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Languages : en
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This report presents the forward sensitivity analysis method as a means for quantification of uncertainty in system analysis. The traditional approach to uncertainty quantification is based on a "black box" approach. The simulation tool is treated as an unknown signal generator, a distribution of inputs according to assumed probability density functions is sent in and the distribution of the outputs is measured and correlated back to the original input distribution. This approach requires large number of simulation runs and therefore has high computational cost. Contrary to the "black box" method, a more efficient sensitivity approach can take advantage of intimate knowledge of the simulation code. In this approach equations for the propagation of uncertainty are constructed and the sensitivity is solved for as variables in the same simulation. This "glass box" method can generate similar sensitivity information as the above "black box" approach with couples of runs to cover a large uncertainty region. Because only small numbers of runs are required, those runs can be done with a high accuracy in space and time ensuring that the uncertainty of the physical model is being measured and not simply the numerical error caused by the coarse discretization. In the forward sensitivity method, the model is differentiated with respect to each parameter to yield an additional system of the same size as the original one, the result of which is the solution sensitivity. The sensitivity of any output variable can then be directly obtained from these sensitivities by applying the chain rule of differentiation. We extend the forward sensitivity method to include time and spatial steps as special parameters so that the numerical errors can be quantified against other physical parameters. This extension makes the forward sensitivity method a much more powerful tool to help uncertainty analysis. By knowing the relative sensitivity of time and space steps with other interested physical parameters, the simulation can be run at appropriate time and space steps that ensure numerical sensitivities will not affect the confidence of the physical parameter sensitivity results. Two well defined benchmark problems, thermal wave and nonlinear diffusion, are utilized to demonstrate the extended forward sensitivity analysis method. All the physical solutions, parameter sensitivity solutions, even time step sensitivity in one case, have analytical forms, which allows the verification to be done in the strictest sense. A pilot code, Extended Forward sensitivity Analysis pilot code (EFA), has been developed to implement the above work.
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Languages : en
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Book Description
This report presents the forward sensitivity analysis method as a means for quantification of uncertainty in system analysis. The traditional approach to uncertainty quantification is based on a "black box" approach. The simulation tool is treated as an unknown signal generator, a distribution of inputs according to assumed probability density functions is sent in and the distribution of the outputs is measured and correlated back to the original input distribution. This approach requires large number of simulation runs and therefore has high computational cost. Contrary to the "black box" method, a more efficient sensitivity approach can take advantage of intimate knowledge of the simulation code. In this approach equations for the propagation of uncertainty are constructed and the sensitivity is solved for as variables in the same simulation. This "glass box" method can generate similar sensitivity information as the above "black box" approach with couples of runs to cover a large uncertainty region. Because only small numbers of runs are required, those runs can be done with a high accuracy in space and time ensuring that the uncertainty of the physical model is being measured and not simply the numerical error caused by the coarse discretization. In the forward sensitivity method, the model is differentiated with respect to each parameter to yield an additional system of the same size as the original one, the result of which is the solution sensitivity. The sensitivity of any output variable can then be directly obtained from these sensitivities by applying the chain rule of differentiation. We extend the forward sensitivity method to include time and spatial steps as special parameters so that the numerical errors can be quantified against other physical parameters. This extension makes the forward sensitivity method a much more powerful tool to help uncertainty analysis. By knowing the relative sensitivity of time and space steps with other interested physical parameters, the simulation can be run at appropriate time and space steps that ensure numerical sensitivities will not affect the confidence of the physical parameter sensitivity results. Two well defined benchmark problems, thermal wave and nonlinear diffusion, are utilized to demonstrate the extended forward sensitivity analysis method. All the physical solutions, parameter sensitivity solutions, even time step sensitivity in one case, have analytical forms, which allows the verification to be done in the strictest sense. A pilot code, Extended Forward sensitivity Analysis pilot code (EFA), has been developed to implement the above work.
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Languages : en
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Book Description
Sensitivity analysis and uncertainty quantificatio.
Author: Dan G. Cacuci
Publisher: CRC Press
ISBN: 020348357X
Category : Mathematics
Languages : en
Pages : 367
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Book Description
As computer-assisted modeling and analysis of physical processes have continued to grow and diversify, sensitivity and uncertainty analyses have become indispensable scientific tools. Sensitivity and Uncertainty Analysis. Volume I: Theory focused on the mathematical underpinnings of two important methods for such analyses: the Adjoint Sensitivity Analysis Procedure and the Global Adjoint Sensitivity Analysis Procedure. This volume concentrates on the practical aspects of performing these analyses for large-scale systems. The applications addressed include two-phase flow problems, a radiative convective model for climate simulations, and large-scale models for numerical weather prediction.
