Sullivan - Introduction to Uncertainty Quantification

Springer International Publishing Switzerland 2015

T. J. Sullivan 1

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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.

The Signal and the Noise: The Art of Science and Prediction

Nate Silver

This book is an introduction to the mathematics of Uncertainty Quantification (UQ), but what is UQ? It is, roughly put, the coming together of probability theory and statistical practice with the real world. These two anecdotes illustrate something of what is meant by this statement:

Thus, a typical UQ problem involves one or more mathematical models for a process of interest, subject to some uncertainty about the correct form of, or parameter values for, those models. Often, though not always, these uncertainties are treated probabilistically.

Perhaps as a result of its history, there are many perspectives on what UQ is, including at the extremes assertions like UQ is just a buzzword for statistics or UQ is just error analysis. These points of view are somewhat extremist, but they do contain a kernel of truth: very often, the probabilistic theory underlying UQ methods is actually quite simple, but is obscured by the details of the application. However, the complications that practical applications present are also part of the essence of UQ: it is all very well giving an accurate prediction for some insurance risk in terms of an elementary mathematical object such as an expected value, but how will you actually go about evaluating that expected value when it is an integral over a million-dimensional parameter space? Thus, it is important to appreciate both the underlying mathematics and the practicalities of implementation, and the presentation here leans towards the former while keeping the latter in mind.

Typical UQ problems of interest include certification, prediction, model and software verification and validation, parameter estimation, data assimilation, and inverse problems. At its very broadest,

UQ studies all sources of error and uncertainty, including the following: systematic and stochastic measurement error; ignorance; limitations of theoretical models; limitations of numerical representations of those models; limitations of the accuracy and reliability of computations, approximations, and algorithms; and human error. A more precise definition is UQ is the end-to-end study of the reliability of scientific inferences. (U.S. Department of Energy, , p. 135)

It is especially important to appreciate the end-to-end nature of UQ studies: one is interested in relationships between pieces of information , not the truth of those pieces of information/assumptions, bearing in mind that they are only approximations of reality. There is always going to be a risk of Garbage In, Garbage Out. UQ cannot tell you that your model is right or true, but only that, if you accept the validity of the model (to some quantified degree), then you must logically accept the validity of certain conclusions (to some quantified degree). In the authors view, this is the proper interpretation of philosophically sound but somewhat unhelpful assertions like Verification and validation of numerical models of natural systems is impossible and The primary value of models is heuristic (Oreskes et al., ). UQ can, however, tell you that two or more of your modelling assumptions are mutually contradictory, and hence that your model is wrong, and a complete UQ analysis will include a meta-analysis examining the sensitivity of the original analysis to perturbations of the governing assumptions.

A prototypical, if rather over-used, example for UQ is an elliptic PDE with uncertainty coefficients: