**Abstract** : This book results from a course developed by the author and reflects both his own and collaborative research regarding the development and implementation of uncertainty quantification (UQ) techniques for large-scale applications over the last two decades.
The objectives of this book are to present fundamental notions for the stochastic modeling of uncertainties and their quantification in computational models encountered in computational sciences and engineering. The text covers basic methods and novel advanced techniques to quantify uncertainties in large-scale engineering and science models. It focuses on aleatory and epistemic uncertainties, parametric uncertainties (associated with the parameters of computational models), and nonparametric uncertainties (induced by modeling errors). The text covers mainly the basic methods and novel advanced methodologies for constructing the stochastic modeling of uncertainties. To this effect, it presents only the fundamental mathematical tools of probability and statistics that are directly useful for uncertainty quantification, and overviews the main approaches for studying the propagation of uncertainties in computational models. Important methods are also presented for performing robust analysis of computational models with respect to uncertainties, robust updating of such computational models, robust optimization and design under uncertainties. It also carries out the calibration and identification of the stochastic model of uncertainties when experimental data is available. The methods are illustrated on advanced applications in computational engineering, such as in computational structural dynamics and vibroacoustics of complex mechanical systems, and in micromechanics and multiscale mechanics of heterogeneous materials.
This book is intended to be a graduate-type textbook for graduate students, engineers, doctoral students, postdocs, researchers, assistant professors, professors, etc. For learning a difficult interdisciplinary domain, such as the UQ domain, the fundamental difficulty is not the access to the knowledge but is related to the understanding of the knowledge organization, to the level of expertise that is required for solving a complex problem, and to the use of methods that are scientifically validated. Such an expertise is obtained by thinking and not by carrying out exercises. The aim of the author is to propose a graduate-type textbook that leads the reader to thorough his/her thinking and is not to train him /her with exercises.
The writing style has deliberately been chosen as short and direct, avoiding unnecessary mathematical details that may prevent access to a quick understanding of the concepts, ideas, and methods. Nevertheless, all the mathematics tools that are presented are absolutely correct and scientifically rigorous. All the required hypotheses for using them are given and explained. Any approximation that is introduced is commented and the limitations are specified. The useful references are given for a reader who is interested in finding the mathematical details for proving some mathematical results. A short paragraph is written at the beginning of each chapter summarizing the objectives of the chapter and explaining the interconnections between the chapters. The chapters have a large number of subsections which will allow readers to clearly find and learn about specific topics that are useful to fulfill the broader chapter objectives.
An objective of this book is to present constructive methods for solving advanced applications and not to re-explain in detail the classical statistical tools, which are already developed in excellent textbooks in which there are academic examples that allow for training the undergraduate students.
The main objective is effectively to not recopy one more time the classical statistical methods for solving academic problems in small dimension with a very small number of scalar random variables. The book presents classical and advanced mathematical tools from the probability theory and novel methodologies in computational statistics that are necessary for solving the large-scale computational models with uncertainty quantification in high dimension. The book presents a set of mathematical tools, their organization, and their interconnections, which allow for constructing efficient methodologies that are required for solving challenging large-scale problems that are encountered in computational engineering. The book proposes to the readers a clear and identified strategy for solving complex problems.
The book includes several topics that are not covered in published research monographs or texts on UQ. This includes random matrix theory for uncertainty quantification, a significant part of the theory on stochastic differential equations, the use of maximum entropy and polynomial chaos techniques to construct prior probability distributions, the identification of non-Gaussian tensor-valued random fields in high stochastic dimension by solving statistical inverse problem related to stochastic boundary value problem, and the robust analysis for design and optimization. Many of these topics appear in research papers, but this is the first time that all of these topics are presented in a unified manner. Another novel aspect of this book is the fact that it addresses uncertainty quantification for large-scale engineering applications, which differentiates the methods proposed in this book from classical approaches. The theory, the methodologies, their implementation, and their experimental validations, are illustrated by large-scale applications, including computational structural dynamics and vibroacoustics, solid mechanics of continuum media.