{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,29]],"date-time":"2026-10-29T17:51:58Z","timestamp":1793296318863,"version":"4.5.0"},"reference-count":42,"publisher":"Institute of Mathematical Statistics","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Ann. Statist."],"published-print":{"date-parts":[[2024,8,1]]},"abstract":"We consider a nonlinear Bayesian data assimilation model for the periodic two-dimensional Navier\u2013Stokes equations with initial condition modelled by a Gaussian process prior. We show that if the system is updated with sufficiently many discrete noisy measurements of the velocity field, then the posterior distribution eventually concentrates near the ground truth solution of the time evolution equation, and in particular that the initial condition is recovered consistently by the posterior mean vector field. We further show that the convergence rate can in general not be faster than inverse logarithmic in sample size, but describe specific conditions on the initial conditions when faster rates are possible. In the proofs, we provide an explicit quantitative estimate for backward uniqueness of solutions of the two-dimensional Navier\u2013Stokes equations.<\/jats:p>","DOI":"10.1214\/24-aos2427","type":"journal-article","created":{"date-parts":[[2024,10,4]],"date-time":"2024-10-04T01:53:47Z","timestamp":1728006827000},"page":"1825-1844","source":"Crossref","is-referenced-by-count":9,"title":["On posterior consistency of data assimilation with Gaussian process priors: The 2D-Navier\u2013Stokes equations"],"prefix":"10.1214","volume":"52","author":[{"given":"Richard","family":"Nickl","sequence":"first","affiliation":[{"name":"Department of Pure Mathematics and Mathematical Statistics, University of Cambridge"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Edriss S.","family":"Titi","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics and Theoretical Physics, University of Cambridge"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"108","reference":[{"key":"10","doi-asserted-by":"publisher","unstructured":"Cotter, S. L.<\/small>, Roberts, G. O.<\/small>, Stuart, A. M.<\/small> and White, D.<\/small> (2013). MCMC Methods for Functions: Modifying Old Algorithms to Make Them Faster. Statist. Sci.<\/i> 28<\/b> 424\u2013446.","DOI":"10.1214\/13-STS421"},{"key":"19","doi-asserted-by":"publisher","unstructured":"Hairer, M.<\/small>, Stuart, A. M.<\/small> and Vollmer, S. J.<\/small> (2014). Spectral gaps for a Metropolis\u2013Hastings algorithm in infinite dimensions. Ann. Appl. Probab.<\/i> 24<\/b> 2455\u20132490.","DOI":"10.1214\/13-AAP982"},{"key":"1","doi-asserted-by":"crossref","unstructured":"Abraham, K.<\/small> and Nickl, R.<\/small> (2019). On statistical Calder\u00f3n problems. Math. Stat. Learn.<\/i> 2<\/b> 165\u2013216.","DOI":"10.4171\/msl\/14"},{"key":"4","doi-asserted-by":"publisher","unstructured":"Beskos, A.<\/small>, Girolami, M.<\/small>, Lan, S.<\/small>, Farrell, P. E.<\/small> and Stuart, A. M.<\/small> (2017). Geometric MCMC for infinite-dimensional inverse problems. J. Comput. Phys.<\/i> 335<\/b> 327\u2013351.","DOI":"10.1016\/j.jcp.2016.12.041"},{"key":"28","doi-asserted-by":"publisher","unstructured":"Monard, F.<\/small>, Nickl, R.<\/small> and Paternain, G. P.<\/small> (2021). Consistent inversion of noisy non-Abelian X-ray transforms. Comm. Pure Appl. Math.<\/i> 74<\/b> 1045\u20131099.","DOI":"10.1002\/cpa.21942"},{"key":"30","doi-asserted-by":"publisher","unstructured":"Nickl, R.<\/small> (2020). Bernstein\u2013von Mises theorems for statistical inverse problems I: Schr\u00f6dinger equation. J. Eur. Math. Soc<\/i>. (JEMS<\/i>) 22<\/b> 2697\u20132750.","DOI":"10.4171\/JEMS\/975"},{"key":"33","doi-asserted-by":"publisher","unstructured":"Nickl, R.<\/small> and Ray, K.<\/small> (2020). Nonparametric statistical inference for drift vector fields of multi-dimensional diffusions. Ann. Statist.<\/i> 48<\/b> 1383\u20131408.","DOI":"10.1214\/19-AOS1851"},{"key":"34","doi-asserted-by":"publisher","unstructured":"Nickl, R.<\/small>, van de Geer, S.<\/small> and Wang, S.<\/small> (2020). Convergence rates for penalized least squares estimators in PDE constrained regression problems. SIAM\/ASA J. Uncertain. Quantificat.<\/i> 8<\/b> 374\u2013413.","DOI":"10.1137\/18M1236137"},{"key":"36","doi-asserted-by":"publisher","unstructured":"Ray, K.<\/small> (2013). Bayesian inverse problems with non-conjugate priors. Electron. J. Stat.<\/i> 7<\/b> 2516\u20132549.","DOI":"10.1214\/13-EJS851"},{"key":"24","doi-asserted-by":"crossref","unstructured":"Knapik, B. T.<\/small>, van der Vaart, A. W.<\/small> and van Zanten, J. H.<\/small> (2013). Bayesian recovery of the initial condition for the heat equation. Comm. Statist. Theory Methods<\/i> 42<\/b> 1294\u20131313.","DOI":"10.1080\/03610926.2012.681417"},{"key":"25","doi-asserted-by":"publisher","unstructured":"LAW, K.<\/small>, STUART, A.<\/small> and ZYGALAKIS, K.<\/small> (2015). Data Assimilation<\/i>: A Mathematical Introduction<\/i>. Texts in Applied Mathematics<\/i> 62<\/b>. Springer, Cham.","DOI":"10.1007\/978-3-319-20325-6"},{"key":"29","doi-asserted-by":"publisher","unstructured":"MONARD, F.<\/small>, NICKL, R.<\/small> and PATERNAIN, G. P.<\/small> (2021). Statistical guarantees for Bayesian uncertainty quantification in nonlinear inverse problems with Gaussian process priors. Ann. Statist.<\/i> 49<\/b> 3255\u20133298.","DOI":"10.1214\/21-aos2082"},{"key":"27","doi-asserted-by":"publisher","unstructured":"MAJDA, A. J.<\/small> and HARLIM, J.<\/small> (2012). Filtering Complex Turbulent Systems<\/i>. Cambridge Univ. Press, Cambridge.","DOI":"10.1017\/CBO9781139061308"},{"key":"37","doi-asserted-by":"publisher","unstructured":"REICH, S.<\/small> and COTTER, C.<\/small> (2015). Probabilistic Forecasting and Bayesian Data Assimilation<\/i>. Cambridge Univ. Press, New York.","DOI":"10.1017\/CBO9781107706804"},{"key":"18","doi-asserted-by":"publisher","unstructured":"GIORDANO, M.<\/small> and RAY, K.<\/small> (2022). Nonparametric Bayesian inference for reversible multidimensional diffusions. Ann. Statist.<\/i> 50<\/b> 2872\u20132898.","DOI":"10.1214\/22-aos2213"},{"key":"42","unstructured":"van de Geer, S. A.<\/small> (2000). Applications of Empirical Process Theory<\/i>. Cambridge Series in Statistical and Probabilistic Mathematics<\/i> 6<\/b>. Cambridge Univ. Press, Cambridge."},{"key":"41","doi-asserted-by":"publisher","unstructured":"Stuart, A. M.<\/small> (2010). Inverse problems: A Bayesian perspective. Acta Numer.<\/i> 19<\/b> 451\u2013559.","DOI":"10.1017\/S0962492910000061"},{"key":"16","doi-asserted-by":"publisher","unstructured":"GIN\u00c9, E.<\/small> and NICKL, R.<\/small> (2016). Mathematical Foundations of Infinite-Dimensional Statistical Models<\/i>. Cambridge Series in Statistical and Probabilistic Mathematics<\/i>, [40]. Cambridge Univ. Press, New York.","DOI":"10.1017\/CBO9781107337862"},{"key":"12","doi-asserted-by":"publisher","unstructured":"Evans, L. C.