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.

PUQ: A code for non-intrusive uncertainty propagation in computer simulations

“…where both q and p can, in general be vectors or even functions [7]. If the input is uncertain and represented by random variable P the output will also be a random variable, denoted Q .…”

Error correction in multi-fidelity molecular dynamics simulations using functional uncertainty quantification

“…Examples of input functions with varying degrees of accuracy include exchange and correlation functionals used in density functional theory calculations [2,15], interatomic potentials for molecular dynamics (MD) [16,17,18], generalized stacking faults used in dislocation dynamics [19], and constitutive laws for micromechanical simulations. In this paper we use functional derivatives (FD), recently proposed as a mathematical framework to quantify uncertainties that arise from constitutive models used in simulations [20], to quantify and correct uncertainties that originate from the interatomic potential used in MD simulations.…”

System-Level Support for Composition of Applications

“…For example, there are an increasing number of examples in the literature of complex simulations in which new computational tasks (components, in this context) are instantiated on demand during a simulation and have a lifetime much shorter than the overall simulation. Examples of this category include [25,30,32,7]. • Application introspection is a novel way of thinking about activities that need deep access to an application such as performance monitoring or debugging.…”

“…As such, the United States Government retains a nonexclusive, royalty-free right to publish or reproduce this article, or to allow others to do so, for Government purposes only. and debugging systems [35,37,27,25,30,32,7]. This emergence is a result of the changing compute/memory/IO balance of high-end computer systems, the push toward increasing physical fidelity and realism in simulations motivating increasing use of coupled multiphysics simulations, and the generally increasing size and richness of data generated.…”