April  2013, 33(4): 1275-1291. doi: 10.3934/dcds.2013.33.1275

On a variational approach for the analysis and numerical simulation of ODEs

1. 

Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, Spain

2. 

E.T.S. Ingenieros Industriales, Universidad de Castilla La Mancha

Received  October 2011 Revised  March 2012 Published  October 2012

This paper is devoted to the study and approximation of systems of ordinary differential equations based on an analysis of a certain error functional associated, in a natural way, with the original problem. We prove that in seeking to minimize the error by using standard descent schemes, the procedure can never get stuck in local minima, but will always and steadily decrease the error until getting to the original solution. One main step in the procedure relies on a very particular linearization of the problem: in some sense, it is like a globally convergent Newton type method. Although our objective here is not to perform a rigorous numerical study of the method, we illustrate its potential for approximation by considering some stiff systems of equations. The performance is astonishingly very good due to the fact that we can use very robust methods to approximate linear stiff problems like implicit collocation schemes. We also include a couple of typical test models for the Lorentz system and the Kepler problem, again confirming a very good performance. We believe that this approach can be used in a systematic way to examine other situations and other types of equations due to its flexibility and its simplicity.
Citation: Sergio Amat, Pablo Pedregal. On a variational approach for the analysis and numerical simulation of ODEs. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1275-1291. doi: 10.3934/dcds.2013.33.1275
References:
[1]

ESAIM-COCV, 15 (2009), 139-148.  Google Scholar

[2]

S. Amat, D. J. López and P. Pedregal, Numerical approximation to ODEs using a variational approach I: The basic framework,, to appear in Optimization., ().   Google Scholar

[3]

Appl. Num. Math., 9, (1992), 91-109.  Google Scholar

[4]

J. Comput. Appl. Math., 45 (1993), 5-16. doi: 10.1016/0378-3782(93)90046-W.  Google Scholar

[5]

J. Comput. Phys., 70 (1987), 1-62.  Google Scholar

[6]

BIT, 3 (1963), 27-43. doi: 10.1007/BF01963532.  Google Scholar

[7]

Springer-Verlag, Berlin, Germany, 1980.  Google Scholar

[8]

Springer Series in Computational Mathematics, Springer, 2006.  Google Scholar

[9]

Springer-Verlag, Berlin, Germany, 1991.  Google Scholar

[10]

John Wiley and Sons Ltd. 1991.  Google Scholar

[11]

Internat. J. Numer. Methods Engrg., 60 (2004), 153-212.  Google Scholar

[12]

Acta Numer., 10 (2001), 357-514.  Google Scholar

[13]

The MathWorks, Inc., MATLAB and SIMULINK,, Natick, ().   Google Scholar

[14]

Numer. Funct. Anal. Opt., 31 (2010), 1532-2467. doi: 10.1080/01630563.2010.497237.  Google Scholar

[15]

Springer-Verlag. Second edition, 1993.  Google Scholar

show all references

References:
[1]

ESAIM-COCV, 15 (2009), 139-148.  Google Scholar

[2]

S. Amat, D. J. López and P. Pedregal, Numerical approximation to ODEs using a variational approach I: The basic framework,, to appear in Optimization., ().   Google Scholar

[3]

Appl. Num. Math., 9, (1992), 91-109.  Google Scholar

[4]

J. Comput. Appl. Math., 45 (1993), 5-16. doi: 10.1016/0378-3782(93)90046-W.  Google Scholar

[5]

J. Comput. Phys., 70 (1987), 1-62.  Google Scholar

[6]

BIT, 3 (1963), 27-43. doi: 10.1007/BF01963532.  Google Scholar

[7]

Springer-Verlag, Berlin, Germany, 1980.  Google Scholar

[8]

Springer Series in Computational Mathematics, Springer, 2006.  Google Scholar

[9]

Springer-Verlag, Berlin, Germany, 1991.  Google Scholar

[10]

John Wiley and Sons Ltd. 1991.  Google Scholar

[11]

Internat. J. Numer. Methods Engrg., 60 (2004), 153-212.  Google Scholar

[12]

Acta Numer., 10 (2001), 357-514.  Google Scholar

[13]

The MathWorks, Inc., MATLAB and SIMULINK,, Natick, ().   Google Scholar

[14]

Numer. Funct. Anal. Opt., 31 (2010), 1532-2467. doi: 10.1080/01630563.2010.497237.  Google Scholar

[15]

Springer-Verlag. Second edition, 1993.  Google Scholar

[1]

Tobias Geiger, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of ODEs with state suprema. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021012

[2]

Sergi Simon. Linearised higher variational equations. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4827-4854. doi: 10.3934/dcds.2014.34.4827

[3]

Bo Tan, Qinglong Zhou. Approximation properties of Lüroth expansions. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2873-2890. doi: 10.3934/dcds.2020389

[4]

Sandrine Anthoine, Jean-François Aujol, Yannick Boursier, Clothilde Mélot. Some proximal methods for Poisson intensity CBCT and PET. Inverse Problems & Imaging, 2012, 6 (4) : 565-598. doi: 10.3934/ipi.2012.6.565

[5]

Takeshi Saito, Kazuyuki Yagasaki. Chebyshev spectral methods for computing center manifolds. Journal of Computational Dynamics, 2021  doi: 10.3934/jcd.2021008

[6]

Zhenbing Gong, Yanping Chen, Wenyu Tao. Jump and variational inequalities for averaging operators with variable kernels. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021045

[7]

Xue-Ping Luo, Yi-Bin Xiao, Wei Li. Strict feasibility of variational inclusion problems in reflexive Banach spaces. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2495-2502. doi: 10.3934/jimo.2019065

[8]

Livia Betz, Irwin Yousept. Optimal control of elliptic variational inequalities with bounded and unbounded operators. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021009

[9]

Tomasz Kosmala, Markus Riedle. Variational solutions of stochastic partial differential equations with cylindrical Lévy noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2879-2898. doi: 10.3934/dcdsb.2020209

[10]

Tao Wang. Variational relations for metric mean dimension and rate distortion dimension. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021050

[11]

Tianyu Liao. The regularity lifting methods for nonnegative solutions of Lane-Emden system. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021036

[12]

Chiun-Chuan Chen, Hung-Yu Chien, Chih-Chiang Huang. A variational approach to three-phase traveling waves for a gradient system. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021055

[13]

Abdeslem Hafid Bentbib, Smahane El-Halouy, El Mostafa Sadek. Extended Krylov subspace methods for solving Sylvester and Stein tensor equations. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021026

[14]

Siting Liu, Levon Nurbekyan. Splitting methods for a class of non-potential mean field games. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021014

[15]

Jianxun Liu, Shengjie Li, Yingrang Xu. Quantitative stability of the ERM formulation for a class of stochastic linear variational inequalities. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021083

[16]

Madalina Petcu, Roger Temam. The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 209-222. doi: 10.3934/dcdss.2011.4.209

[17]

Samir Adly, Oanh Chau, Mohamed Rochdi. Solvability of a class of thermal dynamical contact problems with subdifferential conditions. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 91-104. doi: 10.3934/naco.2012.2.91

[18]

Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533

[19]

Zhihua Zhang, Naoki Saito. PHLST with adaptive tiling and its application to antarctic remote sensing image approximation. Inverse Problems & Imaging, 2014, 8 (1) : 321-337. doi: 10.3934/ipi.2014.8.321

[20]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (50)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]