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2015 | Buch

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Introduction to Uncertainty Quantification

verfasst von: T.J. Sullivan

Verlag: Springer International Publishing

Buchreihe : Texts in Applied Mathematics

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

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Über dieses Buch

This text provides a framework in which the main objectives of the field of uncertainty quantification (UQ) are defined and an overview of the range of mathematical methods by which they can be achieved. Complete with exercises throughout, the book will equip readers with both theoretical understanding and practical experience of the key mathematical and algorithmic tools underlying the treatment of uncertainty in modern applied mathematics. Students and readers alike are encouraged to apply the mathematical methods discussed in this book to their own favorite problems to understand their strengths and weaknesses, also making the text suitable for a self-study.

Uncertainty quantification is a topic of increasing practical importance at the intersection of applied mathematics, statistics, computation and numerous application areas in science and engineering. This text is designed as an introduction to UQ for senior undergraduate and graduate students with a mathematical or statistical background and also for researchers from the mathematical sciences or from applications areas who are interested in the field.

T. J. Sullivan was Warwick Zeeman Lecturer at the Mathematics Institute of the University of Warwick, United Kingdom, from 2012 to 2015. Since 2015, he is Junior Professor of Applied Mathematics at the Free University of Berlin, Germany, with specialism in Uncertainty and Risk Quantification.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

We must think differently about our ideas — and how we test them. We must become more comfortable with probability and uncertainty. We must think more carefully about the assumptions and beliefs that we bring to a problem.

T. J. Sullivan

Chapter 2. Measure and Probability Theory

Abstract

Probability theory, grounded in Kolmogorov’s axioms and the general foundations of measure theory, is an essential tool in the quantitative mathematical treatment of uncertainty. Of course, probability is not the only framework for the discussion of uncertainty: there is also the paradigm of interval analysis, and intermediate paradigms such as Dempster–Shafer theory, as discussed in Section 2.8 and Chapter 5.

T. J. Sullivan

Chapter 3. Banach and Hilbert Spaces

Abstract

This chapter covers the necessary concepts from linear functional analysis on Hilbert and Banach spaces: in particular, we review here basic constructions such as orthogonality, direct sums and tensor products.

T. J. Sullivan

Chapter 4. Optimization Theory

Abstract

This chapter reviews the basic elements of optimization theory and practice, without going into the fine details of numerical implementation. Many UQ problems involve a notion of ‘best fit’, in the sense of minimizing some error function, and so it is helpful to establish some terminology for optimization problems. In particular, many of the optimization problems in this book will fall into the simple settings of linear programming and least squares (quadratic programming), with and without constraints.

T. J. Sullivan

Chapter 5. Measures of Information and Uncertainty

Abstract

This chapter briefly summarizes some basic numerical measures of uncertainty, from interval bounds to information-theoretic quantities such as (Shannon) information and entropy. This discussion then naturally leads to consideration of distances (and distance-like functions) between probability measures.

T. J. Sullivan

Chapter 6. Bayesian Inverse Problems

Abstract

This chapter provides a general introduction, at the high level, to the backward propagation of uncertainty/information in the solution of inverse problems, and specifically a Bayesian probabilistic perspective on such inverse problems. Under the umbrella of inverse problems, we consider parameter estimation and regression. One specific aim is to make clear the connection between regularization and the application of a Bayesian prior. The filtering methods of Chapter 7 fall under the general umbrella of Bayesian approaches to inverse problems, but have an additional emphasis on real-time computational expediency.

T. J. Sullivan

Chapter 7. Filtering and Data Assimilation

Abstract

It is not bigotry to be certain we are right; but it is bigotry to be unable to imagine how we might possibly have gone wrong.

T. J. Sullivan

Chapter 8. Orthogonal Polynomials and Applications

Abstract

Orthogonal polynomials are an important example of orthogonal decompositions of Hilbert spaces. They are also of great practical importance: they play a central role in numerical integration using quadrature rules (Chapter 9) and approximation theory; in the context of UQ, they are also a foundational tool in polynomial chaos expansions (Chapter 11).

T. J. Sullivan

Chapter 9. Numerical Integration

Abstract

The topic of this chapter is the numerical evaluation of definite integrals. Many UQ methods have at their core simple probabilistic constructions such as expected values, and expectations are nothing more than Lebesgue integrals. However, while it is mathematically enough to know that the Lebesgue integral of some function exists, practical applications demand the evaluation of such an integral — or, rather, its approximate evaluation. This usually means evaluating the integrand at some finite collection of sample points. It is important to bear in mind, though, that sampling is not free (each sample of the integration domain, or function evaluation, may correspond to a multi-million-dollar experiment) and that practical applications often involve many dependent and independent variables, i.e. high-dimensional domains of integration. Hence, the accurate numerical integration of integrands over high-dimensional spaces using few samples is something of a ‘Holy Grail’ in this area.

T. J. Sullivan

Chapter 10. Sensitivity Analysis and Model Reduction

Abstract

Le doute n’est pas un état bien agréable, mais l’assurance est un état ridicule.

T. J. Sullivan

Chapter 11. Spectral Expansions

Abstract

This chapter and its sequels consider several spectral methods for uncertainty quantification. At their core, these are orthogonal decomposition methods in which a random variable stochastic process (usually the solution of interest) over a probability space $(\varTheta,\mathcal{F},\mu )$ is expanded with respect to an appropriate orthogonal basis of $L^{2}(\varTheta,\mu; \mathbb{R})$.

T. J. Sullivan

Chapter 12. Stochastic Galerkin Methods

Abstract

Chapter 11 considered spectral expansions of square-integrable random variables, random vectors and random fields of the form

$$\displaystyle{U =\sum _{k\in \mathbb{N}_{0}}u_{k}\varPsi _{k},}$$

where $U \in L^{2}(\varTheta,\mu;\mathcal{U})$, $\mathcal{U}$ is a Hilbert space in which the corresponding deterministic variables/vectors/fields lie, and $\{\varPsi _{k}\mid k \in \mathbb{N}_{0}\}$ is some orthogonal basis for $L^{2}(\varTheta,\mu; \mathbb{R})$.

T. J. Sullivan

Chapter 13. Non-Intrusive Methods

Abstract

Chapter 12 considers a spectral approach to UQ, namely Galerkin expansion, that is mathematically very attractive in that it is a natural extension of the Galerkin methods that are commonly used for deterministic PDEs and (up to a constant) minimizes the stochastic residual, but has the severe disadvantage that the stochastic modes of the solution are coupled together by a large system such as (12.15).

T. J. Sullivan

Chapter 14. Distributional Uncertainty

Abstract

In the previous chapters, it has been assumed that an exact model is available for the probabilistic components of a system, i.e. that all probability distributions involved are known and can be sampled. In practice, however, such assumptions about probability distributions are always wrong to some degree: the distributions used in practice may only be simple approximations of more complicated real ones, or there may be significant uncertainty about what the real distributions actually are. The same is true of uncertainty about the correct form of the forward physical model. In the Bayesian paradigm, similar issues arise if the available information is insufficient for us to specify (or ‘elicit’) a unique prior and likelihood model.

T. J. Sullivan

Backmatter

Titel: Introduction to Uncertainty Quantification
verfasst von: T.J. Sullivan
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-23395-6
Print ISBN: 978-3-319-23394-9
DOI: https://doi.org/10.1007/978-3-319-23395-6