depends on the entire input functions as opposed to its parameters as is common practice. This functional sensitivity can be used to (i) quantify the prediction uncertainty originating from uncertainties in the input functions; (ii) compute a first-order correction to the QoI when a more accurate constitutive law becomes available, and (iii) rank possible high-fidelity simulations in terms of the expected reduction in the error of the predicted QoI. We demonstrate the proposed approach with two examples involving solid mechanics where linear elasticity is used as the low-fidelity constitutive law and a materials model including non-linearities is used as the high-fidelity law. These examples show that functional uncertainty quantification not only provides an exact correction to the coarse prediction if the high-fidelity model is completely known but also a high-accuracy estimate of the correction with only a few evaluations of the high-fidelity model. The proposed approach is generally applicable and we foresee it will be useful to determine where and when high-fidelity information is required in predictive simulations.
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