@article{AyanbayevKlebanovLieetal., author = {Ayanbayev, Birzhan and Klebanov, Ilja and Lie, Han Cheng and Sullivan, T. J.}, title = {Γ-convergence of Onsager-Machlup functionals: II. Infinite product measures on Banach spaces}, series = {Inverse Problems}, volume = {38}, journal = {Inverse Problems}, number = {2}, doi = {10.1088/1361-6420/ac3f82}, pages = {35}, language = {en} } @article{AyanbayevKlebanovLieetal., author = {Ayanbayev, Birzhan and Klebanov, Ilja and Lie, Han Cheng and Sullivan, T. J.}, title = {Γ-convergence of Onsager-Machlup functionals: I. With applications to maximum a posteriori estimation in Bayesian inverse problems}, series = {Inverse Problems}, volume = {38}, journal = {Inverse Problems}, number = {2}, doi = {10.1088/1361-6420/ac3f81}, pages = {32}, language = {en} } @inproceedings{Sullivan, author = {Sullivan, T. J.}, title = {Well-posedness of Bayesian inverse problems in quasi-Banach spaces with stable priors}, series = {88th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM), Weimar 2017}, volume = {17}, booktitle = {88th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM), Weimar 2017}, number = {1}, doi = {10.1002/pamm.201710402}, pages = {871 -- 874}, language = {en} } @article{Sullivan, author = {Sullivan, T. J.}, title = {Well-posed Bayesian inverse problems and heavy-tailed stable quasi-Banach space priors}, series = {Inverse Problems and Imaging}, volume = {11}, journal = {Inverse Problems and Imaging}, number = {5}, doi = {10.3934/ipi.2017040}, pages = {857 -- 874}, language = {en} } @misc{Sullivan2016, author = {Sullivan, T. J.}, title = {Well-posed Bayesian inverse problems and heavy-tailed stable Banach space priors}, issn = {1438-0064}, doi = {10.3934/ipi.2017040}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-59422}, year = {2016}, abstract = {This article extends the framework of Bayesian inverse problems in infinite-dimensional parameter spaces, as advocated by Stuart (Acta Numer. 19:451-559, 2010) and others, to the case of a heavy-tailed prior measure in the family of stable distributions, such as an infinite-dimensional Cauchy distribution, for which polynomial moments are infinite or undefined. It is shown that analogues of the Karhunen-Lo{\`e}ve expansion for square-integrable random variables can be used to sample such measures. Furthermore, under weaker regularity assumptions than those used to date, the Bayesian posterior measure is shown to depend Lipschitz continuously in the Hellinger metric upon perturbations of the misfit function and observed data.}, language = {en} } @article{KlebanovSprungkSullivan, author = {Klebanov, Ilja and Sprungk, Bj{\"o}rn and Sullivan, T. J.}, title = {The linear conditional expectation in Hilbert space}, series = {Bernoulli}, volume = {27}, journal = {Bernoulli}, number = {4}, doi = {10.3150/20-BEJ1308}, pages = {2299 -- 2299}, language = {en} } @article{LieSullivanStuart, author = {Lie, Han Cheng and Sullivan, T. J. and Stuart, Andrew}, title = {Strong convergence rates of probabilistic integrators for ordinary differential equations}, series = {Statistics and Computing}, volume = {29}, journal = {Statistics and Computing}, number = {6}, doi = {10.1007/s11222-019-09898-6}, pages = {1265 -- 1283}, abstract = {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.}, language = {en} } @misc{LieSullivanTeckentrup, author = {Lie, Han Cheng and Sullivan, T. J. and Teckentrup, Aretha}, title = {Random forward models and log-likelihoods in Bayesian inverse problems}, series = {SIAM/ASA Journal on Uncertainty Quantification}, volume = {6}, journal = {SIAM/ASA Journal on Uncertainty Quantification}, number = {4}, issn = {1438-0064}, doi = {10.1137/18M1166523}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-66324}, pages = {1600 -- 1629}, abstract = {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.}, language = {en} } @article{LieSullivan2018, author = {Lie, Han Cheng and Sullivan, T. J.}, title = {Quasi-invariance of countable products of Cauchy measures under non-unitary dilations}, series = {Electronic Communications in Probability}, volume = {23}, journal = {Electronic Communications in Probability}, number = {8}, doi = {10.1214/18-ECP113}, pages = {1 -- 6}, year = {2018}, language = {en} } @inproceedings{CockayneOatesSullivanetal., author = {Cockayne, Jon and Oates, Chris and Sullivan, T. J. and Girolami, Mark}, title = {Probabilistic Numerical Methods for PDE-constrained Bayesian Inverse Problems}, series = {Proceedings of the 36th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering}, volume = {1853}, booktitle = {Proceedings of the 36th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering}, doi = {10.1063/1.4985359}, pages = {060001-1 -- 060001-8}, abstract = {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}, language = {en} }