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Chinese Annals of Mathematics, Series B

, Volume 40, Issue 5, pp 721–764 | Cite as

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

  • Michael S. Jolly
  • Vincent R. Martinez
  • Eric J. Olson
  • Edriss S. TitiEmail author
Article
  • 17 Downloads

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.

Keywords

Data assimilation Nudging Time-Averaged observables Surface quasi-geostrophic equation 

2000 MR Subject Classification

35Q35 35Q86 35Q93 37B55 74H40 93B52 

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Notes

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.

References

  1. [1]
    Albanez, D. A. F., Nussenzveig Lopes, H. J. and Titim, E. S., Continuous data assimilation for the three-dimensional Navier-Stokes-α model, Asymptotic Anal., 97(1-2), 2016, 139–164, DOI: 10.3233/ASY-151351.MathSciNetCrossRefGoogle Scholar
  2. [2]
    Azouani, A., Olson, E. and Titi, E. S., Continuous data assimilation using general interpolant observables, J. Nonlinear Sci., 24(2), 2014, 277–304.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    Azouani, A. and Titi, E. S., Feedback control of nonlinear dissipative systems by finite determining parameters-a reaction diffusion paradigm, Evol. Equ. Control Theory, 3(4), 2014, 579–594.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    Bergemann, K. and Reich, S., An ensemble Kalman-Bucy filter for continuous data assimilation, Meteorl. Z., 21, 2012, 213–219.CrossRefGoogle Scholar
  5. [5]
    Bessaih, H., Olson, E. and Titi, E. S., Continuous assimilation of data with stochastic noise, Nonlinearity, 28, 2015, 729–753.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    Blocher, J., Chaotic Attractors and Synchronization Using Time-Averaged Partial Observations of the Phase Space, Masters Thesis, University of Nevada, Department of Mathematics and Statistics, 2016.Google Scholar
  7. [7]
    Blocher, J., Martinez, V. R. and Olson, E., Data assimilation using noisy time-averaged measurements, Physica D, 376, 2018, 49–59.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    Bloemker, D., Law, K. J. H., Stuart, A. M. and Zygalakis, K., Accuracy and stability of the continuous-time 3DVAR filter for the Navier-Stokes equation, Nonlinearity, 26, 2013, 2193–2219.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    Caffarelli, L. and Vasseur, A., Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Annals of Math., 171(3), 2010, 1903–1930.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    Carrillo, J. A. and Ferreira, C. F., The asymptotic behaviour of subcritical dissipative quasi-geostrophic equations, Nonlinearity, 21, 2008, 1001–1018.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    Charney, J., Halem, M. and Jastrow, R., Use of incomplete historical data to infer the present state of the atmosphere, J. Atmos. Sci., 26, 1969, 1160–1163.CrossRefGoogle Scholar
  12. [12]
    Cheskidov, A. and Dai, M., The existence of a global attractor for the forced critical surface quasi-geostrophic equation in L2, J. Math. Fluid Mech., 20(1), 2018, 213–225.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    Constantin, P., Coti-Zelati, M. and Vicol, V., Uniformly attracting limit sets for the critically dissipative SQG equation, Nonlinearity, 29, 2016, 298–318.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    Constantin, P., Glatt-Holtz, N. and Vicol, V., Unique ergodicity for fractionally dissipated, stochastically forced 2D Euler equations, Comm. Math. Phys., 330(2), 2014, 819–857.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    Constantin, P., Majda, A. and Tabak, E., Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar, Nonlinearity, 7, 1994, 1495–1533.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    Constantin, P., Tarfulea, A. and Vicol, V., Long time dynamics of forced critical SQG, Commun. Math. Phys., 335(1), 2014, 93–141, DOI: 10.1007/s00220-014-2129-3m.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    Constantin, P. and Vicol, V., Nonlinear maximum principles for dissipative linear nonlocal operators and applications, Geom. Fund. Anal., 22(5), 2012, 1289–1321.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    Constantin, P. and Wu, J., Behavior of solutions of 2D quasi-geostrophic equations, SIAM J. Math Anal., 30(5), 1999, 937–948.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    Coti-Zelati, M., Long time behavior of subcritical SQG in scale-invariant Sobolev spaces, J. Nonlinear Sci., 28(1), 2018, 305–335.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    Coti-Zelati, M. and Vicol, V., On the global regularity for the supercritical SQG equation, Indiana Univ. Math. J., 65, 2016, 535–552, DOI: 10.1512/iumj.2016.65.5807.