Publications

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The Polynomial chaos expansion provides a means of representing any L2 random variable as a sum of polynomials that are orthogonal with respect to a chosen measure. Examples include the Hermite polynomials with Gaussian measure on the real line and the Legendre polynomials with uniform measure on an interval. Polynomial chaos can be used to reformulate an uncertain ODE system, using Galerkin projection, as a new, higher-dimensional, deterministic ODE system which describes the evolution of each mode of the polynomial chaos expansion. It is of interest to explore the eigenstructure of the original and reformulated ODE systems by studying the eigenvalues and eigenvectors of their Jacobians. In this talk, we study the distribution of the eigenvalues of the two Jacobians. We outline in general the location of the eigenvalues of the new system with respect to those of the original system, and examine the effect of expansion order on this distribution.

Chemically reacting flow models generally involve inputs and parameters that are determined from empirical measurements, and therefore exhibit a certain degree of uncertainty. Estimating the propagation of this uncertainty into computational model output predictions is crucial for purposes of reacting flow model validation, model exploration, as well as design optimization. Recent years have seen great developments in probabilistic methods and tools for efficient uncertainty quantification (UQ) in computational models. These tools are grounded in the use of Polynomial Chaos (PC) expansions for representation of random variables. The utility and effectiveness of PC methods have been demonstrated in a range of physical models, including structural mechanics, transport in porous media, fluid dynamics, aeronautics, heat transfer, and chemically reacting flow. While high-dimensionality remains nominally an ongoing challenge, great strides have been made in dealing with moderate dimensionality along with non-linearity and oscillatory dynamics. In this talk, I will give an overview of UQ in chemical systems. I will cover both: (1) the estimation of uncertain input parameters from empirical data, and (2) the forward propagation of parametric uncertainty to model outputs. I will cover the basics of forward PC UQ methods with examples of their use. I will also highlight the need for accurate estimation of the joint probability density over the uncertain parameters, in order to arrive at meaningful estimates of model output uncertainties. Finally, I will discuss recent developments on the inference of this density given partial information from legacy experiments, in the absence of raw data.

Discontinuity detection is an important component in many fields: Image recognition, Digital signal processing, and Climate change research. Current methods shortcomings are: Restricted to one- or two-dimensional setting, Require uniformly spaced and/or dense input data, and Give deterministic answers without quantifying the uncertainty. Spectral methods for Uncertainty Quantification with global, smooth bases are challenged by discontinuities in model simulation results. Domain decomposition reduces the impact of nonlinearities and discontinuities. However, while gaining more smoothness in each subdomain, the current domain refinement methods require prohibitively many simulations. Therefore, detecting discontinuities up front and refining accordingly provides huge improvement to the current methodologies.

It is known that, in general, the correlation structure in the joint distribution of model parameters is critical to the uncertainty analysis of that model. Very often, however, studies in the literature only report nominal values for parameters inferred from data, along with confidence intervals for these parameters, but no details on the correlation or full joint distribution of these parameters. When neither posterior nor data are available, but only summary statistics such as nominal values and confidence intervals, a joint PDF must be chosen. Given the summary statistics it may not be reasonable nor necessary to assume the parameters are independent random variables. We demonstrate, using a Bayesian inference procedure, how to construct a posterior density for the parameters exhibiting self consistent correlations, in the absence of data, given (1) the fit-model, (2) nominal parameter values, (3) bounds on the parameters, and (4) a postulated statistical model, around the fit-model, for the missing data. Our approach ensures external Bayesian updating while marginalizing over possible data realizations. We then address the matching of given parameter bounds through the choice of hyperparameters, which are introduced in postulating the statistical model, but are not given nominal values. We discuss some possible approaches, including (1) inferring them in a separate Bayesian inference loop and (2) optimization. We also perform an empirical evaluation of the algorithm showing the posterior obtained with this data free inference compares well with the true posterior obtained from inference against the full data set.