Author: Dan G. Cacuci
Publisher: CRC Press
ISBN: 0203498798
Category : Mathematics
Languages : en
Pages : 304
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Book Description
As computer-assisted modeling and analysis of physical processes have continued to grow and diversify, sensitivity and uncertainty analyses have become indispensable investigative scientific tools in their own right. While most techniques used for these analyses are well documented, there has yet to appear a systematic treatment of the method based
Author: Dan Gabriel Cacuci
Publisher: Springer Nature
ISBN: 303119635X
Category : Science
Languages : en
Pages : 474
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Book Description
This text describes a comprehensive adjoint sensitivity analysis methodology (nth-CASAM), developed by the author, which enablesthe efficient and exact computation of arbitrarily high-order functional derivatives of model responses to model parameters in large-scale systems. The nth-CASAM framework is set in linearly increasing Hilbert spaces, each of state-function-dimensionality, as opposed to exponentially increasing parameter-dimensional spaces, thereby overcoming the so-called “curse of dimensionality” in sensitivity and uncertainty analysis. The nth-CASAM is applicable to any model; the larger the number of model parameters, the more efficient the nth-CASAM becomes for computing arbitrarily high-order response sensitivities. The book will be helpful to those working in the fields of sensitivity analysis, uncertainty quantification, model validation, optimization, data assimilation, model calibration, sensor fusion, reduced-order modelling, inverse problems and predictive modelling. This Volume Two, the second of three, presents the large-scale application of the nth-CASAM to perform a representative fourth-order sensitivity analysis of the Polyethylene-Reflected Plutonium benchmark described in the Nuclear Energy Agency (NEA) International Criticality Safety Benchmark Evaluation Project (ICSBEP) Handbook. This benchmark is modeled mathematically by the Boltzmann particle transport equation, involving 21,976 imprecisely-known parameters, the numerical solution of which requires representative large-scale computations. The sensitivity analysis presented in this volume is the most comprehensive ever performed in the field of reactor physics and the results presented in this book prove, perhaps counter-intuitively, that many of the 4th-order sensitivities are much larger than the corresponding 3rd-order ones, which are, in turn, much larger than the 2nd-order ones, all of which are much larger than the 1st-order sensitivities. Currently, the nth-CASAM is the only known methodology which enables such large-scale computations of exactly obtained expressions of arbitrarily-high-order response sensitivities.
Author: Ralph C. Smith
Publisher: SIAM
ISBN: 1611973228
Category : Computers
Languages : en
Pages : 400
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Book Description
The field of uncertainty quantification is evolving rapidly because of increasing emphasis on models that require quantified uncertainties for large-scale applications, novel algorithm development, and new computational architectures that facilitate implementation of these algorithms. Uncertainty Quantification: Theory, Implementation, and Applications provides readers with the basic concepts, theory, and algorithms necessary to quantify input and response uncertainties for simulation models arising in a broad range of disciplines. The book begins with a detailed discussion of applications where uncertainty quantification is critical for both scientific understanding and policy. It then covers concepts from probability and statistics, parameter selection techniques, frequentist and Bayesian model calibration, propagation of uncertainties, quantification of model discrepancy, surrogate model construction, and local and global sensitivity analysis. The author maintains a complementary web page where readers can find data used in the exercises and other supplementary material.
Author: Andrea Saltelli
Publisher: John Wiley & Sons
ISBN: 9780470725177
Category : Mathematics
Languages : en
Pages : 304
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Book Description
Complex mathematical and computational models are used in all areas of society and technology and yet model based science is increasingly contested or refuted, especially when models are applied to controversial themes in domains such as health, the environment or the economy. More stringent standards of proofs are demanded from model-based numbers, especially when these numbers represent potential financial losses, threats to human health or the state of the environment. Quantitative sensitivity analysis is generally agreed to be one such standard. Mathematical models are good at mapping assumptions into inferences. A modeller makes assumptions about laws pertaining to the system, about its status and a plethora of other, often arcane, system variables and internal model settings. To what extent can we rely on the model-based inference when most of these assumptions are fraught with uncertainties? Global Sensitivity Analysis offers an accessible treatment of such problems via quantitative sensitivity analysis, beginning with the first principles and guiding the reader through the full range of recommended practices with a rich set of solved exercises. The text explains the motivation for sensitivity analysis, reviews the required statistical concepts, and provides a guide to potential applications. The book: Provides a self-contained treatment of the subject, allowing readers to learn and practice global sensitivity analysis without further materials. Presents ways to frame the analysis, interpret its results, and avoid potential pitfalls. Features numerous exercises and solved problems to help illustrate the applications. Is authored by leading sensitivity analysis practitioners, combining a range of disciplinary backgrounds. Postgraduate students and practitioners in a wide range of subjects, including statistics, mathematics, engineering, physics, chemistry, environmental sciences, biology, toxicology, actuarial sciences, and econometrics will find much of use here. This book will prove equally valuable to engineers working on risk analysis and to financial analysts concerned with pricing and hedging.
