Source Identification for the Smoluchowski Equation Using an Ensemble of Adjoint Equation Solutions

ABSTRACT

A source identification algorithm for systems of nonlinear ordinary differential equations of production-destruction type is applied to an inverse problem for a discretized Smoluchowski equation. An unknown source function is estimated by time series of measurements of particle concentrations of a specific size. Based on an ensemble of adjoint equation solutions, a sensitivity operator is constructed that relates perturbations of the sought-for model parameters with perturbations of the measured values. This reduces the inverse problem to a family of quasilinear operator equations. To solve the equations, an algorithm of the Newton–Kantorovich type with \(r\)-pseudoinverse matrices is used. The efficiency and properties of the algorithm are studied numerically.

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REFERENCES

  1. 1

    Pope, C.A., III and Dockery, D.W., Health Effects of Fine Particulate Air Pollution: Lines That Connect, J. Air Waste Manag. Ass., 2006, vol. 56, iss. 6, pp. 709–742.

  2. 2

    Seinfeld, J.H. and Pandis, S.N., Atmospheric Chemistry and Physics, Air Pollution to Climate Change, 2nd ed., Wiley, 2006; ISBN: 978-0-471-7218-8.

  3. 3

    Smoluchowski, M.V., Drei Vortrage über Diffusion, Brownsche Bewegung und Koagulation von Kolloidteilchen, Physik. Zeit., 1916, vol. 17, pp. 557–585.

  4. 4

    Smoluchowski, M.V., Über Brownsche Molekularbewegung unter Einwirkung äußerer Kräfte und deren Zusammenhang mit der verallgemeinerten Diffusionsgleichung, Annalten der Physik, 1916, vol. 353, no. 24, pp. 1103–1112.

  5. 5

    Mueller, H., Zur allgemeinen Theorie ser raschen Koagulation,Kolloidchemische Beihefte, 1928, vol. 27, nos. 6–12, pp. 223–250.

  6. 6

    Galkin, V.A., Uravnenie Smolukhovskogo (Smoluchowski Equation), Moscow: Fizmatlit, 2001.

  7. 7

    Aloyan, A.E., Modelirovanie dinamiki i kinetiki gazovykh primesei i aerozolei v atmosfere (Modeling of Dynamics and Kinetics of Gaseous Admixtures and Aerosols in the Atmosphere), Moscow: Nauka, 2008.

  8. 8

    Matveev, S.A., Smirnov, A.P., and Tyrtyshnikov, E.E., A Fast Numerical Method for the Cauchy Problem for the Smoluchowski Equation,J. Comput. Phys., 2015, vol. 282, pp. 23–32.

  9. 9

    Matveev, S.A., Krapivsky, P.L, Smirnov, A.P., Tyrtyshnikov, E.E., and Brilliantov, N.V., Oscillations in Aggregation-Shattering Processes, Phys. Rev. Lett., 2017, vol. 119, no. 26, pp. 1–10; DOI: 10.1103/PhysRevLett.119.260601.

  10. 10

    Agoshkov, V.I., Metody optimal’nogo upravleniya i sopryazhennykh uravnenii v zadachakh matematicheskoi fiziki (Methods of Optimal Control and Adjoint Equations in Problems of Mathematical Physics), Moscow: Institute of Numerical Mathematics RAS, 2003.

  11. 11

    Mirzaev, I., Byrne, E.C., and Bortz, D.M., An Inverse Problem for a Class of Conditional Probability Measure-Dependent Evolution Equations, Inv. Probl., 2016, vol. 32, no. 9, p. 095005.

  12. 12

    Marchuk, G.I., O postanovke nekotorykh obratnykh zadach (On the Statement of Some Inverse Problems),Dokl. Akad. Nauk SSSR, 1964, vol. 156, no. 3, pp. 503–506.

  13. 13

    Marchuk, G.I., Sopryazhennye uravneniya i analiz slozhnykh sistem (Adjoint Equations and Analysis of Complex Systems), Moscow: Nauka, 1992.

  14. 14

    Issartel, J.P., Rebuilding Sources of Linear Tracers after Atmospheric Concentration Measurements, Atm. Chem. Phys., 2003, vol. 3, no. 6, pp. 2111–2125.

  15. 15

    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.

  16. 16

    Evensen, G., Sequential Data Assimilation with a Nonlinear Quasi-Geostrophic Model Using Monte Carlo Methods to Forecast Error,J. Geophys. Res., vol. 99, no. C5, p. 10143.

  17. 17

    Agoshkov, V.I. and Dubovski, P.B., Solution of the Reconstruction Problem of a Source Function in the Coagulation-Fragmentation Equation, Russ. J. Num. An. Math. Model., 2002, vol. 17, iss. 4, pp. 319–330.

  18. 18

    Bennett, A.F., Inverse Methods in Physical Oceanography (Cambridge Monographs on Mechanics), Cambridge: Cambridge University Press, 1992.

  19. 19

    Karchevsky, A.L., Reformulation of an Inverse Problem Statement that Reduces Computational Costs, Euras. J. Math. Comp. Appl., 2013, vol. 1, no. 2, pp. 4–20.

  20. 20

    Penenko, A.V., Consistent Numerical Schemes for Solving Nonlinear Inverse Source Problems with Gradient-Type Algorithms and Newton–Kantorovich Methods, Sib. Zh. Vych. Mat., 2018, vol. 21, no. 1, pp. 99–116.

  21. 21

    Penenko, A.V., A Newton–Kantorovich Method in Inverse Source Problems for Production-Destruction Models with Time Series-Type Measurement Data, Sib. Zh. Vych. Mat., 2019, vol. 22, no. 1, pp. 57–79.

  22. 22

    Le Dimet, F.-X., Souopgui, I., Titaud, O., et al., Toward the Assimilation of Images, Nonlin. Proc. Geophys., 2015, vol. 22, no. 1, pp. 15–32.

  23. 23

    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.

  24. 24

    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.

  25. 25

    Penenko, A.V., Sorokovoy, A.A., and Sorokovaya, K.E., Numerical Model for Bioaerosol Transformation in the Atmosphere, Opt. Atm. Ok., 2016, vol. 29, no. 6, pp. 462–466.

  26. 26

    Hesstvedt, E., Hov, O., and Isaksen, I.S.A., Quasi-Steady-State Approximations in Air Pollution Modeling: Comparison of Two Numerical Schemes for Oxidant Prediction, Int. J. Chem. Kinet., 1978, vol. 10, no. 9, pp. 971–994.

  27. 27

    GNU Scientific Library Reference Manual Edition 2.2.1, for GSL Version 2.2.1, 2009; Access mode: https://www.gnu.org/software/gsl/manual/html/index

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Acknowledgements

The authors would like to thank E.A. Pyanova and E.A. Tsvetova for their helpful comments and advice.

Funding

This work (the development and investigation of the algorithms) was supported by the Russian Foundation for Basic Research, project no. 17-71-10184.

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Correspondence to A. V. Penenko or A. B. Salimova.

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Penenko, A.V., Salimova, A.B. Source Identification for the Smoluchowski Equation Using an Ensemble of Adjoint Equation Solutions. Numer. Analys. Appl. 13, 152–164 (2020). https://doi.org/10.1134/S1995423920020068

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Keywords

  • Smoluchowski equation
  • inverse source problem
  • Newton--Kantorovich method
  • adjoint equations
  • sensitivity operator