{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,6]],"date-time":"2026-10-06T21:50:27Z","timestamp":1791323427035,"version":"4.3.1"},"reference-count":39,"publisher":"Institute of Mathematical Statistics","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Ann. Statist."],"published-print":{"date-parts":[[2026,10,1]]},"abstract":"<jats:p>A functional Bernstein\u2013von Mises theorem is proved for posterior measures arising in a data assimilation problem with the two-dimensional Navier\u2013Stokes equation where a Gaussian process prior is assigned to the initial condition of the system. The posterior measure, which provides the update in the space of all trajectories arising from a discrete sample of the (deterministic) dynamics, is shown to be approximated by a Gaussian random vector field obtained as the solution to a linear parabolic PDE with Gaussian initial condition. The approximation holds in the strong sense of the supremum norm on the regression functions, showing that predicting future states of Navier\u2013Stokes systems admits 1\/ N-consistent estimators even for commonly used nonparametric models. Consequences for coverage of credible bands and uncertainty quantification are discussed. A local asymptotic minimax theorem is derived that describes the lower bound for estimating the state of the nonlinear system, which is shown to be attained by the Bayesian data assimilation algorithm.<\/jats:p>","DOI":"10.1214\/26-aos2645","type":"journal-article","created":{"date-parts":[[2026,10,2]],"date-time":"2026-10-02T12:50:03Z","timestamp":1790945403000},"page":"2352-2378","source":"Crossref","is-referenced-by-count":0,"title":["Data assimilation with the 2D Navier\u2013Stokes equations: Optimal Gaussian asymptotics for the posterior measure"],"prefix":"10.1214","volume":"54","author":[{"given":"Dimitri","family":"Konen","sequence":"first","affiliation":[{"name":"Department of Pure Mathematics and Mathematical Statistics, University of Cambridge"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Richard","family":"Nickl","sequence":"additional","affiliation":[{"name":"Department of Pure Mathematics and Mathematical Statistics, University of Cambridge"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"108","reference":[{"key":"15","doi-asserted-by":"crossref","unstructured":"<small>Kalman, R. E.<\/small> (1960). A new approach to linear filtering and prediction problems. <i>J. Basic Eng.<\/i> <b>82<\/b> 35\u201345.","DOI":"10.1115\/1.3662552"},{"key":"2","doi-asserted-by":"publisher","unstructured":"<small>Castillo, I.<\/small> and <small>Nickl, R.<\/small> (2013). Nonparametric Bernstein-von Mises theorems in Gaussian white noise. <i>Ann. Statist.<\/i> <b>41<\/b> 1999\u20132028.","DOI":"10.1214\/13-AOS1133"},{"key":"21","doi-asserted-by":"publisher","unstructured":"<small>Monard, F.<\/small>, <small>Nickl, R.<\/small> and <small>Paternain, G. P.<\/small> (2019). Efficient nonparametric Bayesian inference for <i>X<\/i>-ray transforms. <i>Ann. Statist.<\/i> <b>47<\/b> 1113\u20131147.","DOI":"10.1214\/18-AOS1708"},{"key":"24","doi-asserted-by":"publisher","unstructured":"<small>Nickl, R.<\/small> (2020). Bernstein\u2013von Mises theorems for statistical inverse problems I: Schr\u00f6dinger equation. <i>J. Eur. Math. Soc<\/i>. (<i>JEMS<\/i>) <b>22<\/b> 2697\u20132750.","DOI":"10.4171\/JEMS\/975"},{"key":"29","doi-asserted-by":"publisher","unstructured":"<small>Nickl, R.<\/small> and <small>Ray, K.<\/small> (2020). Nonparametric statistical inference for drift vector fields of multi-dimensional diffusions. <i>Ann. Statist.