Uncertainty Quantification and Apportionment in Air Quality Models using the Polynomial Chaos Method

Simulations of large-scale physical systems are often affected by the uncertainties in data, in model parameters, and by incomplete knowledge of the underlying physics. The traditional deterministic simulations do not account for such uncertainties. It is of interest to extend simulation results with ``error bars'' that quantify the degree of uncertainty. This added information provides a confidence level for the simulation result. For example, the air quality forecast with an associated uncertainty information is very useful for making policy decisions regarding environmental protection. Techniques such as Monte Carlo (MC) and response surface are popular for uncertainty quantification, but accurate results require a large number of runs. This incurs a high computational cost, which maybe prohibitive for large-scale models. The polynomial chaos (PC) method was proposed as a practical and efficient approach for uncertainty quantification, and has been successfully applied in many engineering fields. Polynomial chaos uses a spectral representation of uncertainty. It has the ability to handle both linear and nonlinear problems with either Gaussian or non-Gaussian uncertainties. This work extends the functionality of the polynomial chaos method to Source Uncertainty Apportionment (SUA), i.e., we use the polynomial chaos approach to attribute the uncertainty in model results to different sources of uncertainty. The uncertainty quantification and source apportionment are implemented in the Sulfur Transport Eulerian Model (STEM-III). It allows us to assess the combined effects of different sources of uncertainty to the ozone forecast. It also enables to quantify the contribution of each source to the total uncertainty in the predicted ozone levels

Simultaneous Optimal Uncertainty Apportionment and Robust Design Optimization of Systems Governed by Ordinary Differential Equations

The inclusion of uncertainty in design is of paramount practical importance because all real-life systems are affected by it. Designs that ignore uncertainty often lead to poor robustness, suboptimal performance, and higher build costs. Treatment of small geometric uncertainty in the context of manufacturing tolerances is a well studied topic. Traditional sequential design methodologies have recently been replaced by concurrent optimal design methodologies where optimal system parameters are simultaneously determined along with optimally allocated tolerances; this allows to reduce manufacturing costs while increasing performance. However, the state of the art approaches remain limited in that they can only treat geometric related uncertainties restricted to be small in magnitude. This work proposes a novel framework to perform robust design optimization concurrently with optimal uncertainty apportionment for dynamical systems governed by ordinary differential equations. The proposed framework considerably expands the capabilities of contemporary methods by enabling the treatment of both geometric and non-geometric uncertainties in a unified manner. Additionally, uncertainties are allowed to be large in magnitude and the governing constitutive relations may be highly nonlinear. In the proposed framework, uncertainties are modeled using Generalized Polynomial Chaos and are solved quantitatively using a least-square collocation method. The computational efficiency of this approach allows statistical moments of the uncertain system to be explicitly included in the optimization-based design process. The framework formulates design problems as constrained multi-objective optimization problems, thus enabling the characterization of a Pareto optimal trade-off curve that is off-set from the traditional deterministic optimal trade-off curve. The Pareto off-set is shown to be a result of the additional statistical moment information formulated in the objective and constraint relations that account for the system uncertainties. Therefore, the Pareto trade-off curve from the new framework characterizes the entire family of systems within the probability space; consequently, designers are able to produce robust and optimally performing systems at an optimal manufacturing cost. A kinematic tolerance analysis case-study is presented first to illustrate how the proposed methodology can be applied to treat geometric tolerances. A nonlinear vehicle suspension design problem, subject to parametric uncertainty, illustrates the capability of the new framework to produce an optimal design at an optimal manufacturing cost, accounting for the entire family of systems within the associated probability space. This case-study highlights the general nature of the new framework which is capable of optimally allocating uncertainties of multiple types and with large magnitudes in a single calculation