Author: Ruiwen Shu
Publisher:
ISBN:
Category :
Languages : en
Pages : 151
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Book Description
This thesis gives an overview of the current results on uncertainty quantification and sensitivity analysis for multiscale kinetic equations with random inputs, with an emphasis on the author's contribution to this field. In the first part of this thesis we consider a kinetic-fluid model for disperse two-phase flows with uncertainty in the fine particle regime. We propose a stochastic asymptotic-preserving (s-AP) scheme in the generalized polynomial chaos stochastic Galerkin (gPC-sG) framework, which allows the efficient computation of the problem in both kinetic and hydrodynamic regimes. The s-AP property is proved by deriving the equilibrium of the gPC version of the Fokker-Planck operator. The coefficient matrices that arise in a Helmholtz equation and a Poisson equation, essential ingredients of the algorithms, are proved to be positive definite under reasonable and mild assumptions. The computation of the gPC version of a translation operator that arises in the inversion of the Fokker-Planck operator is accelerated by a spectrally accurate splitting method. Numerical examples illustrate the s-AP property and the efficiency of the gPC-sG method in various asymptotic regimes. In the second part of this thesis we consider the same kinetic-fluid model with random initial inputs in the light particle regime. Using energy estimates, we prove the uniform regularity in the random space of the model for random initial data near the global equilibrium in some suitable Sobolev spaces, with the randomness in the initial particle distribution and fluid velocity. By hypocoercivity arguments, we prove that the energy decays exponentially in time, which means that the long time behavior of the solution is insensitive to such randomness in the initial data. Then we consider the gPC-sG method for the same model. For initial data near the global equilibrium and smooth enough in the physical and random spaces, we prove that the gPC-sG method has spectral accuracy, uniformly in time and the Knudsen number, and the error decays exponentially in time. In the third part of this thesis we propose a stochastic Galerkin method using sparse wavelet bases for the Boltzmann equation with multi-dimensional random inputs. The method uses locally supported piecewise polynomials as an orthonormal basis of the random space. By a sparse approach, only a moderate number of basis functions is required to achieve good accuracy in multi-dimensional random spaces. We discover a sparse structure of a set of basis-related coefficients, which allows us to accelerate the computation of the collision operator. Regularity of the solution of the Boltzmann equation in the random space and an accuracy result of the stochastic Galerkin method are proved in multi-dimensional cases. The efficiency of the method is illustrated by numerical examples with uncertainties from the initial data, boundary data and collision kernel. In the fourth part of this thesis we explore the possibility of using Generalized polynomial chaos (gPC) for uncertainty quantification in hyperbolic problems. GPC has been extensively used in uncertainty quantification problems to handle random variables. For gPC to be valid, one requires high regularity on the random space that hyperbolic type problems usually cannot provide, and thus it is believed to behave poorly in those systems. We provide a counter-argument, and show that despite the solution profile itself develops singularities in the random space, which prevents the use of gPC, the physical quantities such as shock emergence time, shock location, and shock width are all smooth functions of random variables in the initial data: with proper shifting, the solution's polynomial interpolation approximates with high accuracy. The studies were inspired by the stability results from hyperbolic systems. We use the Burgers' equation as an example for thorough analysis, and the analysis could be extended to general conservation laws with convex fluxes.
Author: Jihoon Park
Publisher:
ISBN:
Category :
Languages : en
Pages :
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Book Description
Uncertainty quantification in the Earth Sciences forms an integral component in decision making. Such decision has different objectives depending on the subsurface system. For example, the goals include maximizing profits in exploitation of resources or minimizing the effects on the environment. It is often the case that the decision has to balance between multiple conflicting objectives. Because the decision is made on prediction uncertainty, it is crucial to quantify realistic uncertainty which necessitates identification of a variety of sources of model uncertainty. The sources of model uncertainty include different interpretations on subsurface structures and depositional scenarios, unknown spatial distributions of properties, uncertainty in boundary conditions, hydrological/hydraulic properties and errors in measurements. The subsurface system is parameterized to represent model uncertainty. The model variable can be either global (takes scalar value) or spatially distributed. With limited available data, a large number of uncertain model variables exists. One of key tasks is to quantify how each model variable contribute to response uncertainty, which can be achieved by means of sensitivity analysis. Sensitivity analysis plays an important role in geoscientific computer experiments, whether for forecasting, data assimilation or model calibration. Some methods of sensitivity analysis have been used in Earth Sciences but they have clear limitations -- they cannot efficiently deal with multivariate responses, excessive calculations are required, and it is hard to take into account categorical input uncertainty. Overcoming these limitations, we revisit the idea of regionalized sensitivity analysis. In particular, we focus on distance-based global sensitivity analysis to estimate sensitivities of multivariate responses with limited number of samples. We demonstrate how the results from sensitivity analysis can be utilized to reduce model uncertainty with minimal impact on response uncertainty. The results can be used to design second Monte Carlo or building a surrogate model. Uncertainty needs to be updated as more data are required from different sources. In a Bayesian framework, this requires sampling from a posterior density of model and prediction variables. The key components of the workflow are dimensionality reduction of data variables and building of a statistical surrogate model to replace full forward models. It is demonstrated that the methodology successfully performs model inversions with limited number of full forward model runs. In many geoscientific applications, both global and spatial variables are uncertain. For convenience in computations, spatial variables are often converted to a few global variables. Even if the approach is efficient, the inversion results may not be consistent with the stated geological prior which leads to unrealistic uncertainty. In this dissertation, we propose to extend direct forecasting to predict model variables themselves. It is shown that successful inversion can be performed with both global and spatial variables characterizing a field-scale subsurface system. All the methodologies are demonstrated with the case studies. The first case deals with an oil reservoir in Libya. The case is used to study the proposed methods for global sensitivity analysis and approaches for model inversions to integrate dynamic data. The second case deals with the groundwater reservoir in Denmark. The case is used to integrate different sources of data to offer the inputs of decision models for groundwater management.
Author: Sergio Daniel Marques Amaral
Publisher:
ISBN:
Category :
Languages : en
Pages : 175
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Book Description
To support effective decision making, engineers should comprehend and manage various uncertainties throughout the design process. In today's modern systems, quantifying uncertainty can become cumbersome and computationally intractable for one individual or group to manage. This is particularly true for systems comprised of a large number of components. In many cases, these components may be developed by different groups and even run on different computational platforms, making it challenging or even impossible to achieve tight integration of the various models. This thesis presents an approach for overcoming this challenge by establishing a divide-and-conquer methodology, inspired by the decomposition-based approaches used in multidisciplinary analysis and optimization. Specifically, this research focuses on uncertainty analysis, also known as forward propagation of uncertainties, and sensitivity analysis. We present an approach for decomposing the uncertainty analysis task amongst the various components comprising a feed-forward system and synthesizing the local uncertainty analyses into a system uncertainty analysis. Our proposed decomposition-based multicomponent uncertainty analysis approach is shown to converge in distribution to the traditional all-at-once Monte Carlo uncertainty analysis under certain conditions. Our decomposition-based sensitivity analysis approach, which is founded on our decomposition-based uncertainty analysis algorithm, apportions the system output variance among the system inputs. The proposed decomposition-based uncertainty quantification approach is demonstrated on a multidisciplinary gas turbine system and is compared to the traditional all-at-once Monte Carlo uncertainty quantification approach. To extend the decomposition-based uncertainty quantification approach to high dimensions, this thesis proposes a novel optimization formulation to estimate statistics from a target distribution using random samples generated from a (different) proposal distribution. The proposed approach employs the well-defined and determinable empirical distribution function associated with the available samples. The resulting optimization problem is shown to be a single linear equality and box-constrained quadratic program and can be solved efficiently using optimization algorithms that scale well to high dimensions. Under some conditions restricting the class of distribution functions, the solution of the optimization problem yields importance weights that are shown to result in convergence in the Ll-norm of the weighted proposal empirical distribution function to the target distribution function, as the number of samples tends to infinity. Results on a variety of test cases show that the proposed approach performs well in comparison with other well-known approaches. The proposed approaches presented herein are demonstrated on a realistic application; environmental impacts of aviation technologies and operations. The results demonstrate that the decomposition-based uncertainty quantification approach can effectively quantify the uncertainty of a multicomponent system for which the models are housed in different locations and owned by different groups.
