One century-old first-year calculus problem posed by a teacher of that era (and many teachers thereafter) is this: Given a square of side s, what is the approximate change in its area if s is changed by the small amount ds? As the story goes, an overly eager student who was interested in gaining the proverbial extra credit proudly submitted the exact answer, namely, 2s*ds + (ds)2, by subtracting one exact area from the other. Inasmuch as the story is one from the “good old days,” the teacher gave that answer a zero, that is, no partial credit. The required method was, of course, to compute the differential of s2 so as to yield 2s*ds without the second-order term in ds.
An intermediate stage in my buildup to the erudite and authoritative book under review is Yardley Beers’ 67-page book [1] (perhaps pamphlet) on which I cut my proverbial error-analysis, also known as uncertainty-quantification, teeth. The more elaborate rules of thumb therein, substantiated by plausible reasoning if not proofs, have stood the test of time in many quantifications of uncertainty and in sensitivity analyses that I’ve done up to the present. If the above example of the square is a back-of-the-envelope computation, then Beers’ examples of multivariate error quantification are essentially of the same nature, perhaps requiring larger envelopes but (still) relatively simple mathematical expressions and equations.
Fast-forwarding to the present, the textbook under review “is intended for advanced undergraduates, graduate students, and researchers in mathematics, statistics, operations research, computer science, biology, science [sic], and engineering.” It is an excellent book that is based on today’s mature theories of probability and statistics--and seems to me an amazing, single giant leap from the above intuition-intensive, relatively low-complexity paradigms--which, I hasten to add, still have their place.
The book treats uncertainty quantification based on both the frequentist and the Bayesian views of probability/statistics, and has the further specific purpose (writ large) “of quantifying input and response uncertainties in a manner that facilitates predictions with quantified and reduced uncertainties.” (My two simple examples represent raw quantification, without prediction.) The author’s “present novelty,” as he puts it in the preface, “lies in the synthesis of probability, statistics, model development, mathematical and numerical analysis, large-scale simulations, experiments, and disciplinary sciences.” The 15-chapter content of this book fully substantiates the term “synthesis,” and inspires the term “amazing” that I use above. There is far more than novelty here.
As I am in the camp of those who prefer the term “applications of mathematics” over “applied mathematics,” I would judge this book as an outstanding example of a spectrum of mathematics and statistics perfectly applied to their purpose, which includes analysis of research as recent as the last ten years. Its mathematical ingredients include differential and integral equations, linear algebra, numerical analysis, some operator theory, and of course large “use” of probability and statistics. Although the 15 chapters are replete with apposite equations and mathematical expressions, there is very little (pseudo)code. However, the author’s sophistication in the use of computers is evidenced throughout the book, “classical” as the equations might be. Particularly impressive is his reference to the burgeoning field of parallel computer architectures that include field-programmable gate arrays (FPGAs) as boosters of computational efficiency. The existence and nature of modern computers is understood in this work.
The book’s 15 chapters cover the spectrum from uncertainty through models to sensitivity analysis, and include some example applications.
The single appendix, “Concepts from Functional Analysis,” is a superb, seven-page summary of Hilbert space and operators as these are used within the text. It serves as an unobtrusive facilitator of the rest of the book, and is appropriately closer to Friedman’s book [2] than it is to [3].
I am generally quite compulsive about reading a book linearly, that is, from the first page to the last. I did, however, succumb to the temptation of first reading chapter 4, on probability, random processes, and statistics. This 40-page chapter alone is worth the price of the book, and is a crystal-clear review if not review course. The explication of the distinction between random and stochastic differential equations is particularly enlightening, at least to me. I dare say that other readers will have similar experiences with this or other chapters in this uniformly well-written and well-organized book--which has my highest recommendation.