<\/i> <b>48<\/b> 1383\u20131408.","DOI":"10.1214\/19-AOS1851"},{"key":"3","doi-asserted-by":"publisher","unstructured":"<small>Castillo, I.<\/small> and <small>Nickl, R.<\/small> (2014). On the Bernstein-von Mises phenomenon for nonparametric Bayes procedures. <i>Ann. Statist.<\/i> <b>42<\/b> 1941\u20131969.","DOI":"10.1214\/14-AOS1246"},{"key":"34","doi-asserted-by":"publisher","unstructured":"<small>REICH, S.<\/small> and <small>COTTER, C.<\/small> (2015). <i>Probabilistic Forecasting and Bayesian Data Assimilation<\/i>. Cambridge Univ. Press, New York.","DOI":"10.1017\/CBO9781107706804"},{"key":"12","doi-asserted-by":"publisher","unstructured":"<small>GIORDANO, M.<\/small> and <small>RAY, K.<\/small> (2022). Nonparametric Bayesian inference for reversible multidimensional diffusions. <i>Ann. Statist.<\/i> <b>50<\/b> 2872\u20132898.","DOI":"10.1214\/22-aos2213"},{"key":"39","doi-asserted-by":"publisher","unstructured":"<small>VAN DER VAART, A. W.<\/small> and <small>WELLNER, J. A.<\/small> (2023). <i>Weak Convergence and Empirical Processes\u2014with Applications to Statistics<\/i>, 2nd ed. <i>Springer Series in Statistics<\/i>. Springer, Cham.","DOI":"10.1007\/978-3-031-29040-4"},{"key":"37","doi-asserted-by":"publisher","unstructured":"<small>Stuart, A. M.<\/small> (2010). Inverse problems: A Bayesian perspective. <i>Acta Numer.<\/i> <b>19<\/b> 451\u2013559.","DOI":"10.1017\/S0962492910000061"},{"key":"10","doi-asserted-by":"publisher","unstructured":"<small>Ghosal, S.<\/small> and <small>van der Vaart, A.<\/small> (2017). <i>Fundamentals of Nonparametric Bayesian Inference<\/i>. <i>Cambridge Series in Statistical and Probabilistic Mathematics<\/i> <b>44<\/b>. Cambridge Univ. Press, Cambridge.","DOI":"10.1017\/9781139029834"},{"key":"38","doi-asserted-by":"publisher","unstructured":"<small>Taylor, M. E.<\/small> (2011). <i>Partial Differential Equations III. Nonlinear Equations<\/i>, 2nd ed. <i>Applied Mathematical Sciences<\/i> <b>117<\/b>. Springer, New York.","DOI":"10.1007\/978-1-4419-7049-7"},{"key":"32","doi-asserted-by":"publisher","unstructured":"<small>NICKL, R.<\/small> and <small>WANG, S.<\/small> (2024). On polynomial-time computation of high-dimensional posterior measures by Langevin-type algorithms. <i>J. Eur. Math. Soc<\/i>. (<i>JEMS<\/i>) <b>26<\/b> 1031\u20131112.","DOI":"10.4171\/jems\/1304"},{"key":"4","doi-asserted-by":"crossref","unstructured":"<small>CONSTANTIN, P.<\/small> and <small>FOIAS, C.<\/small> (1988). <i>Navier-Stokes Equations<\/i>. <i>Chicago Lectures in Mathematics<\/i>. Univ. Chicago Press, Chicago, IL.","DOI":"10.7208\/chicago\/9780226764320.001.0001"},{"key":"35","doi-asserted-by":"publisher","unstructured":"<small>ROBINSON, J. C.<\/small> (2001). <i>Infinite-Dimensional Dynamical Systems<\/i>: <i>An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors<\/i>. <i>Cambridge Texts in Applied Mathematics<\/i>. Cambridge Univ. Press, Cambridge.","DOI":"10.1007\/978-94-010-0732-0"},{"key":"33","doi-asserted-by":"publisher","unstructured":"<small>RASMUSSEN, A. K.<\/small>, <small>SEIZILLES, F.<\/small>, <small>GIROLAMI, M.<\/small> and <small>KAZLAUSKAITE, I.<\/small> (2024). The Bayesian approach to inverse Robin problems. <i>SIAM\/ASA J. Uncertain. Quantificat.<\/i> <b>12<\/b> 1050\u20131084.","DOI":"10.1137\/23M1620624"},{"key":"25","doi-asserted-by":"publisher","unstructured":"<small>NICKL, R.<\/small> (2023). <i>Bayesian Non-linear Statistical Inverse Problems<\/i>. <i>Zurich Lectures in Advanced Mathematics<\/i>. EMS Press, Berlin.","DOI":"10.4171\/zlam\/30"},{"key":"13","doi-asserted-by":"publisher","unstructured":"<small>GIORDANO, M.