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Ab-initio Functional Decomposition of Kalman Filter: A Feasibility Analysis on Constrained Least Squares Problems

  • Luisa D’AmoreEmail author
  • Rosalba Cacciapuoti
  • Valeria Mele
Conference paper
  • 116 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12044)

Abstract

The standard formulation of Kalman Filter (KF) becomes computationally intractable for solving large scale state space estimation problems as in ocean/weather forecasting due to matrix storage and inversion requirements. We introduce an innovative mathematical/numerical formulation of KF using Domain Decomposition (DD) approach. The proposed DD approach partitions ab-initio the whole KF computational method giving rise to local KF methods that can be solved independently. We present its feasibility analysis using the constrained least square model underlying variational Data Dssimilation problems. Results confirm that the accuracy of solutions of local KF methods are not impaired by DD approach.

Keywords

Kalman Filter Domain Decomposition Data Assimilation Constrained Least Square Problem 

References

  1. 1.
    Antonelli, L., Carracciuolo, L., Ceccarelli, M., D’Amore, L., Murli, A.: Total variation regularization for edge preserving 3D SPECT imaging in high performance computing environments. In: Sloot, P.M.A., Hoekstra, A.G., Tan, C.J.K., Dongarra, J.J. (eds.) ICCS 2002. LNCS, vol. 2330, pp. 171–180. Springer, Heidelberg (2002).  https://doi.org/10.1007/3-540-46080-2_18CrossRefGoogle Scholar
  2. 2.
    Arcucci, R., D’Amore, L., 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
  3. 3.
    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 Prog. 45, 1214–1235 (2017).  https://doi.org/10.1007/s10766-016-0460-3. ISSN 0885–7458CrossRefGoogle Scholar
  4. 4.
    Arcucci, R., D’Amore, L., Carracciuolo, L.: On the problem-decomposition of scalable 4D-Var Data Assimilation model. In: Proceedings of the 2015 International Conference on High Performance Computing and Simulation, HPCS 2015, 2 September 2015, 13th International Conference on High Performance Computing and Simulation, HPCS 2015, Amsterdam, Netherlands, 20 July 2015 through 24 July 2015, pp. 589–594 (2015)Google Scholar
  5. 5.
    Arcucci, R., D’Amore, L., Celestino, S., Laccetti, G., Murli, A.: A scalable numerical algorithm for solving Tikhonov Regularization problems. In: Wyrzykowski, R., Deelman, E., Dongarra, J., Karczewski, K., Kitowski, J., Wiatr, K. (eds.) PPAM 2015. LNCS, vol. 9574, pp. 45–54. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-32152-3_5CrossRefGoogle Scholar
  6. 6.
    Bertero, M., et al.: MedIGrid: a medical imaging application for computational grids. In: Proceedings International Parallel and Distributed Processing Symposium (2003).  https://doi.org/10.1109/IPDPS.2003.1213457
  7. 7.
    D’Amore, L., Campagna, R., Mele, V., Murli, A., Rizzardi, M.: ReLaTIve. An Ansi C90 software package for the Real Laplace Transform Inversion. Numer. Algorithms 63(1), 187–211 (2013).  https://doi.org/10.1007/s11075-012-9636-0
  8. 8.
    D’Amore, L., Mele, V., Laccetti, G., Murli, A.: Mathematical approach to the performance evaluation of matrix multiply algorithm. In: Wyrzykowski, R., Deelman, E., Dongarra, J., Karczewski, K., Kitowski, J., Wiatr, K. (eds.) PPAM 2015. LNCS, vol. 9574, pp. 25–34. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-32152-3_3CrossRefGoogle Scholar
  9. 9.
    D’Amore, L., Campagna, R., Mele, V., Murli, A.: Algorithm 946. ReLIADiff - a C++ software package for Real Laplace transform inversion based on automatic differentiation. ACM Trans. Math. Softw. 40(4), 31:1–31:20 (2014). article 31.  https://doi.org/10.1145/2616971
  10. 10.
    D’Amore, L., Cacciapuoti, R.: A note on domain decomposition approaches for solving 3D variational data assimilation models. Ricerche mat. (2019).  https://doi.org/10.1007/s11587-019-00432-4MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    D’Amore, L., Arcucci, R., Carracciuolo, L., Murli, A.: A scalable approach for variational data assimilation. J. Sci. Comput. 61, 239–257 (2014).  https://doi.org/10.1007/s10915-014-9824-2. ISSN 0885–7474MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    D’Amore, L., Campagna, R., Galletti, A., Marcellino, L., Murli, A.: A smoothing spline that approximates Laplace transform functions only known on measurements on the real axis. Inverse Prob. 28(2) (2012)Google Scholar
  13. 13.
    D’Amore, L., Laccetti, G., Romano, D., Scotti, G., Murli, A.: Towards a parallel component in a GPU–CUDA environment: a case study with the L-BFGS Harwell routine. Int. J. Comput. Math. 92(1) (2015).  https://doi.org/10.1080/00207160.2014.899589
  14. 14.
    D’Amore, L., Mele, V., Romano, D., Laccetti, G., Romano, D.: A multilevel algebraic approach for performance analysis of parallel algorithms. Comput. Inform. 38(4) (2019).  https://doi.org/10.31577/cai_2019_4_817
  15. 15.
    Evensen, G.: The Ensemble Kalman Filter: theoretical formulation and practical implementation. Ocean Dynam. 53, 343–367 (2003) CrossRefGoogle Scholar
  16. 16.
    Gander, M.J.: Schwarz methods over the course of time. ETNA 31, 228–255 (2008)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Gander, W.: Least squares with a quadratic constraint. Numer. Math. 36, 291–307 (1980)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Hannachi, A., Jolliffe, I.T., Stephenson, D.B.: Empirical orthogonal functions and related techniques in atmospheric science: a review. Int. J. Climatol. 1152, 1119–1152 (2007)CrossRefGoogle Scholar
  19. 19.
    Kalman, R.E.: A new approach to linear filtering and prediction problems. Trans. ASME J. Basic Eng. 82, 35–45 (1960)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Murli, A., D’Amore, L., Laccetti, G., Gregoretti, F., Oliva, G.: A multi-grained distributed implementation of the parallel Block Conjugate Gradient algorithm. Concur. Comput. Pract. Exp. 22(15), 2053–2072 (2010)Google Scholar
  21. 21.
    Rozier, D., Birol, F., Cosme, E., Brasseur, P., Brankart, J.M., Verron, J.: A reduced-order Kalman filter for data assimilation in physical oceanography. SIAM Rev. 49(3), 449–465 (2007)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Sorenson, H.W.: Least square estimation:from Gauss to Kalman. IEEE Spectr. 7, 63–68 (1970)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Luisa D’Amore
    • 1
    Email author
  • Rosalba Cacciapuoti
    • 1
  • Valeria Mele
    • 1
  1. 1.University of Naples Federico IINaplesItaly

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