Embedded Model Error Representation for Bayesian Model Calibration

Model error estimation remains one of the key challenges in uncertainty quantification and predictive science. For computational models of complex physical systems, model error, also known as structural error or model inadequacy, is often the largest contributor to the overall predictive uncertainty. This work builds on a recently developed framework of embedded, internal model correction, in order to represent and quantify structural errors, together with model parameters, within a Bayesian inference context. We focus specifically on a Polynomial Chaos representation with additive modification of existing model parameters, enabling a non-intrusive procedure for efficient approximate likelihood construction, model error estimation, and disambiguation of model and data errors' contributions to predictive uncertainty. The framework is demonstrated on several synthetic examples, as well as on a chemical ignition problem.Comment: Preprint 34 pages, 13 figures; v1 submitted on January 19, 2018; v2 submitted on February 5, 2019. v2 changes: addition of various clarifications and references, and minor language edit

Bayesian calibration of interatomic potentials for binary alloys

Developing reliable interatomic potential models with quantified predictive accuracy is crucial for atomistic simulations. Commonly used potentials, such as those constructed through the embedded atom method (EAM), are derived from semi-empirical considerations and contain unknown parameters that must be fitted based on training data. In the present work, we investigate Bayesian calibration as a means of fitting EAM potentials for binary alloys. The Bayesian setting naturally assimilates probabilistic assertions about uncertain quantities. In this way, uncertainties about model parameters and model errors can be updated by conditioning on the training data and then carried through to prediction. We apply these techniques to investigate an EAM potential for a family of gold-copper systems in which the training data correspond to density-functional theory values for lattice parameters, mixing enthalpies, and various elastic constants. Through the use of predictive distributions, we demonstrate the limitations of the potential and highlight the importance of statistical formulations for model error.Comment: Preprint, 28 pages, 18 figures, accepted for publication in Computational Materials Science on 7/11/202

Bayesian Nonlocal Operator Regression (BNOR): A Data-Driven Learning Framework of Nonlocal Models with Uncertainty Quantification

We consider the problem of modeling heterogeneous materials where micro-scale dynamics and interactions affect global behavior. In the presence of heterogeneities in material microstructure it is often impractical, if not impossible, to provide quantitative characterization of material response. The goal of this work is to develop a Bayesian framework for uncertainty quantification (UQ) in material response prediction when using nonlocal models. Our approach combines the nonlocal operator regression (NOR) technique and Bayesian inference. Specifically, we use a Markov chain Monte Carlo (MCMC) method to sample the posterior probability distribution on parameters involved in the nonlocal constitutive law, and associated modeling discrepancies relative to higher fidelity computations. As an application, we consider the propagation of stress waves through a one-dimensional heterogeneous bar with randomly generated microstructure. Several numerical tests illustrate the construction, enabling UQ in nonlocal model predictions. Although nonlocal models have become popular means for homogenization, their statistical calibration with respect to high-fidelity models has not been presented before. This work is a first step towards statistical characterization of nonlocal model discrepancy in the context of homogenization

Compressive sensing adaptation for polynomial chaos expansions

Basis adaptation in Homogeneous Chaos spaces rely on a suitable rotation of the underlying Gaussian germ. Several rotations have been proposed in the literature resulting in adaptations with different convergence properties. In this paper we present a new adaptation mechanism that builds on compressive sensing algorithms, resulting in a reduced polynomial chaos approximation with optimal sparsity. The developed adaptation algorithm consists of a two-step optimization procedure that computes the optimal coefficients and the input projection matrix of a low dimensional chaos expansion with respect to an optimally rotated basis. We demonstrate the attractive features of our algorithm through several numerical examples including the application on Large-Eddy Simulation (LES) calculations of turbulent combustion in a HIFiRE scramjet engine.Comment: Submitted to Journal of Computational Physic