<\/small> (2010). Partial Differential Equations<\/i>, 2nd ed. Graduate Studies in Mathematics<\/i> 19<\/b>. Amer. Math. Soc., Providence, RI.","DOI":"10.1090\/gsm\/019"},{"key":"15","doi-asserted-by":"publisher","unstructured":"Ghosal, S.<\/small> and van der Vaart, A.<\/small> (2017). Fundamentals of Nonparametric Bayesian Inference<\/i>. Cambridge Series in Statistical and Probabilistic Mathematics<\/i> 44<\/b>. Cambridge Univ. Press, Cambridge.","DOI":"10.1017\/9781139029834"},{"key":"23","doi-asserted-by":"publisher","unstructured":"Knapik, B. T.<\/small>, van der Vaart, A. W.<\/small> and van Zanten, J. H.<\/small> (2011). Bayesian inverse problems with Gaussian priors. Ann. Statist.<\/i> 39<\/b> 2626\u20132657.","DOI":"10.1214\/11-AOS920"},{"key":"35","doi-asserted-by":"publisher","unstructured":"NICKL, R.<\/small> and WANG, S.<\/small> (2024). On polynomial-time computation of high-dimensional posterior measures by Langevin-type algorithms. J. Eur. Math. Soc<\/i>. (JEMS<\/i>) 26<\/b> 1031\u20131112.","DOI":"10.4171\/jems\/1304"},{"key":"2","doi-asserted-by":"publisher","unstructured":"BARDOS, C.<\/small> and TARTAR, L.<\/small> (1973). Sur l\u2019unicit\u00e9 r\u00e9trograde des \u00e9quations paraboliques et quelques questions voisines. Arch. Ration. Mech. Anal.<\/i> 50<\/b> 10\u201325.","DOI":"10.1007\/BF00251291"},{"key":"3","doi-asserted-by":"publisher","unstructured":"BENNETT, A. F.<\/small> (2002). Inverse Modeling of the Ocean and Atmosphere<\/i>. Cambridge Univ. Press, Cambridge.","DOI":"10.1017\/CBO9780511535895"},{"key":"5","doi-asserted-by":"publisher","unstructured":"BOHR, J.<\/small> (2021). Stability of the non-Abelian X<\/i>-ray transform in dimension \u2265<\/mo>3<\/mn><\/math>. J. Geom. Anal.<\/i> 31<\/b> 11226\u201311269.","DOI":"10.1007\/s12220-021-00679-0"},{"key":"6","doi-asserted-by":"crossref","unstructured":"BOHR, J.<\/small> and NICKL, R.<\/small> (2023). On log-concave approximations of high-dimensional posterior measures and stability properties in non-linear inverse problems. Ann. Inst. Henri Poincar\u00e9 B, Probab. Stat.<\/i>. (to appear).","DOI":"10.1214\/23-AIHP1397"},{"key":"7","doi-asserted-by":"publisher","unstructured":"BORGGAARD, J.<\/small>, GLATT-HOLTZ, N.<\/small> and KROMETIS, J.<\/small> (2020). On Bayesian consistency for flows observed through a passive scalar. Ann. Appl. Probab.<\/i> 30<\/b> 1762\u20131783.","DOI":"10.1214\/19-AAP1542"},{"key":"8","doi-asserted-by":"crossref","unstructured":"CONSTANTIN, P.<\/small> and FOIAS, C.<\/small> (1988). Navier-Stokes Equations<\/i>. Chicago Lectures in Mathematics<\/i>. Univ. Chicago Press, Chicago, IL.","DOI":"10.7208\/chicago\/9780226764320.001.0001"},{"key":"9","doi-asserted-by":"publisher","unstructured":"COTTER, S. L.<\/small>, DASHTI, M.<\/small>, ROBINSON, J. C.<\/small> and STUART, A. M.<\/small> (2009). Bayesian inverse problems for functions and applications to fluid mechanics. Inverse Probl.<\/i> 25<\/b> 115008, 43.","DOI":"10.1088\/0266-5611\/25\/11\/115008"},{"key":"13","doi-asserted-by":"crossref","unstructured":"EVENSEN, G.<\/small>, VOSSEPOEL, F. C.<\/small> and VAN LEEUWEN, J.<\/small> (2022). Data Assimilation Fundamentals<\/i>. Springer Textbooks in Earth Sciences, Geography and Environment<\/i>. Springer, Cham.","DOI":"10.1007\/978-3-030-96709-3"},{"key":"14","doi-asserted-by":"publisher","unstructured":"FOIAS, C.<\/small> and SAUT, J.-C.<\/small> (1984). Asymptotic behavior, as t<\/mi>\u2192<\/mo>+<\/mo>\u221e<\/mi><\/math>, of solutions of Navier\u2013Stokes equations and nonlinear spectral manifolds. Indiana Univ. Math. J.