Motion Planning of Uncertain Ordinary Differential Equation Systems

This work presents a novel motion planning framework, rooted in nonlinear programming theory, that treats uncertain fully and under-actuated dynamical systems described by ordinary differential equations. Uncertainty in multibody dynamical systems comes from various sources, such as: system parameters, initial conditions, sensor and actuator noise, and external forcing. Treatment of uncertainty in design is of paramount practical importance because all real-life systems are affected by it, and poor robustness and suboptimal performance result if it’s not accounted for in a given design. In this work uncertainties are modeled using Generalized Polynomial Chaos and are solved quantitatively using a least-square collocation method. The computational efficiency of this approach enables the inclusion of uncertainty statistics in the nonlinear programming optimization process. As such, the proposed framework allows the user to pose, and answer, new design questions related to uncertain dynamical systems. Specifically, the new framework is explained in the context of forward, inverse, and hybrid dynamics formulations. The forward dynamics formulation, applicable to both fully and under-actuated systems, prescribes deterministic actuator inputs which yield uncertain state trajectories. The inverse dynamics formulation is the dual to the forward dynamic, and is only applicable to fully-actuated systems; deterministic state trajectories are prescribed and yield uncertain actuator inputs. The inverse dynamics formulation is more computationally efficient as it requires only algebraic evaluations and completely avoids numerical integration. Finally, the hybrid dynamics formulation is applicable to under-actuated systems where it leverages the benefits of inverse dynamics for actuated joints and forward dynamics for unactuated joints; it prescribes actuated state and unactuated input trajectories which yield uncertain unactuated states and actuated inputs. The benefits of the ability to quantify uncertainty when planning the motion of multibody dynamic systems are illustrated through several case-studies. The resulting designs determine optimal motion plans—subject to deterministic and statistical constraints—for all possible systems within the probability space

A Feasible Methodological Framework for Uncertainty Analysis and Diagnosis of Atmospheric Chemical Transport Models

The current state of quantifying uncertainty in chemical transport models (CTM) is often limited and insufficient due to numerous uncertainty sources and inefficient or inaccurate uncertainty propagation methods. In this study, we proposed a feasible methodological framework for CTM uncertainty analysis, featuring sensitivity analysis to filter for important model inputs and a new reduced-form model (RFM) that couples the high-order decoupled direct method (HDDM) and the stochastic response surface model (SRSM) to boost uncertainty propagation. Compared with the SRSM, the new RFM approach is 64% more computationally efficient while maintaining high accuracy. The framework was applied to PM2.5 simulations in the Pearl River Delta (PRD) region and found five precursor emissions, two pollutants in lateral boundary conditions (LBCs), and three meteorological inputs out of 203 model inputs to be important model inputs based on sensitivity analysis. Among these selected inputs, primary PM2.5 emissions, PM2.5 concentrations of LBCs, and wind speed were identified as key uncertainty sources, which collectively contributed 81.4% to the total uncertainty in PM2.5 simulations. Also, when evaluated against observations, we found that there were systematic underestimates in PM2.5 simulations, which can be attributed to the two-product method that describes the formation of secondary organic aerosol

A new hybrid uncertainty optimization method for structures using orthogonal series expansion

© 2017 Elsevier Inc. This paper proposes a new hybrid uncertain design optimization method for structures which contain both random and interval variables simultaneously. The optimization model is formulated with the feasible robustness and the reliability of the worst scenario. The hybrid uncertainty is quantified by using the orthogonal series expansion method that integrates the Polynomial Chaos (PC) expansion method and the Chebyshev interval method within a uniform framework. The design sensitivity of objective and constraints will be developed to greatly facilitate the use of gradient-based optimization algorithms. The numerical results show that this method will be more possible to seek the feasible solution

Evaluation and implementation of measurement uncertainty for determining stationary source emissions: a review

