Partitioned Analysis of Coupled Numerical Models Considering Imprecise Parameters and Inexact Models

Abstract

The present study develops an integrated coupling and uncertainty quantification framework for strongly coupled models that explicitly considers the propagation of uncertainty and bias inherent in model prediction between constituents during the iterative coupling process. Utilizing optimization techniques, three distinct configurations are formulated that differ in sequence of coupling and uncertainty quantification campaigns. Focusing on a controlled structural dynamics problem, the systematic biases from the constituents are quantified, from which the critical components of the model that require further improvement can be identified to aid in the prioritization of future code development efforts.

Experiment-based validation and uncertainty quantification of coupled multi-scale plasticity models

“…Rather, the method by which this bias is quantified is inconsequential to the way in which bias-corrected partitioned analysis of said bias is applied to the prediction, and as such the method selected for computer experiments. What is important, however, is the accuracy with which the method for quantifying bias is able to train the discrepancy function (Stevens and Atamturktur, 2015) as well as assessing the calibration of parameters and inference of bias in a connected manner (Farajpour and Atamturktur, 2014). A variety of methods are available for inferring bias in the constituents, starting with regression-based approaches directly relating bias to tested control settings, be they as simple linear functions (Derber and Wu, 1998), high degree polynomials where coefficients are determined stochastically (Steinberg, 1985), up to non-parametric fits such as a Gaussian process model (Sacks et al, 1989;Kennedy and O'Hagan, 2001;Bayarri et al, 2007), and continuing away to methods for determining relationships between discrepancy and control settings such as a maximum likelihood estimation of parameter distribution characteristics (Xiong et al, 2009;Atamturktur et al, 2015b) or a copula-based approach (Xi et al, 2014).…”

“…Kumar and Ghoniem (2012a) took advantage of separate-effect experiments of dependent outputs, but the information gained from these was limited to tracking the propagation of uncertainties through the coupling process rather than remedying the degrading effects of this propagation. Farajpour and Atamturktur (2014) proposed an integrated coupling-uncertainty quantification framework in which constituent model parameters were calibrated and bias was quantified using separate-effect experiments. However, their study only considered bias for the purpose of avoiding compensation for said bias by uncertain parameters during calibration.…”

“…Earlier studies have implemented integral-effect experiments for the calibration of independent parameters in constituent models (Lin and Yim, 2006) as well as calibration of independent parameters specifically related to coupling, in that these parameters feed into both models (Liu and Muraleetharan, 2012). Using integral-effect experiments, Farajpour and Atamturktur (2014) illustrated the importance of considering both bias and uncertainty in constituent model calibration by training a discrepancy model using integral-effect experiments to bias-correct the coupled predictions. Integral-effect experiments are also highly valuable for validation of the coupled model as the coupled domain is where predictions critical for decision making will occur (Avramova and Ivanov, 2010;Kumar and Ghoniem, 2012b;Liu and Muraleetharan, 2012).…”

“…This process of correcting for the inadequacy in the constituent model, referred to as "bias correction," is implemented to ultimately improve predictions of the coupled model. Determination of the discrepancy function and calibration parameter values with relation to each other prevents unwarranted compensations between parameters, which may otherwise mask model bias (Kennedy and O'Hagan, 2001;Higdon et al, 2008;Farajpour and Atamturktur, 2014;Stevens and Atamturktur, 2015). Approaching this problem in the Bayesian context allows for calibration of the uncertain parameters and discrepancy function simultaneously, while also providing a smooth incorporation of the experimental errors in the calibration.…”

“…Partitioning allows for the calibration and bias correction of constituent models using separate-effect experiments through two unique approaches. The first of these (Figure 4(a)) involves coupling constituent models followed by calibrating their output to separate-effect experiments (see Farajpour and Atamturktur, 2014 for a demonstration of this approach). Herein, we propose a new framework referred to as "bias-corrected partitioned analysis," which operates by first calibrating and bias correcting constituent models with respect to their respective separate-effect experiments following coupling the models in a second step (Figure 4(b)).…”