Uncertainty quantification is recognized as a fundamental task to obtain predictive numerical simulations. However, many realistic engineering applications require complex and computationally expensive high-fidelity numerical simulations for the accurate characterization of the system responses. Moreover, complex physical models and extreme operative conditions can easily lead to hundreds of uncertain parameters that need to be propagated through high-fidelity codes. Under these circumstances, a single fidelity approach, i.e. a workflow that only uses high-fidelity simulations to perform the uncertainty quantification task, is unfeasible due to the prohibitive overall computational cost. In recent years, multifidelity strategies have been introduced to overcome this issue. The core idea of this family of methods is to combine simulations with varying levels of fidelity/accuracy in order to obtain the multifidelity estimators or surrogates with the same accuracy of their single fidelity counterparts at a much lower computational cost. This goal is usually accomplished by defining a prioria sequence of discretization levels or physical modeling assumptions that can be used to decrease the complexity of a numerical realization and thus its computational cost. However ,less attention has been dedicated to low-fidelity models that can be built directly from the small number of high-fidelity simulations available. In this work we focus our attention on Reduced-Order Models that can be considered a particular class of data-driven approaches. Our main goal is to explore the combination of multifidelity uncertainty quantification and reduced-order models to obtain an efficient framework for propagating uncertainties through expensive numerical codes.
Polynomial chaos expansions (PCE) are well-suited to quantifying uncertainty in models parameterized by independent random variables. The assumption of independence leads to simple strategies for building multivariate orthonormal bases and for sampling strategies to evaluate PCE coefficients. In contrast, the application of PCE to models of dependent variables is much more challenging. Three approaches can be used to construct PCE of models of dependent variables. The first approach uses mapping methods where measure transformations, such as the Nataf and Rosenblatt transformation, can be used to map dependent random variables to independent ones; however we show that this can significantly degrade performance since the Jacobian of the map must be approximated. A second strategy is the class of dominating support methods. In these approaches a PCE is built using independent random variables whose distributional support dominates the support of the true dependent joint density; we provide evidence that this approach appears to produce approximations with suboptimal accuracy. A third approach, the novel method proposed here, uses Gram–Schmidt orthogonalization (GSO) to numerically compute orthonormal polynomials for the dependent random variables. This approach has been used successfully when solving differential equations using the intrusive stochastic Galerkin method, and in this paper we use GSO to build PCE using a non-intrusive stochastic collocation method. The stochastic collocation method treats the model as a black box and builds approximations of the input–output map from a set of samples. Building PCE from samples can introduce ill-conditioning which does not plague stochastic Galerkin methods. To mitigate this ill-conditioning we generate weighted Leja sequences, which are nested sample sets, to build accurate polynomial interpolants. We show that our proposed approach, GSO with weighted Leja sequences, produces PCE which are orders of magnitude more accurate than PCE constructed using mapping or dominating support methods.
In the context of the DARPA funded project SEQUOIA we are interested in the design under uncertainty of a jet engine nozzle subject to the performance requirements of a reconnaissance mission for a small unmanned military aircraft. This design task involves complex and expensive aero-thermo-structural computational analyses where it is of a paramount importance to also include the effect of the uncertain variables to obtain reliable predictions of the device’s performance. In this work we focus on the forward propagation analysis which is a key part of the design under uncertainty workflow. This task cannot be tackled directly by means of single fidelity approaches due to the prohibitive computational cost associated to each realization. We report here a summary of our latest advancement regarding several multilevel and multifidelity strategies designed to alleviate these challenges. The overall goal of these techniques is to reduce the computational cost of analyzing a high-fidelity model by resorting to less accurate, but less computationally demanding, lower fidelity models. The features of these multifidelity UQ approaches are initially illustrated and demonstrated on several model problems and afterward for the aero-thermo-structural analysis of the jet engine nozzle.
This study explores a Bayesian calibration framework for the RAMPAGE alloy potential model for Cu-Ni and Cu-Zr systems, respectively. In RAMPAGE potentials, it is proposed that once calibrated potentials for individual elements are available, the inter-species interactions can be described by fitting a Morse potential for pair interactions with three parameters, while densities for the embedding function can be scaled by two parameters from the elemental densities. Global sensitivity analysis tools were employed to understand the impact each parameter has on the MD simulation results. A transitional Markov Chain Monte Carlo algorithm was used to generate samples from the multimodal posterior distribution consistent with the discrepancy between MD simulation results and DFT data. For the Cu-Ni system the posterior predictive tests indicate that the fitted interatomic potential model agrees well with the DFT data, justifying the basic RAMPAGE assumptions. For the Cu-Zr system, where the phase diagram suggests more complicated atomic interactions than in the case of Cu-Ni, the RAMPAGE potential captured only a subset of the DFT data. The resulting posterior distribution for the 5 model parameters exhibited several modes, with each mode corresponding to specific simulation data and a suboptimal agreement with the DFT results.
In this work we propose an approach for accelerating Uncertainty Quantification (UQ) analysis in the context of Multifidelity applications. In the presence of complex multiphysics applications, which often require a prohibitive computational cost for each evaluation, multifidelity UQ techniques try to accelerate the convergence of statistics by leveraging the in- formation collected from a larger number of a lower fidelity model realizations. However, at the-state-of-the-art, the performance of virtually all the multifidelity UQ techniques is related to the correlation between the high and low-fidelity models. In this work we proposed to design a multifidelity UQ framework based on the identification of independent important directions for each model. The main idea is that if the responses of each model can be represented in a common space, this latter can be shared to enhance the correlation when the samples are drawn with respect to it instead of the original variables. There are also two main additional advantages that follow from this approach. First, the models might be correlated even if their original parametrizations are chosen independently. Second, if the shared space between models has a lower dimensionality than the original spaces, the UQ analysis might benefit from a dimension reduction standpoint. In this work we designed this general framework and we also tested it on several test problems ranging from analytical functions for verification purpose, up to more challenging application problems as an aero-thermo-structural analysis and a scramjet flow analysis.
