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Uncertainty Quantification (UQ) is a relatively new research area which describes the methods and approaches used to supply quantitative descriptions of the effects of uncertainty, variability and errors in simulation problems and models. It is rapidly becoming a field of increasing importance, with many real-world applications within statistics, mathematics, probability and engineering, but also within the natural sciences.
Literature on the topic has up until now been largely based on polynomial chaos, which raises difficulties when considering different types of approximation and does not lead to a unified presentation of the methods. Moreover, this description does not consider either deterministic problems or infinite dimensional ones.
This book gives a unified, practical and comprehensive presentation of the main techniques used for the characterization of the effect of uncertainty on numerical models and on their exploitation in numerical problems. In particular, applications to linear and nonlinear systems of equations, differential equations, optimization and reliability are presented. Applications of stochastic methods to deal with deterministic numerical problems are also discussed. Matlab® illustrates the implementation of these methods and makes the book suitable as a textbook and for self-study.
- Discusses the main ideas of Stochastic Modeling and Uncertainty Quantification using Functional Analysis
- Details listings of Matlab® programs implementing the main methods which complete the methodological presentation by a practical implementation
- Construct your own implementations from provided worked examples
Researchers, graduate students at universities in mathematics, physics, engineering and natural sciences fields
- 1. Elements of Probability Theory and Stochastic Processes
- 1.1 Notation
- 1.2 Numerical characteristics of finite populations
- 1.3 Matlab implementation
- 1.4 Couples of numerical characteristics
- 1.5 Matlab implementation
- 1.6 Hilbertian properties of the numerical characteristics
- 1.7 Measure and probability
- 1.8 Construction of measures
- 1.9 Measures, probability and integrals in infinite dimensional spaces
- 1.10 Random variables
- 1.11 Hilbertian properties of random variables
- 1.12 Sequences of random variables
- 1.13 Some usual distributions
- 1.14 Samples of random variables
- 1.15 Gaussian samples
- 1.16 Stochastic processes
- 1.17 Hilbertian structure
- 1.18 Wiener process
- 1.19 Ito integrals
- 1.20 Ito Calculus
- 2. Maximum Entropy and Information
- 2.1 Construction of a stochastic model
- 2.2 The principle of maximum entropy
- 2.3 Generating samples of random variables, random vectors and stochastic processes
- 2.4 Karhunen–Loève expansions and numerical generation of variates from stochastic processes
- 3. Representation of Random Variables
- 3.1 Approximations based on Hilbertian properties
- 3.2 Approximations based on statistical properties (moment matching method)
- 3.3 Interpolation-based approximations (collocation)
- 4. Linear Algebraic Equations Under Uncertainty
- 4.1 Representation of the solution of uncertain linear systems
- 4.2 Representation of eigenvalues and eigenvectors of uncertain matrices
- 4.3 Stochastic methods for deterministic linear systems
- 5. Nonlinear Algebraic Equations Involving Random Parameters
- 5.1 Nonlinear systems of algebraic equations
- 5.2 Numerical solution of noisy deterministic systems of nonlinear equations
- 6. Differential Equations Under Uncertainty
- 6.1 The case of linear differential equations
- 6.2 The case of nonlinear differential equations
- 6.3 The case of partial differential equations
- 6.4 Reduction of Hamiltonian systems
- 6.5 Local solution of deterministic differential equations by stochastic simulation
- 6.6 Statistics of dynamical systems
- 7. Optimization Under Uncertainty
- 7.1 Representation of the solutions in unconstrained optimization
- 7.2 Stochastic methods in deterministic continuous optimization
- 7.3 Population-based methods
- 7.4 Determination of starting points
- 8. Reliability-Based Optimization
- 8.1 The model situation
- 8.2 Reliability index
- 8.3 FORM
- 8.4 The bi-level or double-loop method
- 8.5 One-level or single-loop approach
- 8.6 Safety factors
- No. of pages:
- © ISTE Press - Elsevier 2015
- 25th March 2015
- ISTE Press - Elsevier
- Hardcover ISBN:
- eBook ISBN:
Eduardo Souza De Cursi is Professor at the National Institute for Applied Sciences in Rouen, France, where he is also Dean of International Affairs and Director of the Laboratory for the Optimization and Reliability in Structural Mechanics.
Professor, National Institute for Applied Sciences, Rouen, France
Professor, PUC-Rio, Rio de Janeiro, Brazil
"...a deepening to the mathematics of uncertainty quantification and stochastic modeling through the tools of functional analysis...the perspective on UQ that runs through this book is firmly grounded in probability theory and Hilbert spaces; the elements of linear functional analysis and measure/probability theory are provided." --Zentralblatt MATH"...an excellent introduction for newcomers and a practical reference for established practitioners…Practical techniques are illustrated by well-chosen and thoroughly worked-out examples." --MAA Reviews
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