The Newton--Kantorovich method in inverse source problems for production-destruction models with time series-type measurement data

A. V. Penenko

Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 22 (2019) no. 1, pp. 57-79.

Résumé

The algorithms for solving the inverse source problem for the production–destruction type systems of nonlinear ordinary differential equations with measurement data in the form of time series are presented. The sensitivity operator and its discrete analogue on the basis of adjoint equations are constructed. This operator binds the perturbations in the unknown parameters of the model to those of the measured values. The operator allows one to construct a family of quasi-linear operator equations linking the required unknown parameters and the data of the inverse problem. The Newton–Kantorovich type method with right-hand side $r$-pseudoinverse matrices is used to solve the equations. The algorithm is applied to solving the inverse source problem for the atmospheric impurities transformation model.

Bibliographie

[1] Vasiliev F. P., Metody resheniya ekstremal'nykh zadach, Nauka, M., 1981

[2] Alifanov O. M., Artiukhin E. A., Rumiantsev S. V., Extreme Methods for Solving Ill-Posed Problems With Applications to Inverse Heat Transfer Problems, Begell House, 1995 | MR | Zbl

[3] Lions J., Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971 | MR | Zbl

[4] Shutyaev V. P., “The properties of control operators in one problem on data control and algorithms for its solution”, Mathematical Notes, 57:6 (1995), 668–671 | DOI | MR | Zbl

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

[6] Marchuk G. I., Adjoint Equations and Analysis of Complex Systems, Springer, Netherlands, 1995 | MR

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

[8] Issartel J.-P., “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

[9] Ustinov E. A., “Adjoint sensitivity analysis of atmospheric dynamics: Application to the case of multiple observables”, J. of the Atmospheric Sciences, 58:21 (2001), 3340–3348 | 2.0.CO;2 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR

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

[11] Iglesias M. A., Dawson C., “An iterative representer-based scheme for data inversion in reservoir modeling C. Dawson”, Inverse Problems, 25:3 (2009), 1–34 | DOI | MR

[12] Le Dimet F. X., Souopgui I., Titaud O. et al., “Toward the assimilation of images”, Nonlinear Processes in Geophysics, 22:1 (2015), 15–32 http://www.nonlin-processes-geophys.net/22/15/2015/npg-22-15-2015.pdf | DOI

[13] Penenko A. V., “O reshenii obratnoy koefficientnoy zadachi teploprovodnosti metodom proekcii gradienta”, Sb. Tr. Pervoy mezhdunarodnoy molodezhnoy shkoly-konferencii “Teoriya i chislennye metody resheniya obratnykh i nekorrektnykh zadach”. Chast' I, Sibirskie elektronnye matematicheskie izvestiya, 7 (2010), C.178–C.198

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

[15] Goris N., Elbern H., “Singular vector decomposition for sensitivity analyses of tropospheric chemical scenarios”, Atmospheric Chemistry and Physics, 13:9 (2013), 5063–5087 | DOI

[16] Kantorovich L. V., Akilov G. P., Functional Analysis, Pergamon Press, 1982 | MR | Zbl

[17] Cheverda V. A., “R-pseudoinverses for compact operators in Hilbert spaces: existence and stability”, J. of Inverse and Ill-Posed Problems, 3:2 (1995), 131–148 | DOI | MR | Zbl

[18] Argyros I. K., “Local convergence theorems of Newton's method for nonlinear equations using outer or generalized inverses”, Czechoslovak Mathematical Journal, 50:3 (2000), 603–614 | DOI | MR | Zbl

[19] Penenko A. V., “Consistent numerical schemes for solving nonlinear inverse source problems with gradient-type algorithms and Newton–Kantorovich methods”, Numerical Analysis and Applications, 11:1 (2018), 73–88 | DOI | MR | Zbl

[20] Zhang H., Linford J. C., Sandu A., Sander R., “Chemical mechanism solvers in air quality models”, Atmosphere, 2:3 (2011), 510–532 | DOI

[21] Vainikko G. M., Veretennikov A. Yu., Iteracionnye procedury v nekorrektnykh zadachakh, Nauka, M., 1986

[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

[23] Stockwell W. R., Goliff W. S., “Comment on “simulation of a reacting pollutant puff using an adaptive grid algorithm” by R. K. Srivastava et al.”, J. of Geophysical Research, 107:D22 (2002), 4643–4650 | DOI

[24] Visscher A. De., Air Dispersion Modeling: Foundations and Applications, John Wiley Sons Inc, 2013

[25] Shapiro B., xCellerator User's Guide, , 2012 http://xlr8r.info/usersguide/index.html

[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”, Atmospheric Chemistry and Physics Discussions, 14:23 (2014), 32233–32323 | DOI

[27] Schaap M., Roemer M., Sauter F., Boersen G., Timmermans R., Builtjes P. G. H., Lotos-euros documentation, techreport: B/297/TNO report, 2005