It is known that, in general, the correlation structure in the joint distribution of model parameters is critical to the uncertainty analysis of that model. Very often, however, studies in the literature only report nominal values for parameters inferred from data, along with confidence intervals for these parameters, but no details on the correlation or full joint distribution of these parameters. When neither posterior nor data are available, but only summary statistics such as nominal values and confidence intervals, a joint PDF must be chosen. Given the summary statistics it may not be reasonable nor necessary to assume the parameters are independent random variables. We demonstrate, using a Bayesian inference procedure, how to construct a posterior density for the parameters exhibiting self consistent correlations, in the absence of data, given (1) the fit-model, (2) nominal parameter values, (3) bounds on the parameters, and (4) a postulated statistical model, around the fit-model, for the missing data. Our approach ensures external Bayesian updating while marginalizing over possible data realizations. We then address the matching of given parameter bounds through the choice of hyperparameters, which are introduced in postulating the statistical model, but are not given nominal values. We discuss some possible approaches, including (1) inferring them in a separate Bayesian inference loop and (2) optimization. We also perform an empirical evaluation of the algorithm showing the posterior obtained with this data free inference compares well with the true posterior obtained from inference against the full data set.

Multiscale multiphysics problems arise in a host of application areas of significant relevance to DOE, including electrical storage systems (membranes and electrodes in fuel cells, batteries, and ultracapacitors), water surety, chemical analysis and detection systems, and surface catalysis. Multiscale methods aim to provide detailed physical insight into these complex systems by incorporating coupled effects of relevant phenomena on all scales. However, many sources of uncertainty and modeling inaccuracies hamper the predictive fidelity of multiscale multiphysics simulations. These include parametric and model uncertainties in the models on all scales, and errors associated with coupling, or information transfer, across scales/physics. This presentation introduces our work on the development of uncertainty quantification methods for spatially decomposed atomistic-to-continuum (A2C) multiscale simulations. The key thrusts of this research effort are: inference of uncertain parameters or observables from experimental or simulation data; propagation of uncertainty through particle models; propagation of uncertainty through continuum models; propagation of information and uncertainty across model/scale interfaces; and numerical and computational analysis and control. To enable the bidirectional coupling between the atomistic and continuum simulations, a general formulation has been developed for the characterization of sampling noise due to intrinsic variability in particle simulations, and for the propagation of both this sampling noise and parametric uncertainties through coupled A2C multiscale simulations. Simplified tests of noise quantification in particle computations are conducted through Bayesian inference of diffusion rates in an idealized isothermal binary material system. A proof of concept is finally presented based on application of the present formulation to the propagation of uncertainties in a model plane Couette flow, where the near wall region is handled with molecular dynamics while the bulk region is handled with continuum methods.

Uncertainty quantification in climate models is challenged by the sparsity of the available climate data due to the high computational cost of the model runs. Another feature that prevents classical uncertainty analyses from being easily applicable is the bifurcative behavior in the climate data with respect to certain parameters. A typical example is the Meridional Overturning Circulation in the Atlantic Ocean. The maximum overturning stream function exhibits discontinuity across a curve in the space of two uncertain parameters, namely climate sensitivity and CO2 forcing. We develop a methodology that performs uncertainty quantification in this context in the presence of limited data. © 2009 IEEE.

Uncertainty quantification in climate models is challenged by the sparsity of the available climate data due to the high computational cost of the model runs. Another feature that prevents classical uncertainty analyses from being easily applicable is the bifurcative behavior in the climate data with respect to certain parameters. A typical example is the Meridional Overturning Circulation in the Atlantic Ocean. The maximum overturning stream function exhibits discontinuity across a curve in the space of two uncertain parameters, namely climate sensitivity and CO2 forcing. We develop a methodology that performs uncertainty quantification in this context in the presence of limited data.

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This paper presents recent progress on the use of Computational Singular Perturbation (CSP) techniques for time integration of stiff chemical systems. The CSP integration approach removes fast time scales from the reaction system, thereby enabling integration with explicit time stepping algorithms. For further efficiency improvements, a tabulation strategy was developed to allow reuse of the relevant CSP quantities. This paper outlines the method and demonstrates its use on the simulation of hydrogen-air ignition.

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We present a rapid method for the identification of viruses using microfluidic chip gel electrophoresis (CGE) of high-copy number proteins to generate unique protein profiles. Viral proteins are solubilized by heating at 95°C in borate buffer containing detergent (5 min), then labeled with fluorescamine dye (10 s), and analyzed using the μChemLab CGE system (5 min). Analyses of closely related T2 and T4 bacteriophage demonstrate sufficient assay sensitivity and peak resolution to distinguish the two phage. CGE analyses of four additional viruses - MS2 bacteriophage, Epstein - Barr, respiratory syncytial, and vaccinia viruses - demonstrate reproducible and visually distinct protein profiles. To evaluate the suitability of the method for unique identification of viruses, we employed a Bayesian classification approach. Using a subset of 126 replicate electropherograms of the six viruses and phage for training purposes, successful classification with non-training data was 66/69 or 95% with no false positives. The classification method is based on a single attribute (elution time), although other attributes such as peak width, peak amplitude, or peak shape could be incorporated and may improve performance further. The encouraging results suggest a rapid and simple way to identify viruses without requiring specialty reagents such as PCR probes and antibodies. © 2008 American Chemical Society.