Este artículo presenta una revisión de metodologías comúnmente citadas en la literatura para la estimación de la incertidumbre, como es la metodología no estocástica de la Guía para la Expresión de la Incertidumbre de Medida (GUM), la cual provee una estructura de estimación con limitaciones en su implementación, como son: cálculo de derivadas parciales, suposición de linealidad de los modelos e identificación de las fuentes de incertidumbre y sus distribuciones de probabilidad. Por otro lado, se discuten otros métodos para estimar la incertidumbre, como son: Monte Carlo, Conjuntos Difusos, Intervalo Generalizado, Inferencia Bayesiana, Caos Polinomial y Bootstrap, que a diferencia de la GUM, presentan limitaciones de costo computacional y requieren de conocimientos más especializados para su implementación. El objetivo de este artículo es reportar el grado de aplicación y difusión de los métodos de estimación de la incertidumbre en las emisiones de fuentes fijas, encontrándose que la mayoría se enfoca en estudios usados para la elaboración de inventarios de gases de efecto invernadero (GHG), y son escasos los orientados a la medición de las emisiones de fuentes fijas usando monitoreos de lectura directa, como también los métodos definidos por la Agencia de Protección Ambiental de los Estados Unidos (US EPA). Se discute finalmente las fortalezas y debilidades que dan lugar al fomento de nuevas investigaciones en esta área del conocimiento.This paper presents a review of commonly-cited methods for estimating uncertainty in the literature. One of them is the non-stochastic approach proposed by the Guide to the Expression of Uncertainty in Measurement (GUM), which provides an estimation framework with limitations for the implementation, such as computation of partial derivatives, linear model assumptions, and uncertainty source identification with probability distributions. Other methods to estimate uncertainty are discussed as well; they include Monte Carlo, Fuzzy Sets, Generalized Intervals, Bayesian Inference, Polynomial Chaos, and Bootstrap, which in contrast to GUM present limitations regarding computational cost and require more specialized knowledge to be implemented. The aim of this work is to report the level of application and dissemination of methods for estimating the uncertainty of emissions caused by stationary sources. Most of the works in this field were found to be focused on the creation of inventories of Greenhouse Gases (GHG), and very few of them on the uncertainty associated with measuring the emissions of stationary sources using direct reading monitoring or those defined by the Environmental Protection Agency of the United States (US EPA). Finally, strengths and weaknesses are discussed in order to promote new research in this knowledge area

A Chebyshev interval method for nonlinear dynamic systems under uncertainty

This paper proposes a new interval analysis method for the dynamic response of nonlinear systems with uncertain-but-bounded parameters using Chebyshev polynomial series. Interval model can be used to describe nonlinear dynamic systems under uncertainty with low-order Taylor series expansions. However, the Taylor series-based interval method can only suit problems with small uncertain levels. To account for larger uncertain levels, this study introduces Chebyshev series expansions into interval model to develop a new uncertain method for dynamic nonlinear systems. In contrast to the Taylor series, the Chebyshev series can offer a higher numerical accuracy in the approximation of solutions. The Chebyshev inclusion function is developed to control the overestimation in interval computations, based on the truncated Chevbyshev series expansion. The Mehler integral is used to calculate the coefficients of Chebyshev polynomials. With the proposed Chebyshev approximation, the set of ordinary differential equations (ODEs) with interval parameters can be transformed to a new set of ODEs with deterministic parameters, to which many numerical solvers for ODEs can be directly applied. Two numerical examples are applied to demonstrate the effectiveness of the proposed method, in particular its ability to effectively control the overestimation as a non-intrusive method. © 2012 Elsevier Inc

System level assessment of uncertainty in aviation environmental policy impact analysis

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2010.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student-submitted PDF version of thesis.Includes bibliographical references (p. 83-93).This thesis demonstrates the assessment of uncertainty of a simulation model at the system level, which takes into account the interaction between the modules that comprise the system. Results from this system level assessment process aid policy-makers by identifying the key drivers of uncertainty in model outputs, among the input factors of the various modules that comprise the system. This knowledge can help direct resource allocation for research to reduce the uncertainty in policy outputs. The assessment results can also identify input factors that, when treated as deterministic variables, will not significantly affect the output variability. The system level assessment process is demonstrated on a model that estimates the air quality impacts of aviation. The model comprises two modules: the Aviation Environmental Design Tool (AEDT), which simulates aircraft operations to estimate performance and emissions inventories, and the Aviation environmental Portfolio Management Tool (APMT)- Impacts Air Quality module, which estimates the health and welfare impacts associated with aviation emissions. Global sensitivity analysis is employed to quantify the contribution of uncertainty in each input factor to the variability of system outputs, which here are adult mortality rates and total health cost. The assessment results show that none of the input factors of AEDT contribute significantly to the variability of system outputs. Therefore, if uncertainty reduction in the estimation of adult mortality and total health cost is desired, future research efforts should be directed towards gaining more knowledge on the input factors of the APMT-Impacts Air Quality module. This thesis also demonstrates the application of system level assessment in policy impact analysis, where policy impact is defined as the incremental change between baseline and policy outputs. In such an analysis, it is important to ensure that the uncertainty in policy impacts only accounts for the uncertainty corresponding to the difference between baseline and policy scenarios. Some input factors have a common source of uncertainty between scenarios, in which case the same representation of uncertainty must be used. Other input factors, on the other hand, are assumed to have independent variability between the different scenarios, and therefore need to have independent representation of uncertainty. This thesis demonstrates uncertainty assessment of a technology infusion policy analysis.by Rhea Patricia Liem.S.M

