Nonlinear system identification employing automatic differentiation

Communications in Nonlinear Science and Numerical Simulation

Volumn 18, Issue 10, 2013, Pages 2733-2742

  Schumann Bischoff, Jan ^a,b,c   Luther, Stefan ^a,b,c   Parlitz, Ulrich ^a,b,c  

^a MAX PLANCK INSTITUTE FOR DYNAMICS AND SELF ORGANIZATION   (Germany)

^b UNIVERSITY OF GÖTTINGEN   (Germany)

^c UNIVERSITY MEDICAL CENTER GÖTTINGEN   (Germany)

Author keywords

Data assimilation; Delay differential equations; Nonlinear modelling; Parameter estimation

Indexed keywords

DIFFERENTIAL EQUATIONS; DIFFERENTIATION (CALCULUS); JACOBIAN MATRICES; NONLINEAR EQUATIONS; NONLINEAR SYSTEMS; PARAMETER ESTIMATION;

AUTOMATIC DIFFERENTIATIONS; COST-FUNCTION; DATA ASSIMILATION; DELAY DIFFERENTIAL EQUATIONS; NON-LINEAR MODELLING; NON-LINEAR SYSTEM IDENTIFICATION; OPTIMISATIONS; PARAMETERS ESTIMATION; STATE AND PARAMETERS ESTIMATIONS; STATE ESTIMATION METHODS;

COST FUNCTIONS;

EID: 84877593261     PISSN: 10075704     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cnsns.2013.02.017     Document Type: Article

References (33)

1. Clayton R.H., Bernus O., Cherry E.M., Dierckx H., Fenton F.H., Mirabella L., et al. Challenges and open questions. Prog Biophys Mol Biol 2011, 104:22-48.
2. Parlitz U., Junge L., Kocarev L. Synchronization based parameter estimation from time series. Phys Rev E 1996, 54:6253-6529.
3. Voss H.U., Timmer J., Kurths J. Nonlinear dynamical system identification from uncertain and indirect measurements. Int J Bifurcation Chaos 2004, 14:1905-1933.
4. Ghosh D., Banerjee S. Adaptive scheme for synchronization-based multiparameter estimation from a single chaotic time series and its applications. Phys Rev E 2008, 78:056211.
5. Abarbanel H.D.I., Creveling D.R., Jeanne J.M. Estimation of parameters in nonlinear systems using balanced synchronization. Phys Rev E 2008, 77:016208.
6. Sorrentino F., Ott E. Using synchronization of chaos to identify the dynamics of unknown systems. Chaos 2009, 19:033108.
7. Szendro I.G., Rodríguez M.A., López J.M. On the problem of data assimilation by means of synchronization. J Geophys Res 2009, 114:D20109.
8. Yu W., Chen G., Cao J., Lü J., Parlitz U. Parameter identification of dynamical systems from time series. Phys Rev E 2007, 75:067201.
9. Yu D., Parlitz U. Estimating parameters by autosynchronization with dynamic restrictions. Phys Rev E 2008, 77:066221.
10. Ghosh D. Nonlinear-observer based synchronization scheme for multiparameter estimation. EPL 2008, 84:40012.
11. Fairhurst D., Tyukin I., Nijmeijer H., van Leeuwen C. Observers for canonic models of neural oscillators. Math Model Nat Phenom 2010, 5(3):146-185.
12. Creveling D.R., Gill P.E., Abarbanel H.D.I. State and parameter estimation in nonlinear systems as an optimal tracking problem. Phys Lett A 2008, 372:2640-2644.
13. Quinn J.C., Bryant P.H., Creveling D.R., Klein S.R., Abarbanel H.D.I. Parameter and state estimation of experimental chaotic systems using synchronization. Phys Rev E 2009, 80:016201.
14. Abarbanel H.D.I., Creveling D.R., Farsian R., Kostuk M. Dynamical state and parameter estimation. SIAM J Appl Dyn Syst 2009, 8:1341-1381.
15. Abarbanel H.D.I. Effective actions for statistical data assimilation. Phys Lett A 2009, 373:4044404.
16. Quinn J.C., Abarbanel H.D.I. State and parameter estimation using Monte Carlo evaluation of path integrals. QJR Meteorol Soc 2010, 136(652):1855-1867.
17. Abarbanel H.D.I., Kostuk M., Whartenby W. Data assimilation with regularized nonlinear instabilities. QJR Meteorol Soc 2010, 136(648):769-783.
18. Bröcker J. On variational data assimilation in continuous time. QJR Meteorol Soc 2010, 136:1906-1919.
19. Schumann-Bischoff J., Parlitz U. State and parameter estimation using unconstrained optimization. Phys Rev E 2011, 84:056214.
20. Bröcker J., Szendro I. Sensitivity and out-of-sample error of continuous time data assimilation. QJR Meteorol Soc 2012, 138:785-801.
21. Levenberg K. A method for the solution of certain problems in least squares. Quart Appl Math 1944, 2:164-168.
22. Marquardt D. An algorithm for least-squares estimation of nonlinear parameters. SIAM J Appl Math 1963, 11:431-441.
23. Lourakis MIA. Sparse non-linear least squares optimization for geometric vision. In: European conference on computer vision, vol. 2; 2010. p. 43-56.,. http://www.ics.forth.gr/lourakis/sparseLM/.
24. Rall L.B. Automatic differentiation: techniques and applications. Lect Notes Comput Sci 1981, 120.
25. Walther A, Griewank A. Getting started with ADOL-C, In: Naumann U, Schenk O, editor. Combinatorial Scientific Computing. Chapman-Hall CRC Computational Science; 2012. p. 181-202.
26. Griewank A., Juedes D., Utke J. Algorithm 755: ADOL-C: a package for the automatic differentiation of algorithms written in C/C++. ACM Trans Math Softw 1996, 22:131-167.
27. ColPack:, October 16, 2012. http://www.cscapes.org/coloringpage/software.htm.
28. Lorenz E.N. Predictability - a problem partly solved, ECMWF. Proceedings of the seminar on predictability 1996, vol. 1:1-18. Berkshire, Reading, UK.
29. Mackey M.C., Glass L. Oscillation and chaos in physiological control systems. Science 1977, 197:287-289.
30. Numpy:, October 16, 2012. http://numpy.scipy.org/.
31. Pyadolc:, October 16, 2012. http://https://github.com/b45ch1/pyadolc.
32. A Python wrapper for sparseLM and a Python-C implementation of the estimation method for ODEs can be downloaded from and, Gnu General Public License Version 3, 29 June 2007. http://www.bmp.ds.mpg.de/pyodefit.html.
33. Kuptsov P.V., Parlitz U. Theory and computation of covariant Lyapunov vectors. J Nonlinear Sci 2012, 22:727-762.