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2017 | Buch

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Uncertainty Quantification

An Accelerated Course with Advanced Applications in Computational Engineering

verfasst von: Christian Soize

Verlag: Springer International Publishing

Buchreihe : Interdisciplinary Applied Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book presents the fundamental notions and advanced mathematical tools in the stochastic modeling of uncertainties and their quantification for large-scale computational models in sciences and engineering. In particular, it focuses in parametric uncertainties, and non-parametric uncertainties with applications from the structural dynamics and vibroacoustics of complex mechanical systems, from micromechanics and multiscale mechanics of heterogeneous materials.
Resulting from a course developed by the author, the book begins with a description of the fundamental mathematical tools of probability and statistics that are directly useful for uncertainty quantification. It proceeds with a well carried out description of some basic and advanced methods for constructing stochastic models of uncertainties, paying particular attention to the problem of calibrating and identifying a stochastic model of uncertainty when experimental data is available.
This book is intended to be a graduate-level textbook for students as well as professionals interested in the theory, computation, and applications of risk and prediction in science and engineering fields.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Fundamental Notions in Stochastic Modeling of Uncertainties and Their Propagation in Computational Models

Abstract

This first chapter deals with a very short overview of fundamental notions, of concepts, and of vocabulary, concerning the aleatory uncertainties and the epistemic uncertainties, the sources of uncertainties and the variabilities (that are illustrated by showing experimental measurements for a real system), the role played by the model-parameter uncertainties and by the modeling errors in a nominal computational model, the major challenges for the computational models, and finally, concerning the fundamental methodologies.

Christian Soize

Chapter 2. Elements of Probability Theory

Abstract

This chapter deals with fundamental elements of the probability theory, which must be understood for understanding the other chapters of the course. It presents a short overview on the probability distributions, on the second-order vector-valued random variables, and on some important mathematical results concerning the convergence of sequences of random variables and the central limit theorem.

Christian Soize

Chapter 3. Markov Process and Stochastic Differential Equation

Abstract

The developments that are presented in this chapter are fundamental for understanding some important tools for UQ, in particular those presented in Chapter 4 , which are devoted to the Markov Chain Monte Carlo (MCMC) methods that are a class of algorithms for constructing realizations from a probability distribution.

Christian Soize

Chapter 4. MCMC Methods for Generating Realizations and for Estimating the Mathematical Expectation of Nonlinear Mappings of Random Vectors

Abstract

The Markov Chain Monte Carlo (MCMC) method constitutes a fundamental tool for UQ and in computational statistics, which allows for generating realizations of a random vector for which the pdf is given, and then for computing the mathematical expectation of nonlinear mappings of this random vector. The tved in Chapter 3

Christian Soize

Chapter 5. Fundamental Probabilistic Tools for Stochastic Modeling of Uncertainties

Abstract

This chapter is central and is devoted to the main tools that are necessary for constructing the stochastic modeling of uncertainties and for studying their propagations in computational models.

Christian Soize

Chapter 6. Brief Overview of Stochastic Solvers for the Propagation of Uncertainties

Abstract

The course is focused on the stochastic modeling of uncertainties and their identification by solving statistical inverse problems. The course is not devoted to the mathematical developments related to the stochastic solvers (there are many textbooks devoted to this subject). In this context, this chapter is limited to a list of principal approaches and to a brief description of the Galerkin method (spectral approach) and to the Monte Carlo method.

Christian Soize

Chapter 7. Fundamental Tools for Statistical Inverse Problems

Abstract

This chapter deals with the fundamental statistical tools for solving statistical inverse problems that allow for identifying the stochastic models of uncertainties through the computational models. This part of the mathematical statistics is very well developed and there are a huge number of textbooks with which one can easily get lost when trying to learn. We have voluntarily limited the presentation to the basic ideas of the statistical inversion theory. Consequently, the nonstationary inverse problems such as the Bayesian filtering leading, for instance, to the linear Kalman filters and to the extended Kalman filters will not be presented.

Christian Soize

Chapter 8. Uncertainty Quantification in Computational Structural Dynamics and Vibroacoustics

Abstract

This chapter deals with an ensemble of methodologies for stochastic modeling of both the model-parameter uncertainties and the model uncertainties induced by modeling errors, in computational structural dynamics and in computational vibroacoustics, for which basic illustrations and experimental validations are given. The developments use all the previous chapters and in particular, the random matrix theory (Chapter 5 ) for constructing the nonparametric probabilistic approach of uncertainties, the stochastic solvers (Chapter 6 ), and the tools for statistical inverse problems (Chapter 7 with Chapter 4 ) in order to perform the identification of the stochastic models.

Christian Soize

Chapter 9. Robust Analysis with Respect to the Uncertainties for Analysis, Updating, Optimization, and Design

Abstract

The objective of this chapter is to present some applications in different areas in order to show the importance to perform robust computation with respect to the uncertainties that exist in the computational models. The developments given in this chapter use all the tools and methods presented in the previous chapters and in particular, the random matrix theory (Chapter 5), the stochastic solvers (Chapters 4 and 6), the statistical inverse methods (Chapter 7), and the parametric probabilistic approach of model-parameter uncertainties, the nonparametric probabilistic approach of both the model-parameter uncertainties and the model uncertainties induced by the modeling errors, and the generalized probabilistic approaches of uncertainties, that have been presented in Chapter 8.

Christian Soize

Chapter 10. Random Fields and Uncertainty Quantification in Solid Mechanics of Continuum Media

Abstract

The statistical inverse problem for the experimental identification of a non-Gaussian matrix-valued random field that is the model parameter of a boundary value problem, using some partial and limited experimental data related to a model observation, is a very difficult and challenging problem. A complete advanced methodology is presented and is based on the use of all the developments presented in the previous chapters and in particular, the random matrix theory (Chapter 5 ), the stochastic solvers (Chapters 4 and 6 ), the statistical inverse methods (Chapter 7 ). However, we will start this chapter by presenting new mathematical results concerning the random fields and their polynomial chaos representations, which constitute the extension in infinite dimension of the tools presented in Sections 5.5 to 5.7 for the finite dimension, and which are necessary for solving the statistical inverse problems related to the non-Gaussian random fields.

Christian Soize

Backmatter

Titel: Uncertainty Quantification
verfasst von: Christian Soize
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-54339-0
Print ISBN: 978-3-319-54338-3
DOI: https://doi.org/10.1007/978-3-319-54339-0