PICARD-LINDELOF ITERATIONS AND
MULTIPLE SHOOTING METHOD FOR
PARAMETER ESTIMATION
Stanislav Slavov1, Tsvetelin Tsvetkov2 1Department of Physics
University of Chemical Technology and Metallurgy
8, Kliment Ohridski, Blvd., Sofia, 1756, BULGARIA 2Department of Mathematics
University of Chemical Technology and Metallurgy
8, Kliment Ohridski, Blvd., Sofia, 1756, BULGARIA
In this article, we modify the Picard-Lindelöf iteration scheme in order to show an iteration algorithm for parameter estimation of ordinary differential equations. The proposed algorithm inherited the advantages exhibited in the classical algorithms and, moreover, the parameters can be transformed to a form that are convenient and suitable for computation. In the end, a numerical example has also been discused to highlight the results.
You will need Adobe Acrobat reader. For more information and free download of the reader, please follow this link.
References
[1] J.P.N. Bishwal, Parameter Estimation in Stochastic Differential Equations,
Lecture Notes in Math., 1923, Springer, Berlin, 2008.
[2] H.G. Bock, Numerical treatment of inverse problems in chemical reaction
kinetics, In: Modelling of Chemical Reaction Systems (Ed-s: K. Ebert, P.
Deuflhard, and W. Jager), Springer, 1981, 102-125.
[3] H.G. Bock, Recent advances in parameter identification techniques for ordinary
differential equations, In: Numerical Treatment of Inverse Problems
in Differential and Integral Equations (Ed-s: P. Deuflhard and E. Hairer),
Birkha¨user, 1983, 95-121.
[4] H.G. Bock, Randwertproblemmethoden zur Parameteridenti Fizierung in
Systemen Nichtlinearer Differentialgleichungen, PhD Thesis, Universitat
Bonn, 1987.
[5] N. J-B. Brunel, Parameter estimation of ODE's via nonparametric estimators,
Electronic Journal of Statistics, 2 (2008), 1242-1267, doi: 10.1214/07-
EJS132.
[6] A. Dishliev, K. Dishlieva, S. Nenov, Specific Asymptotic Properties of the
Solutions of Impulsive Differential Equations. Methods and Applications,
Academic Publication, 2011.
[7] N. Kyurkchiev, On a sigmoidal growth function generated by reaction
networks. some extensions and applications, Communications in Applied
Analysis, 23, No. 3 (2019), 383-400.
[8] N. Kyurkchiev, A. Iliev, and A. Rahnev, On a special choice of nutrient
supply for cell growth in a continuous bioreactor. Some modeling and approximation
aspects, Dynamic Systems and Applications, 28, No. 3 (2019), 587-606.
[9] V. Kyurkchiev, A. Iliev, A. Rahnev and N. Kyurkchiev, Some New Logistic
Differential Models: Properties and Applications, Lambert, 2019.
[10] H. Miao, X. Xia, A.S. Perelson, Hulin Wu, On identifiability of nonlinear
ode models and applications in viral dynamics, SIAM Review, 53, No. 1
(2011), 3-39, doi: 10.1137/090757009.
[11] S.I. Nenov, Impulsive controllability and optimization problems in population
dynamics, Nonlinear Analysis: Theory, Methods & Applications,
Pergamon, 36, No. 7 (1999).
[12] A. Papavasiliou, Ch. Ladroue, Parameter estimation for rough differential
equations, The Annals of Statistics, 39, No. 4 (2011), 2047-2073, doi:
10.1214/11-AOS893.
[13] D. Rocha, S. Gouveia, C. Pinto, M. Scotto, J.N. Tavares, E. Valadas,
and L.F. Caldeira, On the parameters estimation of HIV dynamic models,
REVSTAT – Statistical Journal, 17, No. 2 (2019), 209-222.
[14] D.S. Stratiev, R.K. Dinkov, I.K. Shishkova, A.D. Nedelchev, T. Tsaneva,
E. Nikolaychuk, I.M. Sharafutdinov, N. Rudney, S. Nenov, M. Mitkova,
M. Skunov, D. Yordanov, An investigation on the feasibility of simulating
the distribution of the boiling point and molecular Weight of heavy oils,
Petroleum Science and Technology, 33, No. 5 (2015), 527-541.