Functional derivatives for uncertainty quantification and error estimation and reduction via optimal high-fidelity simulations

Abstract

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.