Model reduction of stochastic groundwater flow and transport processes

This work presents a comprehensive methodology for the reduction of analytical or numerical stochastic models characterized by uncertain input parameters or boundary conditions. The technique, based on the Polynomial Chaos Expansion (PCE) theory, represents a versatile solution to solve direct or inverse problems related to propagation of uncertainty. The potentiality of the methodology is assessed investigating different applicative contexts related to groundwater flow and transport scenarios, such as global sensitivity analysis, risk analysis and model calibration. This is achieved by implementing a numerical code, developed in the MATLAB environment, presented here in its main features and tested with literature examples. The procedure has been conceived under flexibility and efficiency criteria in order to ensure its adaptability to different fields of engineering; it has been applied to different case studies related to flow and transport in porous media. Each application is associated with innovative elements such as (i) new analytical formulations describing motion and displacement of non-Newtonian fluids in porous media, (ii) application of global sensitivity analysis to a high-complexity numerical model inspired by a real case of risk of radionuclide migration in the subsurface environment, and (iii) development of a novel sensitivity-based strategy for parameter calibration and experiment design in laboratory scale tracer transport

Methodology for technology evaluation under uncertainty and its application in advanced coal gasification processes

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 273-287).Integrated gasification combined cycle (IGCC) technology has attracted interest as a cleaner alternative to conventional coal-fired power generation processes. While a number of pilot projects have been launched to experimentally test IGCC technologies, mathematical simulation remains a central part of the ongoing research efforts. A major challenge in modeling an IGCC power plant is the lack of real experience and reliable data. It is critical to properly understand the state of knowledge and evaluate the impact of uncertainty in every phase of the R&D process. A rigorous investigation of the effect of uncertainty on IGCC system requires accurate quantification of input uncertainty and efficient propagation of uncertainty through system models. This thesis proposes several uncertainty quantification methods which expand the sources of information that can be used for parameter estimation. Key features of these methods include the use of entropy maximization to translate subjective opinions to probability distribution functions, and a more flexible probability model that easily captures anomaly associated with small sample data. In addition, Bayesian estimation is extended to dynamic models. Aided by a computationally efficient algorithm, termed sequential Monte Carlo method, the Bayesian approach is shown to be an effective way to estimate time-variant parameters. Uncertainty propagation is performed using the deterministic equivalent modeling method (DEMM) which is based on polynomial chaos representation of random variables and probabilistic collocation algorithm. One major issue often overlooked in the analysis of IGCC models is to represent correlation in the input parameters. This thesis proposes the use of principal component analysis (PCA) to represent correlated random variables. The resulting formulation is the same as the truncated Karhunen-Lodve expansions. Explicit incorporation of correlation not only improves accuracy of the approximation but also reduces the overall computational time. A comprehensive study of the MIT-BP IGCC model is carried out to determine uncertainties of the key measures of performance and cost, including energy output, thermal efficiency, CO 2 emission, plant capital cost, and cost of electricity. Whenever possible, the probability distributions of input parameters are estimated based on realistic data. Experts' judgments are solicited if data acquisition is infeasible. Uncertainty analysis is conducted in a three-step approach. First, technology-related input parameters are taken into account to determine uncertainties of plant performance. Second, cost uncertainties are determined with only economic inputs in order to identify important economic parameters. Finally, the plant model is integrated with cost model and they are evaluated with the key technical and economic inputs identified in the previous steps. Our study indicates the property of coal feed has a substantial impact on the energy production of the IGCC plant, and subsequently on the cost of electricity. Immature technologies such as gasification and gas turbine have important bearing on model performance hence need to be addressed in future research.by Bo Gong.Ph.D