Tim Sullivan

Errata for Introduction to Uncertainty Quantification

This page lists corrections and clarifications to the text of Introduction to Uncertainty Quantification, published in 2015 by Springer as volume 63 of the Texts in Applied Mathematics series. Many thanks to all those who have pointed out these mistakes; please get in touch if you spot more.

  1. Introduction
  2. Measure and Probability Theory
  3. Banach and Hilbert Spaces
  4. Optimization Theory
  5. Measures of Information and Uncertainty
  6. Bayesian Inverse Problems
  7. Filtering and Data Assimilation
  8. Orthogonal Polynomials and Applications
  9. Numerical Integration
  10. Sensitivity Analysis and Model Reduction
  11. Spectral Expansions
  12. Stochastic Galerkin Methods
  13. Non-Intrusive Methods
  14. Distributional Uncertainty

Chapter 1: Introduction

  • p.3: In the second display, half-way down the page, the velocity field given by Darcy's law should be \( v = - \kappa \nabla u\), i.e. the minus sign is missing.
  • p.6: In the second paragraph, “August” should be “august”.

Chapter 2: Measure and Probability Theory

  • p.14: Just before the statement of Theorem 2.10, a closing parenthesis “)” is missing.

Chapter 3: Banach and Hilbert Spaces

  • p.37: Example 3.3(c) should begin “The analogous inner product on the space \( \mathbb{K}^{m \times n} \) of...”

Chapter 4: Optimization Theory

  • p.65: In Theorem 4.17 (Jensen's inequality), it is implicit that the set \(K \subseteq \mathcal{X}\) is a convex set, otherwise it makes no sense to claim that \(f \colon K \to \mathbb{R} \cup \{ \pm \infty \}\) is a convex function — and, worse, \(\mathbb{E}_{\mu}[X]\) might not lie in \(K\) even though all of \(X\)'s values lie in \(K\). Incidentally, the definition of \(\mathbb{E}_{\mu}[X]\) given here — that \(\langle \ell \vert \mathbb{E}_{\mu}[X] \rangle = \mathbb{E}_{\mu}[\langle \ell \vert X \rangle]\) for all \(\ell \in \mathcal{X}'\) — corresponds to the weak/Pettis integral.

Chapter 5: Measures of Information and Uncertainty

  • pp.81 and 88: Definition 5.4 defines the Kullback–Leibler divergence from \(\mu\) to \(\nu\) for \(\sigma\)-finite measures \(\mu\) and \(\nu\), but Exercise 5.3 only checks non-negativity for the case that \(\mu\) and \(\nu\) are probability measures. Naturally, the proof is much the same for the \(\sigma\)-finite case.
  • p.85: Proposition 5.12 should have the 2 inside the square root.
  • p.89: As in Proposition 5.12, Exercise 5.7 should have the 2 inside the square root.

Chapter 6: Bayesian Inverse Problems

  • p.95: The second displayed equation is missing an adjoint/transpose on the second appearance of \(K\). It should read \(\mathbb{E}[(\hat{u} - u) \otimes (\hat{u} - u)] = K \mathbb{E}[\eta \otimes \eta] K^{\ast} = (A^{\ast} Q^{-1} A)^{-1}\).
  • p.101: In the sixth line, in the definition of the measure \(\nu\), the “\(\mathbb{Q}_{0}\)” should be “\(\mathbb{Q}\)”.

Chapter 7: Filtering and Data Assimilation

  • p.115: In the fifth bullet point, the observational noise vector \(\eta_{k}\) should hace covariance \(R_{k}\) not \(Q_{k}\).
  • p.116: At the bottom of the page, the argmin should be over \(z_{k} \in \mathcal{X}^{k + 1}\), not over \(z_{k} \in \mathcal{X}\).
  • p.117: In equation (7.6), recall that \(m_{0}\) is the mean of the random initial condition of the system at time \(t_{0}\), as defined two pages previously but not used until now.
  • p.119: In the second line of the variational derivation of the prediction step, “\(\hat{x}_{m|k-1}\)” should be “\(\hat{x}_{k|k-1}\)”. Two lines later, the reference to a “\(k\)-tuple of states” should refer to a “\((k + 1)\)-tuple of states”.
  • p.122: Just before the paragraph on the Kálmán gain, the reference to equation (7.13) should be a reference to equation (7.12).

Chapter 8: Orthogonal Polynomials and Applications

  • p.133: In the second paragraph, “taken and as the primary definition” should be “taken as the primary definition”.
  • p.135: The integral at the top of the page expressing orthogonality for the Jacobi polynomials is missing the weight function \((1 - x)^{\alpha} (1 + x)^{\beta}\) before the \(\mathrm{d} x\). Similarly, the \((1 - x)^{\beta}\) in the text just before the integral should be \((1 + x)^{\beta}\). The appearances of the weight function in Table 8.2 on p.162 are correct.
  • p.138: In the proof of Lemma 8.4, delete the words “this is” after “By Sylvester's criterion,”.
  • p.141: In the definition of \(\beta_{0}\), some readers might prefer the explicit statement of the integrand, i.e. \(\beta_{0} = \int_{\mathbb{R}} 1 \, \mathrm{d} \mu\).
  • p.152: In the middle of the page, \(H^{k}(\mathcal{X}, \mu)\) should be \(H^{k}(I, \mu)\).
  • p.162: The normalisation constant for the Chebyshev polynomials of the first kind is incorrect. It should be \(\pi\) for \(n = 0\) and \(\pi / 2\) for \(n > 0\), i.e. \(\| q_{n} \|_{L^{2}(\mu)}^{2} = (\pi / 2) ( 1 + \delta_{0 n} )\).

