Novel Uncertainty Quantification Techniques for Problems Described by Stochastic Partial Differential Equations - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Novel Uncertainty Quantification Techniques for Problems Described by Stochastic Partial Differential Equations write by Peng Chen. This book was released on 2014. Novel Uncertainty Quantification Techniques for Problems Described by Stochastic Partial Differential Equations available in PDF, EPUB and Kindle. Uncertainty propagation (UP) in physical systems governed by PDEs is a challenging problem. This thesis addresses the development of a number of innovative techniques that emphasize the need for high-dimensionality modeling, resolving discontinuities in the stochastic space and considering the computational expense of forward solvers. Both Bayesian and non-Bayesian approaches are considered. Applications demonstrating the developed techniques are investigated in the context of flow in porous media and reservoir engineering applications. An adaptive locally weighted projection method (ALWPR) is firstly developed. It adaptively selects the needed runs of the forward solver (data collection) to maximize the predictive capability of the method. The methodology effectively learns the local features and accurately quantifies the uncertainty in the prediction of the statistics. It could provide predictions and confidence intervals at any query input and can deal with multi-output responses. A probabilistic graphical model framework for uncertainty quantification is next introduced. The high dimensionality issue of the input is addressed by a local model reduction framework. Then the conditional distribution of the multi-output responses on the low dimensional representation of the input field is factorized into a product of local potential functions that are represented non-parametrically. A nonparametric loopy belief propagation algorithm is developed for studying uncertainty quantification directly on the graph. The nonparametric nature of the model is able to efficiently capture non-Gaussian features of the response. Finally an infinite mixture of Multi-output Gaussian Process (MGP) models is presented to effectively deal with many of the difficulties of current UQ methods. This model involves an infinite mixture of MGP's using Dirichlet process priors and is trained using Variational Bayesian Inference. The Bayesian nature of the model allows for the quantification of the uncertainties due to the limited number of simulations. The automatic detection of the mixture components by the Variational Inference algorithm is able to capture discontinuities and localized features without adhering to ad hoc constructions. Finally, correlations between the components of multi-variate responses are captured by the underlying MGP model in a natural way. A summary of suggestions for future research in the area of uncertainty quantification field are given at the end of the thesis.
Spectral Methods for Uncertainty Quantification
Spectral Methods for Uncertainty Quantification - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Spectral Methods for Uncertainty Quantification write by Olivier Le Maitre. This book was released on 2010-03-11. Spectral Methods for Uncertainty Quantification available in PDF, EPUB and Kindle. This book deals with the application of spectral methods to problems of uncertainty propagation and quanti?cation in model-based computations. It speci?cally focuses on computational and algorithmic features of these methods which are most useful in dealing with models based on partial differential equations, with special att- tion to models arising in simulations of ?uid ?ows. Implementations are illustrated through applications to elementary problems, as well as more elaborate examples selected from the authors’ interests in incompressible vortex-dominated ?ows and compressible ?ows at low Mach numbers. Spectral stochastic methods are probabilistic in nature, and are consequently rooted in the rich mathematical foundation associated with probability and measure spaces. Despite the authors’ fascination with this foundation, the discussion only - ludes to those theoretical aspects needed to set the stage for subsequent applications. The book is authored by practitioners, and is primarily intended for researchers or graduate students in computational mathematics, physics, or ?uid dynamics. The book assumes familiarity with elementary methods for the numerical solution of time-dependent, partial differential equations; prior experience with spectral me- ods is naturally helpful though not essential. Full appreciation of elaborate examples in computational ?uid dynamics (CFD) would require familiarity with key, and in some cases delicate, features of the associated numerical methods. Besides these shortcomings, our aim is to treat algorithmic and computational aspects of spectral stochastic methods with details suf?cient to address and reconstruct all but those highly elaborate examples.
Quantification of Uncertainty: Improving Efficiency and Technology
Quantification of Uncertainty: Improving Efficiency and Technology - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Quantification of Uncertainty: Improving Efficiency and Technology write by Marta D'Elia. This book was released on 2020-07-30. Quantification of Uncertainty: Improving Efficiency and Technology available in PDF, EPUB and Kindle. This book explores four guiding themes – reduced order modelling, high dimensional problems, efficient algorithms, and applications – by reviewing recent algorithmic and mathematical advances and the development of new research directions for uncertainty quantification in the context of partial differential equations with random inputs. Highlighting the most promising approaches for (near-) future improvements in the way uncertainty quantification problems in the partial differential equation setting are solved, and gathering contributions by leading international experts, the book’s content will impact the scientific, engineering, financial, economic, environmental, social, and commercial sectors.
Spectral Methods for Uncertainty Quantification
Spectral Methods for Uncertainty Quantification - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Spectral Methods for Uncertainty Quantification write by Olivier Le Maitre. This book was released on 2010-12-02. Spectral Methods for Uncertainty Quantification available in PDF, EPUB and Kindle. This book deals with the application of spectral methods to problems of uncertainty propagation and quanti?cation in model-based computations. It speci?cally focuses on computational and algorithmic features of these methods which are most useful in dealing with models based on partial differential equations, with special att- tion to models arising in simulations of ?uid ?ows. Implementations are illustrated through applications to elementary problems, as well as more elaborate examples selected from the authors’ interests in incompressible vortex-dominated ?ows and compressible ?ows at low Mach numbers. Spectral stochastic methods are probabilistic in nature, and are consequently rooted in the rich mathematical foundation associated with probability and measure spaces. Despite the authors’ fascination with this foundation, the discussion only - ludes to those theoretical aspects needed to set the stage for subsequent applications. The book is authored by practitioners, and is primarily intended for researchers or graduate students in computational mathematics, physics, or ?uid dynamics. The book assumes familiarity with elementary methods for the numerical solution of time-dependent, partial differential equations; prior experience with spectral me- ods is naturally helpful though not essential. Full appreciation of elaborate examples in computational ?uid dynamics (CFD) would require familiarity with key, and in some cases delicate, features of the associated numerical methods. Besides these shortcomings, our aim is to treat algorithmic and computational aspects of spectral stochastic methods with details suf?cient to address and reconstruct all but those highly elaborate examples.
Stochastic Systems
Stochastic Systems - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Stochastic Systems write by Mircea Grigoriu. This book was released on 2012-05-15. Stochastic Systems available in PDF, EPUB and Kindle. Uncertainty is an inherent feature of both properties of physical systems and the inputs to these systems that needs to be quantified for cost effective and reliable designs. The states of these systems satisfy equations with random entries, referred to as stochastic equations, so that they are random functions of time and/or space. The solution of stochastic equations poses notable technical difficulties that are frequently circumvented by heuristic assumptions at the expense of accuracy and rigor. The main objective of Stochastic Systems is to promoting the development of accurate and efficient methods for solving stochastic equations and to foster interactions between engineers, scientists, and mathematicians. To achieve these objectives Stochastic Systems presents: A clear and brief review of essential concepts on probability theory, random functions, stochastic calculus, Monte Carlo simulation, and functional analysis Probabilistic models for random variables and functions needed to formulate stochastic equations describing realistic problems in engineering and applied sciences Practical methods for quantifying the uncertain parameters in the definition of stochastic equations, solving approximately these equations, and assessing the accuracy of approximate solutions Stochastic Systems provides key information for researchers, graduate students, and engineers who are interested in the formulation and solution of stochastic problems encountered in a broad range of disciplines. Numerous examples are used to clarify and illustrate theoretical concepts and methods for solving stochastic equations. The extensive bibliography and index at the end of the book constitute an ideal resource for both theoreticians and practitioners.