<\/small> and <small>WANG, S.<\/small> (2025). Statistical algorithms for low-frequency diffusion data: A PDE approach. <i>Ann. Statist.<\/i> <b>53<\/b> 1150\u20131175.","DOI":"10.1214\/25-aos2496"},{"key":"14","doi-asserted-by":"publisher","unstructured":"<small>HOFFMANN, M.<\/small> and <small>RAY, K.<\/small> (2025). Nonparametric Bayesian estimation in a multidimensional diffusion model with high frequency data. <i>Probab. Theory Related Fields<\/i> <b>191<\/b> 103\u2013180.","DOI":"10.1007\/s00440-024-01317-w"},{"key":"1","doi-asserted-by":"publisher","unstructured":"<small>CARRILLO, J. A.<\/small>, <small>HOFFMANN, F.<\/small>, <small>STUART, A. M.<\/small> and <small>VAES, U.<\/small> (2024). The mean-field ensemble Kalman filter: Near-Gaussian setting. <i>SIAM J. Numer. Anal.<\/i> <b>62<\/b> 2549\u20132587.","DOI":"10.1137\/24M1628207"},{"key":"5","doi-asserted-by":"publisher","unstructured":"<small>COTTER, S. L.<\/small>, <small>DASHTI, M.<\/small>, <small>ROBINSON, J. C.<\/small> and <small>STUART, A. M.<\/small> (2009). Bayesian inverse problems for functions and applications to fluid mechanics. <i>Inverse Probl.<\/i> <b>25<\/b> 115008.","DOI":"10.1088\/0266-5611\/25\/11\/115008"},{"key":"6","unstructured":"<small>DOUMECHE, N.<\/small>, <small>BACH, F.<\/small>, <small>BEDEK, E.<\/small>, <small>BIAU, G.<\/small>, <small>BOYER, C.<\/small> and <small>GOUDE, Y.<\/small> (2025). Forecasting time series with constraints. arXiv preprint."},{"key":"7","doi-asserted-by":"publisher","unstructured":"<small>EVANS, L. C.<\/small> (1998). <i>Partial Differ. Equ.<\/i>. <i>Graduate Studies in Mathematics<\/i> <b>19<\/b>. Amer. Math. Soc., Providence, RI.","DOI":"10.1090\/gsm\/019"},{"key":"8","doi-asserted-by":"publisher","unstructured":"<small>EVENSEN, G.<\/small> (2009). <i>Data Assimilation<\/i>, 2nd ed. Springer, Berlin. The ensemble Kalman filter.","DOI":"10.1007\/978-3-642-03711-5"},{"key":"9","doi-asserted-by":"crossref","unstructured":"<small>EVENSEN, G.<\/small>, <small>VOSSEPOEL, F. C.<\/small> and <small>VAN LEEUWEN, J.<\/small> (2022). <i>Data Assimilation Fundamentals<\/i>. <i>Springer Textbooks in Earth Sciences, Geography and Environment.<\/i> Springer, Cham.","DOI":"10.1007\/978-3-030-96709-3"},{"key":"11","doi-asserted-by":"publisher","unstructured":"<small>GIN\u00c9, E.<\/small> and <small>NICKL, R.<\/small> (2016). <i>Mathematical Foundations of Infinite-Dimensional Statistical Models<\/i>. Cambridge Univ. Press, New York.","DOI":"10.1017\/CBO9781107337862"},{"key":"16","doi-asserted-by":"publisher","unstructured":"<small>KEKKONEN, H.<\/small> (2022). Consistency of Bayesian inference with Gaussian process priors for a parabolic inverse problem. <i>Inverse Probl.<\/i> <b>38<\/b> 035002.","DOI":"10.1088\/1361-6420\/ac4839"},{"key":"17","unstructured":"<small>KOERS, G.<\/small>, <small>SZABO, B.<\/small> and <small>VAN DER VAART, A. W.<\/small> (2025). Linear methods for non-linear inverse problems. arXiv preprint."},{"key":"18","unstructured":"<small>KONEN, D.<\/small> (2026). Inverting the Fisher information operator in non-linear models. arXiv preprint. Available at arXiv:<a href=\"https:\/\/arxiv.org\/abs\/2601.13254\">2601.13254<\/a>."},{"key":"19","doi-asserted-by":"crossref","unstructured":"<small>KONEN, D.<\/small> and <small>NICKL, R.<\/small> (2026). Supplement to \u201cData assimilation with the <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\" id=\"eq_4\"><mn>2<\/mn><mi mathvariant=\"italic\">D<\/mi><\/math> Navier\u2013Stokes equations: Optimal Gaussian asymptotics for the posterior measure.