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Continuous Data Assimilation with Blurred-in-Time Measurements of the Surface Quasi-Geostrophic Equation

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Abstract

An intrinsic property of almost any physical measuring device is that it makes observations which are slightly blurred in time. The authors consider a nudging-based approach for data assimilation that constructs an approximate solution based on a feedback control mechanism that is designed to account for observations that have been blurred by a moving time average. Analysis of this nudging model in the context of the subcritical surface quasi-geostrophic equation shows, provided the time-averaging window is sufficiently small and the resolution of the observations sufficiently fine, that the approximating solution converges exponentially fast to the observed solution over time. In particular, the authors demonstrate that observational data with a small blur in time possess no significant obstructions to data assimilation provided that the nudging properly takes the time averaging into account. Two key ingredients in our analysis are additional bounded-ness properties for the relevant interpolant observation operators and a non-local Gronwall inequality.

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Acknowledgements

The authors would like to thank the Institute of Pure and Applied Mathematics (IPAM) at UCLA for the warm hospitality where this collaboration was conceived. The authors are also thankful to Thomas Bewley, Aseel Farhat and Hakima Bessaih for the insightful discussions.

Author information

Authors and Affiliations

  1. Department of Mathematics, Indiana University, Bloomington, IN 47405, USA

    Michael S. Jolly

  2. Department of Mathematics and Statistics, CUNY-Hunter College, New York, NY, 10065, USA

    Vincent R. Martinez

  3. Department of Mathematics and Statistics, University of Nevada-Reno, Reno, NV, 89557, USA

    Eric J. Olson

  4. Department of Mathematics, Texas A & M University, College Station, TX, 77843-3368, USA

    Edriss S. Titi

  5. Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, 76100, Israel

    Edriss S. Titi

Authors
  1. Michael S. Jolly
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  2. Vincent R. Martinez
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  3. Eric J. Olson
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  4. Edriss S. Titi
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Corresponding author

Correspondence to Edriss S. Titi.

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Dedicated to Professor Andrew J. Majda on the occasion of his 70th birthday

This work was supported by NSF Grants DMS-1418911, DMS-1418928, ONR Grant N00014-15-1-2333, the Einstein Stiftung/Foundation-Berlin, through the Einstein Visiting Fellow Program and the John Simon Guggenheim Memorial Foundation.

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Jolly, M.S., Martinez, V.R., Olson, E.J. et al. Continuous Data Assimilation with Blurred-in-Time Measurements of the Surface Quasi-Geostrophic Equation. Chin. Ann. Math. Ser. B 40, 721–764 (2019). https://doi.org/10.1007/s11401-019-0158-0

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  • DOI: https://doi.org/10.1007/s11401-019-0158-0

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