@inproceedings{TeymurLieSullivanetal.2018, author = {Teymur, Onur and Lie, Han Cheng and Sullivan, T. J. and Calderhead, Ben}, title = {Implicit probabilistic integrators for ODEs}, booktitle = {Advances in Neural Information Processing Systems 31 (NIPS 2018)}, arxiv = {http://arxiv.org/abs/1805.07970}, year = {2018}, language = {en} } @article{KlebanovSchusterSullivan2020, author = {Klebanov, Ilja and Schuster, Ingmar and Sullivan, T. J.}, title = {A rigorous theory of conditional mean embeddings}, volume = {2}, journal = {SIAM Journal on Mathematics of Data Science}, number = {3}, arxiv = {http://arxiv.org/abs/1912.00671}, doi = {10.1137/19M1305069}, pages = {583 -- 606}, year = {2020}, language = {en} } @article{GirolamiIpsenOatesetal.2019, author = {Girolami, Mark A. and Ipsen, Ilse C. F. and Oates, Chris and Owen, Art B. and Sullivan, T. J.}, title = {Editorial: Special edition on probabilistic numerics}, volume = {29}, journal = {Statistics and Computing}, number = {6}, doi = {doi:10.1007/s11222-019-09892-y}, pages = {1181 -- 1183}, year = {2019}, language = {en} } @article{NavaYazdaniHegeSullivanetal.2020, author = {Nava-Yazdani, Esfandiar and Hege, Hans-Christian and Sullivan, T. J. and von Tycowicz, Christoph}, title = {Geodesic Analysis in Kendall's Shape Space with Epidemiological Applications}, volume = {62}, journal = {Journal of Mathematical Imaging and Vision}, number = {4}, arxiv = {http://arxiv.org/abs/1906.11950}, doi = {10.1007/s10851-020-00945-w}, pages = {549 -- 559}, year = {2020}, abstract = {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.}, language = {en} } @article{KlebanovSprungkSullivan2021, author = {Klebanov, Ilja and Sprungk, Bj{\"o}rn and Sullivan, T. J.}, title = {The linear conditional expectation in Hilbert space}, volume = {27}, journal = {Bernoulli}, number = {4}, arxiv = {http://arxiv.org/abs/2008.12070}, doi = {10.3150/20-BEJ1308}, pages = {2299 -- 2299}, year = {2021}, language = {en} } @article{BonnetAkianSavinetal.2020, author = {Bonnet, Luc and Akian, Jean-Luc and Savin, {\´E}ric and Sullivan, T. J.}, title = {Adaptive reconstruction of imperfectly-observed monotone functions, with applications to uncertainty quantification}, volume = {13}, journal = {Algorithms}, number = {8}, arxiv = {http://arxiv.org/abs/2007.05236}, doi = {10.3390/a13080196}, pages = {196}, year = {2020}, abstract = {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.}, language = {en} } @inproceedings{Sullivan2017, author = {Sullivan, T. J.}, title = {Well-posedness of Bayesian inverse problems in quasi-Banach spaces with stable priors}, volume = {17}, booktitle = {88th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM), Weimar 2017}, number = {1}, arxiv = {http://arxiv.org/abs/1710.05610}, doi = {10.1002/pamm.201710402}, pages = {871 -- 874}, year = {2017}, language = {en} } @article{Sullivan2017, author = {Sullivan, T. J.}, title = {Well-posed Bayesian inverse problems and heavy-tailed stable quasi-Banach space priors}, volume = {11}, journal = {Inverse Problems and Imaging}, number = {5}, arxiv = {http://arxiv.org/abs/1605.05898}, doi = {10.3934/ipi.2017040}, pages = {857 -- 874}, year = {2017}, language = {en} } @article{LieSullivanStuart2019, author = {Lie, Han Cheng and Sullivan, T. J. and Stuart, Andrew}, title = {Strong convergence rates of probabilistic integrators for ordinary differential equations}, volume = {29}, journal = {Statistics and Computing}, number = {6}, arxiv = {http://arxiv.org/abs/1703.03680}, doi = {10.1007/s11222-019-09898-6}, pages = {1265 -- 1283}, year = {2019}, 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} } @inproceedings{CockayneOatesSullivanetal.2017, author = {Cockayne, Jon and Oates, Chris and Sullivan, T. J. and Girolami, Mark}, title = {Probabilistic Numerical Methods for PDE-constrained Bayesian Inverse Problems}, volume = {1853}, booktitle = {Proceedings of the 36th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering}, arxiv = {http://arxiv.org/abs/1701.04006}, doi = {10.1063/1.4985359}, pages = {060001-1 -- 060001-8}, year = {2017}, 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} }