\u201d <a href=\"https:\/\/doi.org\/10.1214\/26-AOS2645SUPP\">https:\/\/doi.org\/10.1214\/26-AOS2645SUPP<\/a>","DOI":"10.1214\/26-AOS2645SUPP"},{"key":"20","doi-asserted-by":"publisher","unstructured":"<small>LAW, K.<\/small>, <small>STUART, A.<\/small> and <small>ZYGALAKIS, K.<\/small> (2015). <i>Data Assimilation<\/i>. <i>Texts in Applied Mathematics<\/i> <b>62<\/b>. Springer, Cham.","DOI":"10.1007\/978-3-319-20325-6"},{"key":"22","doi-asserted-by":"publisher","unstructured":"<small>MONARD, F.<\/small>, <small>NICKL, R.<\/small> and <small>PATERNAIN, G. P.<\/small> (2021a). Consistent inversion of noisy non-Abelian X-ray transforms. <i>Comm. Pure Appl. Math.<\/i> <b>74<\/b> 1045\u20131099.","DOI":"10.1002\/cpa.21942"},{"key":"23","doi-asserted-by":"publisher","unstructured":"<small>MONARD, F.<\/small>, <small>NICKL, R.<\/small> and <small>PATERNAIN, G. P.<\/small> (2021b). Statistical guarantees for Bayesian uncertainty quantification in nonlinear inverse problems with Gaussian process priors. <i>Ann. Statist.<\/i> <b>49<\/b> 3255\u20133298.","DOI":"10.1214\/21-aos2082"},{"key":"26","unstructured":"<small>NICKL, R.<\/small> (2024a). Bernstein-von Mises theorems for time evolution equations. arXiv preprint. Available at arXiv:<a href=\"https:\/\/arxiv.org\/abs\/2407.14781\">2407.14781<\/a>."},{"key":"27","doi-asserted-by":"publisher","unstructured":"<small>NICKL, R.<\/small> (2024b). Consistent inference for diffusions from low frequency measurements. <i>Ann. Statist.<\/i> <b>52<\/b> 519\u2013549.","DOI":"10.1214\/24-aos2357"},{"key":"28","doi-asserted-by":"publisher","unstructured":"<small>NICKL, R.<\/small>, <small>PAVLIOTIS, G. A.<\/small> and <small>RAY, K.<\/small> (2025). Bayesian nonparametric inference in McKean\u2013Vlasov models. <i>Ann. Statist.<\/i> <b>53<\/b> 170\u2013193.","DOI":"10.1214\/24-aos2459"},{"key":"30","unstructured":"<small>NICKL, R.<\/small> and <small>SEIZILLES, F.<\/small> (2025). Inferring diffusivity from killed diffusion. arXiv preprint. Available at arXiv:<a href=\"https:\/\/arxiv.org\/abs\/2503.14978\">2503.14978<\/a>."},{"key":"31","doi-asserted-by":"publisher","unstructured":"<small>NICKL, R.<\/small> and <small>TITI, E. S.<\/small> (2024b). On posterior consistency of data assimilation with Gaussian process priors: The 2D-Navier\u2013Stokes equations. <i>Ann. Statist.<\/i> <b>52<\/b> 1825\u20131844.","DOI":"10.1214\/24-aos2427"},{"key":"36","doi-asserted-by":"publisher","unstructured":"<small>SANZ-ALONSO, D.<\/small> and <small>WANIOREK, N.<\/small> (2024). Analysis of a computational framework for Bayesian inverse problems: Ensemble Kalman updates and MAP estimators under mesh refinement. <i>SIAM\/ASA J. Uncertain. Quantificat.<\/i> <b>12<\/b> 30\u201368.","DOI":"10.1137\/23M1567035"}],"container-title":["The Annals of Statistics"],"original-title":[],"deposited":{"date-parts":[[2026,10,6]],"date-time":"2026-10-06T21:05:33Z","timestamp":1791320733000},"score":1,"resource":{"primary":{"URL":"https:\/\/projecteuclid.org\/journals\/annals-of-statistics\/volume-54\/issue-5\/Data-assimilation-with-the-2D-NavierStokes-equations--Optimal-Gaussian\/10.1214\/26-AOS2645.full"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,10,1]]},"references-count":39,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2026,10,1]]}},"URL":"https:\/\/doi.org\/10.1214\/26-aos2645","relation":{},"ISSN":["0090-5364","2168-8966"],"issn-type":[{"value":"0090-5364","type":"print"},{"value":"2168-8966","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,10,1]]}}}