Ricerche di Matematica

, Volume 68, Issue 2, pp 679–691 | Cite as

A note on domain decomposition approaches for solving 3D variational data assimilation models

  • Luisa D’AmoreEmail author
  • Rosalba Cacciapuoti


Data assimilation (DA) is a methodology for combining mathematical models simulating complex systems (the background knowledge) and measurements (the reality or observational data) in order to improve the estimate of the system state (the forecast). The DA is an inverse and ill posed problem usually used to handle a huge amount of data, so, it is a big and computationally expensive problem. In the present work we prove that the functional decomposition of the 3D variational data assimilation (3D Var DA) operator, previously introduced by the authors, is equivalent to apply multiplicative parallel Schwarz (MPS) method, to the Euler–Lagrange equations arising from the minimization of the data assimilation functional. It results that convergence issues as well as mesh refininement techniques and coarse grid correction—issues of the functional decomposition not previously addressed—could be employed to improve performance and scalability of the 3D Var DA functional decomposition in real cases.


3D VarDA DD methods Multiplicative Schwarz method 

Mathematics Subject Classification

65F22 65K15 65M55 65N55 8608 



  1. 1.
    Ghil, M., Malanotte-Rizzoli, P.: Data Assimilation in Meteorology and Oceanography, Advances in Geophysics, vol. 33. Academic Press, New York (1991)Google Scholar
  2. 2.
    Kalnay, E.: Atmospheric Modeling, Data Assimilation and Predictability. Cambridge University Press, Cambridge (2003)Google Scholar
  3. 3.
    Navon, I.M.: Data assimilation for numerical weather prediction: a review. In: Park, S.K., Xu, L. (eds.) Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications. Springer, Berlin (2009)Google Scholar
  4. 4.
    Blum, J., Le Dimet, F.X., Navon, I.M.: Data Assimilation for Geophysical Fluids, Volume XIV of Handbook of Numerical Analysis, vol. 9. Elsevier, Amsterdam (2005)Google Scholar
  5. 5.
    D’Elia, M., Perego, M., Veneziani, A.: A variational data assimilation procedure for the incompressible Navier-Stokes equations in hemodynamics. J. Sci. Comput. 52(2), 340–359 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Murli, A., Cuomo, S., D’Amore, L., Galletti, A.: Numerical regularization of a real inversion formula based on the Laplace transform’s eigenfunction expansion of the inverse function. Inverse Probl. 23(2), 713–731 (2007)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Le Dimet, F.X., Talagrand, O.: Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects. Tellus 38A, 97–110 (1986)CrossRefGoogle Scholar
  8. 8.
    Kalman, R.E.: A new approach to linear filtering and prediction problems. Trans. ASME J. Basic Eng. 82(Series D), 35–45 (1960)MathSciNetCrossRefGoogle Scholar
  9. 9.
    D’Amore, L., Casaburi, D., Galletti, A., Marcellino, L., Murli, A.: Integration of emerging computer technologies for an efficient image sequences analysis. Integr. Comput. Aided Eng. 18(4), 365–378 (2011)CrossRefGoogle Scholar
  10. 10.
    Miyoshi, T.: Computational Challenges in Big Data Assimilation with Extreme-Scale Simulations, Talk at BDEC Workshop. Charleston, SC (2013)Google Scholar
  11. 11.
    Tremolet, Y., Le Dimet, F.X.: Parallel algorithms for variational data assimilation and coupling models. Parallel Comput. 22, 657–674 (1986)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Arcucci, R., D’Amore, L., Celestino, S., Laccetti, G., Murli, A.: A scalable numerical algorithm for solving Tikhonov regularization problems. In: Wiatr, K., Karczewski, K., Wyrzykowski, R., Deelman, E., Kitowski, J., Dongarra, J. (eds.) 11th International Conference on Parallel Processing and Applied Mathematics, PPAM 2015. Lecture Notes in Computer Science, vol. 9574, pp. 45–54. Springer (2016).
  13. 13.
    Arcucci, R., D’Amore, L., Carracciuolo, L., Scotti, G., Laccetti, G.: A decomposition of the tikhonov regularization functional oriented to exploit hybrid multilevel parallelism. Int. J. Parallel Program. 45(5), 1214–1235 (2017). CrossRefGoogle Scholar
  14. 14.
    D’Amore, L., Arcucci, R., Carracciuolo, L., Murli, A.: A scalable approach for variational data assimilation. J. Sci. Comput. 61, 239–257 (2014)MathSciNetCrossRefGoogle Scholar
  15. 15.
    D’Amore, L., Arcucci, R., Carracciuolo, L., Murli, A.: DD-ocean var: a domain decomposition fully parallel data assimilation software for the mediterranean forecasting system. In: Procedia Computer Science, 13th Annual International Conference on Computational Science, ICCS, vol. 8, pp. 1235–1244. Elsevier (2013). CrossRefGoogle Scholar
  16. 16.
    Chan, T., Mathew, T.P.: Domain decomposition algorithms. Acta Numer. 3, 64–143 (1994)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Dolean V., Jolivet P., Nataf, F.: An Introduction to Domain Decomposition Methods: Algorithms, Theory and Parallel Implementation, HAL Id: cel-01100932 (2016)Google Scholar
  18. 18.
    Toselli, A., Widlund, O.: Domain Decomposition Methods. Algorthms and Theory. Springer, New York (2005)CrossRefGoogle Scholar
  19. 19.
    Gunzburger, M.D., Lee, J.: A domain decomposition method for optimization problems for partial differential equations. Comput. Math. Appl. 40, 177–192 (2000)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Tai, X.-C., Tseng, P.: Convergence rate analysis of an asynchronous space decomposition method for convex minimization. Math. Comput. 71(239), 1105–1135 (2001)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Xu, J.: Iterative methods by space decomposition and subspace correction. SIAM Rev. 34(4), 581–613 (1992)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Tai, X.-C., Xu, J.: Global convergence of subspace correction methods for convex optimization problems. Math. Comput. 71(237), 105–124 (2002)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Fornasier, M., Schonlieb, C.B.: Subspace correction methods for total variation and \(l_1\) minimization. SIAM J. Numer. Anal. 47(5), 3397–3428 (2009)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Cardin, F., Favrett, M., Lovison, A., Masci, L.: Stochastic and geometric aspects of reduced reactiondiffusion dynamics. Ricerche Matematica 8, 1–16 (2018). CrossRefGoogle Scholar
  25. 25.
    Zayed, E.M.E., Rahman, H.M.A.: On using the modified variational iteration method for solving the nonlinear coupled equations in the mathematical physics. Ricerche di Matematica 59, 137–159 (2010)MathSciNetCrossRefGoogle Scholar
  26. 26.
    D’Amore, L., Arcucci, R., Pistoia, J., Toumi, R., Murli, A.: On the variational data assimilation problem solving and sensitivity analysis. J. Comput. Phys. 335, 311–326 (2017)MathSciNetCrossRefGoogle Scholar
  27. 27.
    D’Amore, L., Murli, A.: Regularization of a Fourier series method for the Laplace transform inversion with real data. Inverse Prob. 18(4), 1185–1205 (2002)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Arcucci, R. D’Amore, L., Carracciuolo, L.: On the problem-decomposition of scalable 4D-Var Data Assimilation models. In: Proceedings of the 2015 International Conference on High Performance Computing and Simulation, HPCS 2015, 2 September 2015, Article number 7237097, pp. 589–594 (2015)Google Scholar

Copyright information

© Università degli Studi di Napoli "Federico II" 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Applications “R.Caccioppoli”University of Naples Federico IINaplesItaly

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