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    Dong, H., Dissipative quasi-geostrophic equations in critical Sobolev spaces: smoothing effect and global well-posedness, Discrete Gontin. Dyn. Syst. Series A, 26(4), 2010, 1197–1211.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    Farhat, A., Jolly, M. S. and Titi, E. S., Continuous data assimilation for the 2D Benard convection through velocity measurements alone, Phys. D, 303, 2015, 59–66.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    Farhat, A., Lunasin, E. and Titi, E. S., Abridged dynamic continuous data assimilation for the 2D Navier-Stokes equations, J. Math. Fluid Mech., 18(1), 2016, DOI: 10.1007/s00021-015-0225-6.Google Scholar
  24. [24]
    Farhat, A., Lunasin, E. and Titi, E. S., Data assimilation algorithm for 3D Benard convection in porous media employing only temperature measurements, J. Math. Anal. Appl., 438(1), 2016, 492–506.MathSciNetzbMATHCrossRefGoogle Scholar
  25. [25]
    Farhat, A., Lunasin, E. and Titi, E. S., On the Charney conjecture employing temperature measurements alone: the paradigm of 3D planetary geostrophic model, Math. Glim. Weather Forecast, 2, 2016, 61–74.zbMATHGoogle Scholar
  26. [26]
    Foias, C., Mondaini, C. and Titi, E. S., A discrete data assimiliation scheme for the solutions of the 2D Navier-Stokes equations and their statistics, SIAM J. Appl. Dyn. Syst., 15(4), 2016, 2109–2142.MathSciNetzbMATHCrossRefGoogle Scholar
  27. [27]
    Ibdah, H. A., Mondaini, C. F. and Titi, E. S., Uniform in time error estimates for fully discrete numerical schemes of a data assimilation algorithm, 2018, arXiv:1805.01595vl.Google Scholar
  28. [28]
    Jolly, M. S., Martinez, V. R. and Titi, E. S., A data assimilation algorithm for the subcritical surface quasi-geostrophic equation, Adv. Nonlinear Stud., 17(1), 2017, 167–192.MathSciNetzbMATHCrossRefGoogle Scholar
  29. [29]
    Ju, N., The maximum principle and the global attractor for the dissipative 2D quasi-g eostrophic equations, Comm. Math. Phys., 255, 2005, 161–181.MathSciNetzbMATHCrossRefGoogle Scholar
  30. [30]
    Kalnay, E., Atmospheric Modeling, Data Assimilation, and Predictability, Cambridge University Press, New York, 2003.Google Scholar
  31. [31]
    Kato, T. and Ponce, G., Commutator estimates and the Euler and Navier-Stokes equation, Comm. Pure. Appl. Math., 41(7), 1988, 891–907.MathSciNetzbMATHCrossRefGoogle Scholar
  32. [32]
    Kenig, C. E., Ponce, G. and Vega, L., Well-posedness of the initial value problem for the Korteweg-de-Vries equation, J. Am. Math. Soc., 4(2), 1991, 323–347.MathSciNetzbMATHCrossRefGoogle Scholar
  33. [33]
    Khouider, B. and Titi, E. S., An inviscid regularization for the surface quasi-geostrophic equation, Commun. Pure. Appl. Math., 61, 2008, 1331–1346.MathSciNetzbMATHCrossRefGoogle Scholar
  34. [34]
    Kiselev, A. and Nazarov, F., Variation on a theme of Caffarelli and Vasseur, J. Math. Sci., 166(1), 2010, 31–39.MathSciNetzbMATHCrossRefGoogle Scholar
  35. [35]
    Kiselev, A., Nazarov, F. and Volberg, A., Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math., 167, 2007, 445–453.MathSciNetzbMATHCrossRefGoogle Scholar
  36. [36]
    Markowich, P. A., Titi, E. S. and Trabelsi, S., Continuous data assimilation for the three-dimensional Brinkman-Forchheimer-extended Darcy model, Nonlinearity, 24(4), 2016, 1292–1328.MathSciNetzbMATHCrossRefGoogle Scholar
  37. [37]
    Mondaini, C. F. and Titi, E. S., Uniform-in-time error estimates for the postprocessing Galerkin method applied to a data assimilation algorithm, SIAM J. Numer. Anal., 56(1), 2018, 78–110.MathSciNetzbMATHCrossRefGoogle Scholar
  38. [38]
    Pedlosky, J., Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987.zbMATHCrossRefGoogle Scholar
  39. [39]
    Resnick, S. G., Dynamical problems in non-linear advective partial differential equations, PhD Thesis, The University of Chicago, ProQuest LLC, Ann. Arbor., MI, 1995, 1–86.Google Scholar

Copyright information

© The Editorial Office of CAM and Springer-Verlag Berlin Heidelberg 2019

Authors and Affiliations

  • Michael S. Jolly
    • 1
  • Vincent R. Martinez
    • 2
  • Eric J. Olson
    • 3
  • Edriss S. Titi
    • 4
    • 5
    Email author
  1. 1.Department of MathematicsIndiana UniversityBloomingtonUSA
  2. 2.Department of Mathematics and StatisticsCUNY-Hunter CollegeNew YorkUSA
  3. 3.Department of Mathematics and StatisticsUniversity of Nevada-RenoRenoUSA
  4. 4.Department of MathematicsTexas A & M UniversityCollege StationUSA
  5. 5.Department of Computer Science and Applied MathematicsWeizmann Institute of ScienceRehovotIsrael

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