Source indentification for the Smoluchowski equation

A. V. Penenko ; A. B. Salimova

Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 23 (2020) no. 2, pp. 183-199.

Résumé

A source identification algorithm for the systems of nonlinear ordinary differential equations of the production-destruction type is applied to the inverse problem for the discretized Smoluchowski equation. An unknown source function is estimated by time series of measurements of the specific size particles concentration. Based on an ensemble of adjoint equations solutions, the sensitivity operator is constructed that links the 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 is used with $r$-pseudoinverse matrices. The efficiency and properties of the algorithm are numerically studied.

Bibliographie

[1] Pope C. Arden, W. Dockery Douglas, “Health effects of fine particulate air pollution: lines that connect”, J. of the Air Waste Management Association, 56:6 (2006), 709–742 | DOI

[2] J. H. Seinfeld, S. N. Pandis, “Atmospheric chemistry and physics”, Air Pollution to Climate Change, 2nd edition, John Wiley Sons, 2006

[3] M. V. Smoluchowski, “Drei vortrage uber diffusion, brownsche bewegung und koagulation von kolloidteilchen”, Physik. Zeit., 17 (1916), 557–585

[4] M. V. Smoluchowski, “Uber brownsche molekularbewegung unter einwirkung äußerer Kräfte und deren zusammenhang mit der verallgemeinerten diffusionsgleichung”, Annalten der Physik, 353:24 (1916), 1103–1112 | DOI

[5] Hans Mueller, “Zur allgemeinen theorie ser raschen koagulation”, Kolloidchemische Beihefte, 27:6–12 (1928), 223–250

[6] V. A. Galkin, “Uravnenie Smolukhovskogo”, Fizmatlit, M., 2001

[7] A. E. Aloyan, “Modelirovanie dinamiki i kinetiki gazovykh primesei i aerozolei v atmosfere”, Nauka, M., 2008

[8] S. A. Matveev, A. P. Smirnov, E. E. Tyrtyshnikov, “A fast numerical method for the Cauchy problem for the Smoluchowski equation”, J. of Computational Physics, 282 (2015), 23–32 | DOI | MR | Zbl

[9] Matveev S.A, Krapivsky P.L, A. P. Smirnov, E. E. Tyrtyshnikov, N. V. Brilliantov, “Oscillations in aggregation-shattering processes”, Physical Review Letters, 119:26 (2017), 1–10 | DOI | MR

[10] V. I. Agoshkov, Metody optimal'nogo upravleniya i sopryazhennykh uravnenii v zadachakh matematicheskoi fiziki, IVM RAN, M., 2003

[11] Mirzaev Inom, C. Byrne Erin, Bortz David, “An inverse problem for a class of conditional probability measure-dependent evolution equations”, Inverse Problems, 32:9 (2016), 095005 | DOI | MR | Zbl

[12] G. I. Marchuk, “O postanovke nekotorykh obratnykh zadach”, DAN SSSR, 156:3 (1964), 503–506 | Zbl

[13] G. I. Marchuk, Sopryazhennye uravneniya i analiz slozhnykh sistem, Nauka, M., 1992

[14] J. P. Issartel, “Rebuilding sources of linear tracers after atmospheric concentration measurements”, Atmospheric Chemistry and Physics, 3:6 (2003), 2111–2125 | DOI

[15] J. P. Issartel, “Emergence of a tracer source from air concentration measurements, a new strategy for linear assimilation”, Atmospheric Chemistry and Physics, 5:1 (2005), 249–273 | DOI

[16] Evensen Geir, “Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error”, J. of Geophysical Research, 99:C5, 10143 | MR

[17] V. I. Agoshkov, P. B. Dubovski, “Solution of the reconstruction problem of a source function in the coagulation-fragmentation equation”, Russ. J. of Numerical Analysis and Mathematical Modelling, 17:4 (2002), 319–330 | DOI | MR | Zbl

[18] F. Bennett Andrew, Inverse Methods in Physical Oceanography, Cambridge Monographs on Mechanics, Cambridge University Press, Cambridge, 1992 | MR | Zbl

[19] A. L. Karchevsky, “Reformulation of an inverse problem statement that reduces computational costs”, Eurasian J. of Mathematical and Computer Applications, 1:2 (2013), 4–20 | DOI | MR

[20] A. V. Penenko, “Soglasovannye chislennye skhemy dlya resheniya nelineinykh obratnykh zadach identifikatsii istochnikov gradientnymi algoritmami i metodami N'yutona–Kantorovicha”, Sib. zhurn. vychisl. matematiki, 21:1 (2018), 99–116 | MR | Zbl

[21] A. V. Penenko, “Metod N'yutona–Kantorovicha dlya resheniya obratnykh zadach identifikatsii istochnikov v modelyakh produktsii-destruktsii s dannymi tipa vremennykh ryadov”, Sib. zhurn. vychisl. matematiki, 22:1 (2019), 57–79 | MR

[22] F. X. Le Dimet, I. Souopgui, O. Titaud et al., “Toward the assimilation of images”, Nonlinear Processes in Geophysics, 22:1 (2015), 15–32 | DOI

[23] A. V. Penenko, “O reshenii obratnoi koeffitsientnoi zadachi teploprovodnosti metodom proektsii gradienta”, Sibirskie elektronnye matematicheskie izvestiya, 23, Tr. Pervoi mezhdunarodnoi molodezhnoi shkoly-konferentsii «Teoriya i chislennye metody resheniya obratnykh i nekorrektnykh zadach». Chast' I (2010), 178–198

[24] A. V. Penenko, S. V. Nikolaev, S. K. i dr. Golushko, “Chislennye algoritmy identifikatsii koeffitsienta diffuzii v zadachakh tkanevoi inzhenerii”, Mat. biol. i bioinf., 11:2 (2016), 426–444 | DOI

[25] A. V. Penenko, A. A. Sorokovoi, K. E. Sorokovaya, “Chislennaya model' transformatsii bioaerozolei v atmosfere”, Optika atmosfery i okeana, 29:6 (2016), 462–466

[26] Hesstvedt Eigil, Hov Oystein, Ivar S. A. Isaksen, “Quasi-steady-state approximations in air pollution modeling: Comparison of two numerical schemes for oxidant prediction”, Int. J. of Chemical Kinetics, 10:9 (1978), 971–994 | DOI

[27] 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