Dokument-ID Dokumenttyp Verfasser/Autoren Herausgeber Haupttitel Abstract Auflage Verlagsort Verlag Erscheinungsjahr Seitenzahl Schriftenreihe Titel Schriftenreihe Bandzahl ISBN Quelle der Hochschulschrift Konferenzname Quelle:Titel Quelle:Jahrgang Quelle:Heftnummer Quelle:Erste Seite Quelle:Letzte Seite URN DOI Abteilungen OPUS4-5808 Buch (Monographie) Sullivan, T. J. Introduction to Uncertainty Quantification Springer 2015 63 978-3-319-23394-9 10.1007/978-3-319-23395-6 Numerical Mathematics OPUS4-5807 Wissenschaftlicher Artikel Owhadi, Houman; Scovel, Clint; Sullivan, T. J. On the Brittleness of Bayesian Inference With the advent of high-performance computing, Bayesian methods are becoming increasingly popular tools for the quantification of uncertainty throughout science and industry. Since these methods can impact the making of sometimes critical decisions in increasingly complicated contexts, the sensitivity of their posterior conclusions with respect to the underlying models and prior beliefs is a pressing question to which there currently exist positive and negative answers. We report new results suggesting that, although Bayesian methods are robust when the number of possible outcomes is finite or when only a finite number of marginals of the data-generating distribution are unknown, they could be generically brittle when applied to continuous systems (and their discretizations) with finite information on the data-generating distribution. If closeness is defined in terms of the total variation (TV) metric or the matching of a finite system of generalized moments, then (1) two practitioners who use arbitrarily close models and observe the same (possibly arbitrarily large amount of) data may reach opposite conclusions; and (2) any given prior and model can be slightly perturbed to achieve any desired posterior conclusion. The mechanism causing brittleness/robustness suggests that learning and robustness are antagonistic requirements, which raises the possibility of a missing stability condition when using Bayesian inference in a continuous world under finite information. 2015 16 SIAM Review 57 4 566 582 10.1137/130938633 Numerical Mathematics OPUS4-6014 Wissenschaftlicher Artikel Lie, Han Cheng; Sullivan, T. J. Cameron--Martin theorems for sequences of Cauchy-distributed random variables Given a sequence of Cauchy-distributed random variables defined by a sequence of location parameters and a sequence of scale parameters, we consider another sequence of random variables that is obtained by perturbing the location or scale parameter sequences. Using a result of Kakutani on equivalence of infinite product measures, we provide sufficient conditions for the equivalence of laws of the two sequences. 2016 arXiv 1608.03784 Numerical Mathematics OPUS4-6023 misc Lie, Han Cheng; Sullivan, T. J. Cameron--Martin theorems for sequences of Cauchy-distributed random variables Given a sequence of Cauchy-distributed random variables defined by a sequence of location parameters and a sequence of scale parameters, we consider another sequence of random variables that is obtained by perturbing the location or scale parameter sequences. Using a result of Kakutani on equivalence of infinite product measures, we provide sufficient conditions for the equivalence of laws of the two sequences. 2016 urn:nbn:de:0297-zib-60230 Numerical Mathematics OPUS4-6632 misc Lie, Han Cheng; Sullivan, T. J.; Teckentrup, Aretha Random forward models and log-likelihoods in Bayesian inverse problems We consider the use of randomised forward models and log-likelihoods within the Bayesian approach to inverse problems. Such random approximations to the exact forward model or log-likelihood arise naturally when a computationally expensive model is approximated using a cheaper stochastic surrogate, as in Gaussian process emulation (kriging), or in the field of probabilistic numerical methods. We show that the Hellinger distance between the exact and approximate Bayesian posteriors is bounded by moments of the difference between the true and approximate log-likelihoods. Example applications of these stability results are given for randomised misfit models in large data applications and the probabilistic solution of ordinary differential equations. 2018 29 SIAM/ASA Journal on Uncertainty Quantification 6 4 1600 1629 urn:nbn:de:0297-zib-66324 10.1137/18M1166523 Numerical Mathematics OPUS4-6616 Wissenschaftlicher Artikel Cockayne, Jon; Oates, Chris; Sullivan, T. J.; Girolami, Mark Bayesian Probabilistic Numerical Methods 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. 2019 33 SIAM Review 61 4 756 789 10.1137/17M1139357 Numerical Mathematics OPUS4-6657 Wissenschaftlicher Artikel Lie, Han Cheng; Sullivan, T. J.; Stuart, Andrew Strong convergence rates of probabilistic integrators for ordinary differential equations Probabilistic integration of a continuous dynamical system is a way of systematically introducing model error, at scales no larger than errors inroduced by standard numerical discretisation, in order to enable thorough exploration of possible responses of the system to inputs. It is thus a potentially useful approach in a number of applications such as forward uncertainty quantification, inverse problems, and data assimilation. We extend the convergence analysis of probabilistic integrators for deterministic ordinary differential equations, as proposed by Conrad et al.\ (\textit{Stat.\ Comput.}, 2016), to establish mean-square convergence in the uniform norm on discrete- or continuous-time solutions under relaxed regularity assumptions on the driving vector fields and their induced flows. Specifically, we show that randomised high-order integrators for globally Lipschitz flows and randomised Euler integrators for dissipative vector fields with polynomially-bounded local Lipschitz constants all have the same mean-square convergence rate as their deterministic counterparts, provided that the variance of the integration noise is not of higher order than the corresponding deterministic integrator. 2019 18 Statistics and Computing 29 6 1265 1283 10.1007/s11222-019-09898-6 Numerical Mathematics OPUS4-6658 Konferenzveröffentlichung Cockayne, Jon; Oates, Chris; Sullivan, T. J.; Girolami, Mark Probabilistic Numerical Methods for PDE-constrained Bayesian Inverse Problems This paper develops meshless methods for probabilistically describing discretisation error in the numerical solution of partial differential equations. This construction enables the solution of Bayesian inverse problems while accounting for the impact of the discretisation of the forward problem. In particular, this drives statistical inferences to be more conservative in the presence of significant solver error. Theoretical results are presented describing rates of convergence for the posteriors in both the forward and inverse problems. This method is tested on a challenging inverse problem with a nonlinear forward model 2017 NaN Proceedings of the 36th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering 1853 060001-1 060001-8 10.1063/1.4985359 Numerical Mathematics OPUS4-6659 Wissenschaftlicher Artikel Schäfer, Florian; Sullivan, T. J.; Owhadi, Houman Compression, inversion, and approximate PCA of dense kernel matrices at near-linear computational complexity 2021 42 Multiscale Modeling and Simulation 19 2 688 730 10.1137/19M129526X Numerical Mathematics OPUS4-6660 Wissenschaftlicher Artikel Lie, Han Cheng; Sullivan, T. J. Equivalence of weak and strong modes of measures on topological vector spaces Modes of a probability measure on an infinite-dimensional Banach space X are often defined by maximising the small-radius limit of the ratio of measures of norm balls. Helin and Burger weakened the definition of such modes by considering only balls with centres in proper subspaces of X, and posed the question of when this restricted notion coincides with the unrestricted one. We generalise these definitions to modes of arbitrary measures on topological vector spaces, defined by arbitrary bounded, convex, neighbourhoods of the origin. We show that a coincident limiting ratios condition is a necessary and sufficient condition for the equivalence of these two types of modes, and show that the coincident limiting ratios condition is satisfied in a wide range of topological vector spaces. 2018 Inverse Problems 34 11 115013 10.1088/1361-6420/aadef2 Numerical Mathematics