This book is designed as a broad introduction to the mathematics of Un- certainty Quantification (UQ) at the fourth year (senior) undergraduate or beginning postgraduate level. It is aimed primarily at readers from a math- ematical or statistical (rather than, say, engineering) background. The main mathematical prerequisite is familiarity with the language of linear functional analysis and measure / probability theory, and some familiarity with basic optimization theory. Chapters 2–5 of the text provide a review of this mate- rial, generally without detailed proof. The aim of this book has been to give a survey of the main objectives in the field of UQ and a few of the mathematical methods by which they can be achieved. However, this book is no exception to the old saying that books are never completed, only abandoned. There are many more UQ problems and solution methods in the world than those covered here. For any grievous omissions, I ask for your indulgence, and would be happy to receive sugges- tions for improvements. With the exception of the preliminary material on measure theory and functional analysis, this book should serve as a basis for a course comprising 30–45 hours’ worth of lectures, depending upon the instructor’s choices in terms of selection of topics and depth of treatment. The examples and exercises in this book aim to be simple but informative about individual components of UQ studies: practical applications almost always require some ad hoc combination of multiple techniques (e.g., Gaus- sian process regression plus quadrature plus reduced-order modelling). Such compound examples have been omitted in the interests of keeping the pre- sentation of the mathematical ideas clean, and in order to focus on examples and exercises that will be more useful to instructors and students. Each chapter concludes with a bibliography, the aim of which is threefold: to give sources for results discussed but not proved in the text; to give some historical overview and context; and, most importantly, to give students a jumping-off point for further reading and research. This has led to a large bibliography, but hopefully a more useful text for budding researchers. I would like to thank Achi Dosanjh at Springer for her stewardship of this project, and the anonymous reviewers for their thoughtful comments, which prompted many improvements to the manuscript. Frominitial conception to nearly finished product, this book has benefitted from interactions with many people: they have given support and encourage- ment, offered stimulating perspectives on the text and the field of UQ, and pointed out the inevitable typographical mistakes. In particular, I would like to thank Paul Constantine, Zach Dean, Charlie Elliott, Zydrunas Gimbutas, Calvin Khor, Ilja Klebanov, Han Cheng Lie, Milena Kremakova, David Mc- Cormick, Damon McDougall, Mike McKerns, Akil Narayan, Michael Ortiz, Houman Owhadi, Adwaye Rambojun, Asbjørn Nilsen Riseth, Clint Scovel, Colin Sparrow, Andrew Stuart, Florian Theil, Joy Tolia, Florian Wechsung, Thomas Whitaker, and Aim´ ee Williams. Finally, since the students on the 2013–14 iteration of the University of Warwick mathematics module MA4K0 Introduction to Uncertainty Quantifi- cation were curious and brave enough to be the initial ‘guinea pigs’ for this material, they deserve a special note of thanks.
Sullivan, T. J. (2015). Introduction to Uncertainty. Springer.