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Fundamentals of ion transport in nanopores were studied through a joint experimental and computational effort. The study evaluated both nanoporous polymer membranes and track-etched nanoporous polycarbonate membranes. The track-etched membranes provide a geometrically well characterized platform, while the polymer membranes are more closely related to ion exchange systems currently deployed in RO and ED applications. The experimental effort explored transport properties of the different membrane materials. Poly(aniline) membranes showed that flux could be controlled by templating with molecules of defined size. Track-etched polycarbonate membranes were modified using oxygen plasma treatments, UV-ozone exposure, and UV-ozone with thermal grafting, providing an avenue to functionalized membranes, increased wettability, and improved surface characteristic lifetimes. The modeling effort resulted in a novel multiphysics multiscale simulation model for field-driven transport in nanopores. This model was applied to a parametric study of the effects of pore charge and field strength on ion transport and charge exclusion in a nanopore representative of a track-etched polycarbonate membrane. The goal of this research was to uncover the factors that control the flux of ions through a nanoporous material and to develop tools and capabilities for further studies. Continuation studies will build toward more specific applications, such as polymers with attached sulfonate groups, and complex modeling methods and geometries.

Many systems involving chemical reactions between small numbers of molecules exhibit inherent stochastic variability. Such stochastic reaction networks are at the heart of processes such as gene transcription, cell signaling or surface catalytic reactions, which are critical to bioenergy, biomedical, and electrical storage applications. The underlying molecular reactions are commonly modeled with chemical master equations (CMEs), representing jump Markov processes, or stochastic differential equations (SDEs), rather than ordinary differential equations (ODEs). As such reaction networks are often inferred from noisy experimental data, it is not uncommon to encounter large parametric uncertainties in these systems. Further, a wide range of time scales introduces the need for reduced order representations. Despite the availability of mature tools for uncertainty/sensitivity analysis and reduced order modeling in deterministic systems, there is a lack of robust algorithms for such analyses in stochastic systems. In this talk, we present advances in algorithms for predictability and reduced order representations for stochastic reaction networks and apply them to bistable systems of biochemical interest. To study the predictability of a stochastic reaction network in the presence of both parametric uncertainty and intrinsic variability, an algorithm was developed to represent the system state with a spectral polynomial chaos (PC) expansion in the stochastic space representing parametric uncertainty and intrinsic variability. Rather than relying on a non-intrusive collocation-based Galerkin projection [1], this PC expansion is obtained using Bayesian inference, which is ideally suited to handle noisy systems through its probabilistic formulation. To accommodate state variables with multimodal distributions, an adaptive multiresolution representation is used [2]. As the PC expansion directly relates the state variables to the uncertain parameters, the formulation lends itself readily to sensitivity analysis. Reduced order modeling in the time dimension is accomplished using a Karhunen-Loeve (KL) decomposition of the stochastic process in terms of the eigenmodes of its covariance matrix. Subsequently, a Rosenblatt transformation relates the random variables in the KL decomposition to a set of independent random variables, allowing the representation of the system state with a PC expansion in those independent random variables. An adaptive clustering method is used to handle multimodal distributions efficiently, and is well suited for high-dimensional spaces. The spectral representation of the stochastic reaction networks makes these systems more amenable to analysis, enabling a detailed understanding of their functionality, and robustness under experimental data uncertainty and inherent variability.

While models of combustion processes have been successful in developing engines with improved fuel economy, more costly simulations are required to accurately model pollution chemistry. These simulations will also involve significant parametric uncertainties. Computational singular perturbation (CSP) and polynomial chaos-uncertainty quantification (PC-UQ) can be used to mitigate the additional computational cost of modeling combustion with uncertain parameters. PC-UQ was used to interrogate and analyze the Davis-Skodje model, where the deterministic parameter in the model was replaced with an uncertain parameter. In addition, PC-UQ was combined with CSP to explore how model reduction could be combined with uncertainty quantification to understand how reduced models are affected by parametric uncertainty.

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