TY - JOUR A1 - Cockayne, Jon A1 - Oates, Chris A1 - Sullivan, T. J. A1 - Girolami, Mark T1 - Bayesian Probabilistic Numerical Methods JF - SIAM Review N2 - Over forty years ago average-case error was proposed in the applied mathematics literature as an alternative criterion with which to assess numerical methods. In contrast to worst-case error, this criterion relies on the construction of a probability measure over candidate numerical tasks, and numerical methods are assessed based on their average performance over those tasks with respect to the measure. This paper goes further and establishes Bayesian probabilistic numerical methods as solutions to certain inverse problems based upon the numerical task within the Bayesian framework. This allows us to establish general conditions under which Bayesian probabilistic numerical methods are well defined, encompassing both the non-linear and non-Gaussian context. For general computation, a numerical approximation scheme is proposed and its asymptotic convergence established. The theoretical development is extended to pipelines of computation, wherein probabilistic numerical methods are composed to solve more challenging numerical tasks. The contribution highlights an important research frontier at the interface of numerical analysis and uncertainty quantification, and a challenging industrial application is presented. Y1 - 2019 U6 - https://doi.org/10.1137/17M1139357 VL - 61 IS - 4 SP - 756 EP - 789 ER - TY - GEN A1 - Oates, Chris A1 - Cockayne, Jon A1 - Prangle, Dennis A1 - Sullivan, T. J. A1 - Girolami, Mark ED - Hickernell, F. J. ED - Kritzer, P. T1 - Optimality criteria for probabilistic numerical methods T2 - Multivariate Algorithms and Information-Based Complexity N2 - It is well understood that Bayesian decision theory and average case analysis are essentially identical. However, if one is interested in performing uncertainty quantification for a numerical task, it can be argued that the decision-theoretic framework is neither appropriate nor sufficient. To this end, we consider an alternative optimality criterion from Bayesian experimental design and study its implied optimal information in the numerical context. This information is demonstrated to differ, in general, from the information that would be used in an average-case-optimal numerical method. The explicit connection to Bayesian experimental design suggests several distinct regimes in which optimal probabilistic numerical methods can be developed. Y1 - 2020 U6 - https://doi.org/10.1515/9783110635461-005 VL - 27 SP - 65 EP - 88 PB - De Gruyter ER - TY - JOUR A1 - Kersting, Hans A1 - Sullivan, T. J. A1 - Hennig, Philipp T1 - Convergence rates of Gaussian ODE filters JF - Statistics and Computing Y1 - 2020 U6 - https://doi.org/10.1007/s11222-020-09972-4 VL - 30 SP - 1791 EP - 1816 PB - Springer CY - Statistics and Computing ER - TY - JOUR A1 - Klebanov, Ilja A1 - Schuster, Ingmar A1 - Sullivan, T. J. T1 - A rigorous theory of conditional mean embeddings JF - SIAM Journal on Mathematics of Data Science Y1 - 2020 U6 - https://doi.org/10.1137/19M1305069 VL - 2 IS - 3 SP - 583 EP - 606 ER - TY - JOUR A1 - Nava-Yazdani, Esfandiar A1 - Hege, Hans-Christian A1 - Sullivan, T. J. A1 - von Tycowicz, Christoph T1 - Geodesic Analysis in Kendall's Shape Space with Epidemiological Applications JF - Journal of Mathematical Imaging and Vision N2 - We analytically determine Jacobi fields and parallel transports and compute geodesic regression in Kendall’s shape space. Using the derived expressions, we can fully leverage the geometry via Riemannian optimization and thereby reduce the computational expense by several orders of magnitude over common, nonlinear constrained approaches. The methodology is demonstrated by performing a longitudinal statistical analysis of epidemiological shape data. As an example application we have chosen 3D shapes of knee bones, reconstructed from image data of the Osteoarthritis Initiative (OAI). Comparing subject groups with incident and developing osteoarthritis versus normal controls, we find clear differences in the temporal development of femur shapes. This paves the way for early prediction of incident knee osteoarthritis, using geometry data alone. Y1 - 2020 U6 - https://doi.org/10.1007/s10851-020-00945-w VL - 62 IS - 4 SP - 549 EP - 559 ER - TY - JOUR A1 - Bonnet, Luc A1 - Akian, Jean-Luc A1 - Savin, Éric A1 - Sullivan, T. J. T1 - Adaptive reconstruction of imperfectly-observed monotone functions, with applications to uncertainty quantification JF - Algorithms N2 - Motivated by the desire to numerically calculate rigorous upper and lower bounds on deviation probabilities over large classes of probability distributions, we present an adaptive algorithm for the reconstruction of increasing real-valued functions. While this problem is similar to the classical statistical problem of isotonic regression, the optimisation setting alters several characteristics of the problem and opens natural algorithmic possibilities. We present our algorithm, establish sufficient conditions for convergence of the reconstruction to the ground truth, and apply the method to synthetic test cases and a real-world example of uncertainty quantification for aerodynamic design. Y1 - 2020 U6 - https://doi.org/10.3390/a13080196 VL - 13 IS - 8 SP - 196 ER - TY - JOUR A1 - Lie, Han Cheng A1 - Sullivan, T. J. A1 - Teckentrup, Aretha T1 - Error bounds for some approximate posterior measures in Bayesian inference JF - Numerical Mathematics and Advanced Applications ENUMATH 2019 Y1 - 2020 U6 - https://doi.org/10.1007/978-3-030-55874-1_26 SP - 275 EP - 283 PB - Springer ER - TY - JOUR A1 - Klebanov, Ilja A1 - Sprungk, Björn A1 - Sullivan, T. J. T1 - The linear conditional expectation in Hilbert space JF - Bernoulli Y1 - 2021 U6 - https://doi.org/10.3150/20-BEJ1308 VL - 27 IS - 4 SP - 2299 EP - 2299 ER - TY - JOUR A1 - Schäfer, Florian A1 - Sullivan, T. J. A1 - Owhadi, Houman T1 - Compression, inversion, and approximate PCA of dense kernel matrices at near-linear computational complexity JF - Multiscale Modeling and Simulation Y1 - 2021 U6 - https://doi.org/10.1137/19M129526X VL - 19 IS - 2 SP - 688 EP - 730 ER - TY - JOUR A1 - Wang, Junyang A1 - Cockayne, Jon A1 - Chkrebtii, Oksana A1 - Sullivan, T. J. A1 - Oates, Chris T1 - Bayesian numerical methods for nonlinear partial differential equations JF - Statistics and Computing Y1 - 2021 U6 - https://doi.org/10.1007/s11222-021-10030-w VL - 31 IS - 5 ER -