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Numerical Analysis and Applications

, Volume 12, Issue 1, pp 51–69 | Cite as

A Newton–Kantorovich Method in Inverse Source Problems for Production-Destruction Models with Time Series-Type Measurement Data

  • A. V. PenenkoEmail author
Article
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Abstract

Algorithms for solving an inverse source problem for production–destruction systems of nonlinear ordinary differential equations with measurement data in the form of time series are presented. A sensitivity operator and its discrete analogue are constructed on the basis of adjoint equations. This operator relates perturbations of the sought-for parameters of the model to those of the measured values. The operator generates a family of quasi-linear operator equations linking the required unknown parameters and the data of the inverse problem. A Newton–Kantorovich method with right-hand side r-pseudo-inverse matrices is used to solve the equations. The algorithm is applied to solving an inverse source problem for an atmospheric pollution transformation model.

Keywords

inverse source problem big data Newton–Kantorovich method adjoint equations sensitivity operator r-pseudoinverse matrix right inverse 

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References

  1. 1.
    Vasiliev, F.P., Metody resheniya ekstremal’nykh zadach (Methods for Solving Extreme Problems), Moscow: Nauka, 1981.Google Scholar
  2. 2.
    Alifanov, O.M., Artyukhin, E.A., and Rumyantsev, S.V., ExtremeMethods for Solving Ill-Posed Problems with Applications to Inverse Heat Transfer Problems, New York: Begell House, 1995.zbMATHGoogle Scholar
  3. 3.
    Lions, J-L., Optimal Control of Systems Governed by Partial Differential Equations, Berlin: Springer-Verlag, 1971.CrossRefzbMATHGoogle Scholar
  4. 4.
    Shutyaev, V.P., The Properties of Control Operators in One Problem on Data Control and Algorithms for Its Solution, Math. Notes, 1995, vol. 57, no. 6, pp. 668–671.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Marchuk, G.I., On the Statement of Some Inverse Problems, Dokl. Akad. Nauk SSSR, 1964, vol. 156, no. 3, pp. 503–506.MathSciNetGoogle Scholar
  6. 6.
    Marchuk, G.I., Adjoint Equations and Analysis of Complex Systems, Amsterdam: Springer, 1995.CrossRefzbMATHGoogle Scholar
  7. 7.
    Issartel, J.-P., Rebuilding Sources of Linear Tracers after Atmospheric Concentration Measurements, Atm. Chem. Phys., 2003, vol. 3, no. 6, pp. 2111–2125.CrossRefGoogle Scholar
  8. 8.
    Issartel, J.-P., Emergence of a Tracer Source from Air Concentration Measurements, A New Strategy for Linear Assimilation, Atm. Chem. Phys., 2005, vol. 5, no. 1, pp. 249–273.CrossRefGoogle Scholar
  9. 9.
    Ustinov, E.A., Adjoint Sensitivity Analysis of Atmospheric Dynamics: Application to the Case of Multiple Observables, J. Atm. Sci., 2001, vol. 58, no. 21, pp. 3340–3348.MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bennett, A.F., Inverse Methods in Physical Oceanography, Cambridge: Cambridge Univ. Press, 1992.CrossRefzbMATHGoogle Scholar
  11. 11.
    Iglesias, M.A. and Dawson, C., An Iterative Representer-Based Scheme for Data Inversion in Reservoir Modeling, Inv. Problems, 2009, vol. 25, no. 3, pp. 1–34.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Le Dimet, F.X., Souopgui, I., Titaud, O., et al., Toward the Assimilation of Images, Nonlin. Process. Geophys., 2015, vol. 22, no. 1, pp. 15–32; http://www.nonlin-processes–geophys.net/22/15/2015/npg–22–15–2015.pdf.CrossRefGoogle Scholar
  13. 13.
    Penenko, A.V., On Solution of the Inverse Coefficient Heat Conduction Problem with a Gradient Projection Method, Sib. El. Mat. Izv., 2010, vol. 7, pp. 178–198.Google Scholar
  14. 14.
    Penenko, A.V., Nikolaev, S.V., Golushko, S.K., Romashchenko, A.V., and Kirilova, I.A., Numerical Algorithms for Diffusion Coefficient Identification in Problems of Tissue Engineering, Mat. Biol. Bioinform., 2016, vol. 11, no. 2, pp. 426–444.CrossRefGoogle Scholar
  15. 15.
    Goris, N. and Elbern, H., Singular Vector Decomposition for Sensitivity Analyses of Tropospheric Chemical Scenarios, Atm. Chem. Phys., 2013, vol. 13, no. 9, pp. 5063–5087.CrossRefGoogle Scholar
  16. 16.
    Kantorovich, L.V. and Akilov, G.P., Funktsional’nyi analiz (Functional Analysis),Moscow: Nauka, 1984.zbMATHGoogle Scholar
  17. 17.
    Cheverda, V.A., R-Pseudoinverses for Compact Operators in Hilbert Spaces: Existence and Stability, J. Inv. Ill-Posed Problems, 1995, vol. 3, no. 2, pp. 131–148.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Argyros, I.K., Local Convergence Theorems of Newton’s Method for Nonlinear Equations Using Outer or Generalized Inverses, Czechoslovak Math. J., 2000, vol. 50, iss. 3, pp. 603–614.Google Scholar
  19. 19.
    Penenko, A.V., Consistent Numerical Schemes for Solving Nonlinear Inverse Source Problems with Gradient-Type Algorithms and Newton–Kantorovich Methods, Num. An. Appl., 2018, vol. 11, no. 1, pp. 73–88.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Zhang, H., Linford, J.C., Sandu, A., and Sander, R., Chemical Mechanism Solvers in Air Quality Models, Atmosphere, 2011, vol. 2, no. 3, pp. 510–532.CrossRefGoogle Scholar
  21. 21.
    Vainikko, G.M. and Veretennikov, A.Yu., Iteratsionnye protsedury v nekorrektnykh zadachakh (Iteration Procedures in Ill-Posed Problems),Moscow: Nauka, 1986.zbMATHGoogle Scholar
  22. 22.
    GNU Scientific Library Reference Manual Edition 2.2.1, for GSL Version 2.2.1, 2009, https://www. gnu.org/software/gsl/manual/html_node/index_old.html.Google Scholar
  23. 23.
    Stockwell, W.R., Comment on “Simulation of a Reacting Pollutant Puff Using an Adaptive Grid Algorithm” by R.K. Srivastava et al., J. Geophys. Res., 2002, vol. 107, no. D22, pp. 4643–4650.Google Scholar
  24. 24.
    Visscher, A.De., Air Dispersion Modeling: Foundations and Applications,Wiley, 2013.CrossRefGoogle Scholar
  25. 25.
    Shapiro, B., xCellerator User’s Guide, 2012, http://xlr8r.info/usersguide/index.html.Google Scholar
  26. 26.
    Bocquet, M., Elbern, H., Eskes, H., et al., Data Assimilation in Atmospheric Chemistry Models: Current Status and Future Prospects for Coupled Chemistry Meteorology Models, Atm. Chem. Phys. Discuss., 2014, vol. 14, no. 23, pp. 32233–32323.CrossRefGoogle Scholar
  27. 27.
    Schaap, M., Roemer, M., Sauter, F., Boersen, G., Timmermans, R., and Builtjes, P.G.H., Lotos-Euros Documentation, 2005 (techreport: B&O / 297 / TNO report).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Institute of Computational Mathematics and Mathematical Geophysics, Siberian BranchRussian Academy of SciencesNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia

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