Consistent numerical schemes for solving nonlinear inverse source problems with the gradient-type algorithms and the Newton--Kantorovich methods

A. V. Penenko

Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 21 (2018) no. 1, pp. 99-116.

Résumé

The algorithms of solving the inverse source problem for systems of the production-destruction equations are considered. Consistent in the sense of the Lagrangian identity numerical schemes for solving direct and conjugate problems have been built. With the adjoint equations, the sensitivity operator and its discrete analogue have been constructed. It links the measured values perturbations with the perturbations of the model parameters. This operator transforms the inverse problem to a quasilinear form and allows applying the Newton–Kantorovich methods to it. The paper provides a numerical comparison of the gradient algorithms based on the consistent and inconsistent numerical schemes and the Newton–Kantorovich algorithm applied to solving the inverse source problem for the nonlinear Lorenz model.

Bibliographie

[1] Vasilev F. P., Metody resheniya ekstremalnykh zadach, Nauka, M., 1981 | MR

[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, New York, 1995 | MR

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

[4] Bocquet M., Elbern H., Eskes H., Hirtl M., Zabkar R., Carmichael G. R., Flemming J., Inness A., Pagowski M., Camano J. L. P., Saide P. E., Jose R. S., Sofiev M., Vira J., Baklanov A., Carnevale C., Grell G., Seigneur C., “Data assimilation in atmospheric chemistry models: current status and future prospects for coupled chemistry meteorology models”, Atmospheric Chemistry and Physics Discussions, 15 (2015), 5325–5358 | DOI

[5] Penenko V. V., Metody chislennogo modelirovaniya atmosfernykh protsessov, Gidrometeoizdat, L., 1981 | MR

[6] Ahrens B., “Variational data assimilation for a Lorenz model using a non-standard genetic algorithm”, Meteorology and Atmospheric Physics, 70:3 (1999), 227–238 | DOI | MR

[7] 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

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

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

[10] Pudykiewicz J. A., “Application of adjoint tracer transport equations for evaluating source parameters”, Atmospheric Environment, 32 (1998), 3039–3050 | DOI

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

[12] Penenko A. V., “O reshenii obratnoi koeffitsientnoi zadachi teploprovodnosti metodom proektsii gradienta”, Tr. Pervoi mezhdunarodnoi molodezhnoi shkoly-konferentsii “Teoriya i chislennye metody resheniya obratnykh i nekorrektnykh zadach”, Chast I, Sibirskie elektronnye matematicheskie izvestiya, 7 (2010), C.178–C.198

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

[14] Kantorovich L. V., Akilov G. P., Funktsionalnyi analiz, Nauka, Glavnaya redaktsiya fiziko-matematicheskoi literatury, M., 1984 | MR

[15] Daescu D., Carmichael G. R., Sandu A., “Adjoint implementation of rosenbrock methods applied to variational data assimilation problems”, J. of Computational Physics, 165 (2000), 496–510 | DOI

[16] Karchevsky A. L., “A proper flow chart for a numerical solution of an inverse problem by an optimization method”, Numerical Analysis and Applications, 1:2 (2008), 114–122 | DOI

[17] Lorenz V., “Deterministic nonperiodic flow”, J. of the Atmospheric Sciences, 20 (1963), 130–141 | 2.0.CO;2 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI

[18] Evensen G., Fario N., “Solving for the generalized inverse of the Lorenz model”, J. of the Meteorological Society of Japan, 75:1B (1997), 229–243 | DOI

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

[20] Penenko V., Tsvetova E., Penenko A., “Variational approach and Euler's integrating factors for environmental studies”, Computers Mathematics with Applications, 67 (2014), 2240–2256 | DOI | MR

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

[22] Jay L. O., Sandu A., Potra F. A., Carmichael G. R., “Improved QSSA methods for atmospheric chemistry integration”, SIAM J. on Scientific Computing, 18:1 (1997), 182–202 | DOI | MR

[23] Elbern H., Schmidt H., Ebel A., “Variational data assimilation for tropospheric chemistry modeling”, J. of Geophysical Research: Atmospheres, 102 (1997), 15967–15985 | DOI

[24] Wolfram language system documentation center: Nonlinear conjugate gradient methods, http://reference.wolfram.com/language/tutorial/UnconstrainedOptimizationConjugateGradientwMethods.html

[25] Wolfram language system documentation center: Newton's method, http://reference.wolfram.com/language/tutorial/UnconstrainedOptimizationNewtonsMethodRoot.html