Chapter 9: Numerical Integration

  • p.168: At the top of the page, after the displayed equation, “\(h = \tfrac{1}{n}\)” should be “\(h = \tfrac{b - a}{n}\)”.
  • p.176: Just before the discussion of sparse quadrature formulae, “and ‘using’ derivative” should be “and ‘using’ one derivative”.
  • p.177: In the multi-line displayed equation, the subscript \(\ell\) in \(Q_{\ell - i + 1}^{(1)}\) is \(\ell = 2\).
  • pp.179–181: The discussion of the variance-based error bound for the Monte Carlo estimator is, of course, predicated on the assumption that \(f(X)\) has finite variance.
  • p.180: In the caption of Figure 9.2, “\(\mathbb{E}[(a + X^{(1)})^{-1}]\)” should be “\(\mathbb{E}[(a - X^{(1)})^{-1}]\)”.
  • p.185: At the beginning of the second paragraph on Multi-Level Monte Carlo, “have at our disposal hierarchy” should be “have at our disposal a hierarchy”.
  • p.188: The definition of the Hardy–Krause variation of \(f \colon [0, 1]^{d} \to \mathbb{R}\) should make no mention of \(s\); it is simply the sum of all the Vitali variations \(V^{\mathrm{Vit}}(f|_{F})\) where \(F\) runs over all faces of \([0, 1]^{d}\), with dimension between \(1\) and \(d\) inclusive.
  • p.190: In Theorem 9.23, in the second displayed equation, \(x_{N}\) should be \(x_{n}\).

Chapter 10: Sensitivity Analysis and Model Reduction

  • p.200: The last displayed equation in the proof should end with a full stop / period.
  • p.205: After equation (10.5), the next sentence should read “Equation (10.5) can be re-written as \(\frac{\mathrm{d} q}{\mathrm{d} \theta} (\bar{u}, \bar{\theta}) = \lambda \frac{\partial F}{\partial \theta} (\bar{u}, \bar{\theta}) + \frac{\partial q}{\partial \theta} (\bar{u}, \bar{\theta})\)” instead of being a reference to and rearrangement of equation (10.4).
  • p.211: In the first line of the statement of Theorem 10.15, “\(i \subseteq \mathcal{N}\)” should read “\(I \subseteq \mathcal{N}\)”. Later in the statement of the same theorem, in equation (10.14), the sum over \(I \subseteq \mathcal{D}\) should also be a sum over \(I \subseteq \mathcal{N}\).

Chapter 11: Spectral Expansions

  • p.233: Just before equation (11.4), there should be a comma between “\(U\)” and “defined”.
  • pp.241–242: At the bottom of p.241, after taking the \(L^{2}(\nu)\) inner product with \(\Phi_{\ell}\), the right-hand side of the equation should be \(v_{\ell} \langle \Phi_{\ell}^{2} \rangle_{\nu}\) instead of \(v_{\ell} \langle \Psi_{\ell}^{2} \rangle_{\nu}\). This mistake is carried over the page: the denominator is \(\langle \Phi_{\ell}^{2} \rangle_{\nu}\) not \(\langle \Psi_{\ell}^{2} \rangle_{\nu}\). The denominator in the sum for \(u_{\ell}\) is correct as is, i.e. \(\langle \Psi_{\ell}^{2} \rangle_{\mu}\).

Chapter 12: Stochastic Galerkin Methods

  • p.256: At the top of the page, “multiplication can fail to commutative” should be “multiplication can fail to be commutative”.
  • p.257: In the third line of Section 12.2, “the approach is as simple is multiplying” should be “the approach is as simple as multiplying”.
  • p.263: At the top of the page, “uniqueness of solutions problems like” should be “uniqueness of solutions to problems like”.
  • p.267: On the third line, the Galerkin solution should be denoted \(u = u^{(M)}\), not \(u = u_{\Gamma}\). Also, in the second displayed equation, there is an extra right angle bracket just before the word “for”.
  • p.269: In the final paragraph, which begins the discussion of stochastic Galerkin projection, it would have been clearer to say explicitly that \(\Psi_{1}, \dots, \Psi_{K}\) are the chosen polynomial chaos basis (or other orthogonal basis) of \(\mathcal{S}_{K}\).

Chapter 13: Non-Intrusive Methods

  • p.278: In the footnote at the bottom of the page, the sum should read “\(U(t, x; \theta) = \sum_{k \in \mathbb{N}_{0}} u_{k}(t, x) \Psi_{k}(\theta)\)” instead of “\(U(t, x; \theta) = \sum_{k \in \mathbb{N}_{0}} (t, x) \Psi_{k}(\theta)\)”.
  • p.281: In Remark 13.3(a), “the approximation the stochastic modes” should read “the approximation of the stochastic modes”.
  • p.286: In the middle of the page, “has with the undesirable property” should read “has the undesirable property”.

Chapter 14: Distributional Uncertainty

  • p.299: On the fifth line of Section 14.3, “particular” should be “particularly”.
  • p.308: In the statement of Theorem 14.19, on the line after the definition of \(\mathcal{A}\), the domain of \(\varphi_{k, j}\) should be \(\mathcal{X}_{k}\) and not \(\mathcal{X}\).
  • p.315: “not just no impact” should be “not just little impact”.
  • p.315: There is an extra closing parenthesis at the end of “(and hence pass to a smaller feasible set \(\mathcal{A}' \subsetneq \mathcal{A}\))”.