Publications

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Uncertainty quantificatio 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 CO{sub 2} forcing. We develop a methodology that performs uncertainty quantificatio in the presence of limited data that have discontinuous character. Our approach is two-fold. First we detect the discontinuity location with a Bayesian inference, thus obtaining a probabilistic representation of the discontinuity curve location in presence of arbitrarily distributed input parameter values. Furthermore, we developed a spectral approach that relies on Polynomial Chaos (PC) expansions on each sides of the discontinuity curve leading to an averaged-PC representation of the forward model that allows efficient uncertainty quantification and propagation. The methodology is tested on synthetic examples of discontinuous data with adjustable sharpness and structure.

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Recent advances in high frame rate complementary metal-oxide-semiconductor (CMOS) cameras coupled with high repetition rate lasers have enabled laser-based imaging measurements of the temporal evolution of turbulent reacting flows. This measurement capability provides new opportunities for understanding the dynamics of turbulence-chemistry interactions, which is necessary for developing predictive simulations of turbulent combustion. However, quantitative imaging measurements using high frame rate CMOS cameras require careful characterization of the their noise, non-linear response, and variations in this response from pixel to pixel. We develop a noise model and calibration tools to mitigate these problems and to enable quantitative use of CMOS cameras. We have demonstrated proof of principle for image de-noising using both wavelet methods and Bayesian inference. The results offer new approaches for quantitative interpretation of imaging measurements from noisy data acquired with non-linear detectors. These approaches are potentially useful in many areas of scientific research that rely on quantitative imaging measurements.

Uncertainty quantification in climate models is challenged by the prohibitive cost of a large number of model evaluations for sampling. Another feature that often prevents classical uncertainty analysis from being readily 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 a discontinuity across a curve in the space of two uncertain parameters, namely climate sensitivity and CO2 forcing. In order to propagate uncertainties from model parameters to model output we use polynomial chaos (PC) expansions to represent the maximum overturning stream function in terms of the uncertain climate sensitivity and CO2 forcing parameters. Since the spectral methodology assumes a certain degree of smoothness, the presence of discontinuities suggests that separate PC expansions on each side of the discontinuity will lead to more accurate descriptions of the climate model output compared to global PC expansions. We propose a methodology that first finds a probabilistic description of the discontinuity given a number of data points. Assuming the discontinuity curve is a polynomial, the algorithm is based on Bayesian inference of its coefficients. Markov chain Monte Carlo sampling is used to obtain joint distributions for the polynomial coefficients, effectively parameterizing the distribution over all possible discontinuity curves. Next, we apply the Rosenblatt transformation to the irregular parameter domains on each side of the discontinuity. This transformation maps a space of uncertain parameters with specific probability distributions to a space of i.i.d standard random variables where orthogonal projections can be used to obtain PC coefficients. In particular, we use uniform random variables that are compatible with PC expansions based on Legendre polynomials. The Rosenblatt transformation and the corresponding PC expansions for the model output on either side of the discontinuity are applied successively for several realizations of the discontinuity curve. The climate model output and its associated uncertainty at specific design points is then computed by taking a quadrature-based integration average over PC expansions corresponding to possible realizations of the discontinuity curve.

Conventional methods for uncertainty quantification are generally challenged in the 'tails' of probability distributions. This is specifically an issue for many climate observables since extensive sampling to obtain a reasonable accuracy in tail regions is especially costly in climate models. Moreover, the accuracy of spectral representations of uncertainty is weighted in favor of more probable ranges of the underlying basis variable, which, in conventional bases does not particularly target tail regions. Therefore, what is ideally desired is a methodology that requires only a limited number of full computational model evaluations while remaining accurate enough in the tail region. To develop such a methodology, we explore the use of surrogate models based on non-intrusive Polynomial Chaos expansions and Galerkin projection. We consider non-conventional and custom basis functions, orthogonal with respect to probability distributions that exhibit fat-tailed regions. We illustrate how the use of non-conventional basis functions, and surrogate model analysis, improves the accuracy of the spectral expansions in the tail regions. Finally, we also demonstrate these methodologies using precipitation data from CCSM simulations.

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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.

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.

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.

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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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An automated procedure has been previously developed to generate simplified skeletal reaction mechanisms for the combustion of n-heptane/air mixtures at equivalence ratios between 0.5 and 2.0 and different pressures. The algorithm is based on a Computational Singular Perturbation (CSP)-generated database of importance indices computed from homogeneous n-heptane/air ignition solutions. In this paper, we examine the accuracy of these simplified mechanisms when they are used for modeling laminar n-heptane/air premixed flames. The objective is to evaluate the accuracy of the simplified models when transport processes lead to local mixture compositions that are not necessarily part of the comprehensive homogeneous ignition databases. The detailed mechanism was developed by Curran et al. and involves 560 species and 2538 reactions. The smallest skeletal mechanism considered consists of 66 species and 326 reactions. We show that these skeletal mechanisms yield good agreement with the detailed model for premixed n-heptane flames, over a wide range of equivalence ratios and pressures, for global flame properties. They also exhibit good accuracy in predicting certain elements of internal flame structure, especially the profiles of temperature and major chemical species. On the other hand, we find larger errors in the concentrations of many minor/radical species, particularly in the region where low-temperature chemistry plays a significant role. We also observe that the low-temperature chemistry of n-heptane can play an important role at very lean or very rich mixtures, reaching these limits first at high pressure. This has implications to numerical simulations of non-premixed flames where these lean and rich regions occur naturally. © 2009 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

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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.

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.

Terrorist attacks using an aerosolized pathogen preparation have gained credibility as a national security concern since the anthrax attacks of 2001. The ability to characterize the parameters of such attacks, i.e., to estimate the number of people infected, the time of infection, the average dose received, and the rate of disease spread in contemporary American society (for contagious diseases), is important when planning a medical response. For non-contagious diseases, we address the characterization problem by formulating a Bayesian inverse problem predicated on a short time-series of diagnosed patients exhibiting symptoms. To keep the approach relevant for response planning, we limit ourselves to 3.5 days of data. In computational tests performed for anthrax, we usually find these observation windows sufficient, especially if the outbreak model employed in the inverse problem is accurate. For contagious diseases, we formulated a Bayesian inversion technique to infer both pathogenic transmissibility and the social network from outbreak observations, ensuring that the two determinants of spreading are identified separately. We tested this technique on data collected from a 1967 smallpox epidemic in Abakaliki, Nigeria. We inferred, probabilistically, different transmissibilities in the structured Abakaliki population, the social network, and the chain of transmission. Finally, we developed an individual-based epidemic model to realistically simulate the spread of a rare (or eradicated) disease in a modern society. This model incorporates the mixing patterns observed in an (American) urban setting and accepts, as model input, pathogenic transmissibilities estimated from historical outbreaks that may have occurred in socio-economic environments with little resemblance to contemporary society. Techniques were also developed to simulate disease spread on static and sampled network reductions of the dynamic social networks originally in the individual-based model, yielding faster, though approximate, network-based epidemic models. These reduced-order models are useful in scenario analysis for medical response planning, as well as in computationally intensive inverse problems.

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.

We present computations of a methane-air edge flame stabilized against an incoming flow mixing layer, using detailed methane-air chemistry. We analyze the computed edge flame, with a focus on NO-structure. We examine the spatial distribution of NO and its production/consumption rate. We investigate the breakdown of the NO source term among the thermal, prompt, N{sub 2}O, and NO{sub 2} pathways. We examine the contributions of the four pathways at different locations, as the edge flame structure changes with downstream distance, tending to a classical diffusion flame structure. We also examine the dominant reaction flux contributions in each pathway. We compare the results to those in premixed, non-premixed, and opposed-jet triple flames.

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Bayesian statistics provides a foundation for inference from noisy and incomplete data, a natural mechanism for regularization in the form of prior information, and a quantitative assessment of uncertainty in the inferred results. Inverse problems - representing indirect estimation of model parameters, inputs, or structural components - can be fruitfully cast in this framework. Complex and computationally intensive forward models arising in physical applications, however, can render a Bayesian approach prohibitive. This difficulty is compounded by high-dimensional model spaces, as when the unknown is a spatiotemporal field. We present new algorithmic developments for Bayesian inference in this context, showing strong connections with the forward propagation of uncertainty. In particular, we introduce a stochastic spectral formulation that dramatically accelerates the Bayesian solution of inverse problems via rapid evaluation of a surrogate posterior. We also explore dimensionality reduction for the inference of spatiotemporal fields, using truncated spectral representations of Gaussian process priors. These new approaches are demonstrated on scalar transport problems arising in contaminant source inversion and in the inference of inhomogeneous material or transport properties. We also present a Bayesian framework for parameter estimation in stochastic models, where intrinsic stochasticity may be intermingled with observational noise. Evaluation of a likelihood function may not be analytically tractable in these cases, and thus several alternative Markov chain Monte Carlo (MCMC) schemes, operating on the product space of the observations and the parameters, are introduced.

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