<\/i> 33<\/b> 459\u2013477.","DOI":"10.1512\/iumj.1984.33.33025"},{"key":"17","doi-asserted-by":"publisher","unstructured":"GIORDANO, M.<\/small> and NICKL, R.<\/small> (2020). Consistency of Bayesian inference with Gaussian process priors in an elliptic inverse problem. Inverse Probl.<\/i> 36<\/b> 085001, 35.","DOI":"10.1088\/1361-6420\/ab7d2a"},{"key":"20","unstructured":"HOFFMANN, M.<\/small> and RAY, K.<\/small> (2022). Bayesian estimation in a multidimensional diffusion model with high frequency data. arXiv."},{"key":"21","unstructured":"KALNAY, E.<\/small> (2003). Atmospheric Modelling, Data Assimilation, and Predictability<\/i>. Cambridge Univ. Press, Cambridge, UK."},{"key":"22","doi-asserted-by":"publisher","unstructured":"KEKKONEN, H.<\/small> (2022). Consistency of Bayesian inference with Gaussian process priors for a parabolic inverse problem. Inverse Probl.<\/i> 38<\/b> Paper No. 035002, 29.","DOI":"10.1088\/1361-6420\/ac4839"},{"key":"31","doi-asserted-by":"crossref","unstructured":"NICKL, R.<\/small> (2024). Inference for diffusions from low frequency measurements. Ann. Statist.<\/i> 52<\/b> 519\u2013549.","DOI":"10.1214\/24-AOS2357"},{"key":"38","doi-asserted-by":"publisher","unstructured":"ROBINSON, J. C.<\/small> (2001). Infinite-Dimensional Dynamical Systems<\/i>: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors<\/i>. Cambridge Texts in Applied Mathematics<\/i>. Cambridge Univ. Press, Cambridge.","DOI":"10.1007\/978-94-010-0732-0"},{"key":"39","doi-asserted-by":"publisher","unstructured":"SCHWAB, C.<\/small> and STUART, A. M.<\/small> (2012). Sparse deterministic approximation of Bayesian inverse problems. Inverse Probl.<\/i> 28<\/b> 045003, 32.","DOI":"10.1088\/0266-5611\/28\/4\/045003"},{"key":"40","doi-asserted-by":"crossref","unstructured":"ST-AMANT, S.<\/small> (2022). Stability estimate for the broken non-Abelian X-ray transform in Minkowski space. Inverse Probl.<\/i> 38<\/b> Paper No. 105007, 36.","DOI":"10.1088\/1361-6420\/ac88f2"},{"key":"26","doi-asserted-by":"publisher","unstructured":"LORENZ, E. N.<\/small> (1963). Deterministic nonperiodic flow. J. Atmos. Sci.<\/i> 20<\/b> 130\u2013141.","DOI":"10.1175\/1520-0469(1963)020<0130:DNF>2.0.CO;2"},{"key":"11","doi-asserted-by":"publisher","unstructured":"CUI, T.<\/small>, LAW, K. J. H.<\/small> and MARZOUK, Y. M.<\/small> (2016). Dimension-independent likelihood-informed MCMC. J. Comput. Phys.<\/i> 304<\/b> 109\u2013137.","DOI":"10.1016\/j.jcp.2015.10.008"},{"key":"32","doi-asserted-by":"publisher","unstructured":"NICKL, R.<\/small> (2023). Bayesian Non-linear Statistical Inverse Problems<\/i>. Zurich Lectures in Advanced Mathematics<\/i>. EMS Press, Berlin.","DOI":"10.4171\/zlam\/30"}],"container-title":["The Annals of Statistics"],"original-title":[],"deposited":{"date-parts":[[2026,10,7]],"date-time":"2026-10-07T17:35:19Z","timestamp":1791394519000},"score":1,"resource":{"primary":{"URL":"https:\/\/projecteuclid.org\/journals\/annals-of-statistics\/volume-52\/issue-4\/On-posterior-consistency-of-data-assimilation-with-Gaussian-process-priors\/10.1214\/24-AOS2427.full"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,1]]},"references-count":42,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2024,8,1]]}},"URL":"https:\/\/doi.org\/10.1214\/24-aos2427","relation":{},"ISSN":["0090-5364","2168-8966"],"issn-type":[{"value":"0090-5364","type":"print"},{"value":"2168-8966","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,8,1]]}}}