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 - A, 2013, 33 (4) : 1275-1291. doi: 10.3934/dcds.2013.33.1275
References:
[1]

S. Amat and P. Pedregal, A variational approach to implicit ODEs and differential inclusions,, ESAIM-COCV, 15 (2009), 139.   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]

W. Auzinger, R. Frank and G. Kirlinger, An extension of $B$-convergence for Runge-Kutta methods,, Appl. Num. Math., 9 (1992), 91.   Google Scholar

[4]

W. Auzinger, R. Frank and G. Kirlinger, Modern convergence theory for stiff initial value problems,, J. Comput. Appl. Math., 45 (1993), 5.  doi: 10.1016/0378-3782(93)90046-W.  Google Scholar

[5]

G. D. Byrne and A. C. Hindmarsh, Stift ODE solvers: A review of current and coming attractions,, J. Comput. Phys., 70 (1987), 1.   Google Scholar

[6]

G. Dahlquist, A special stability problem for linear multistep methods,, BIT, 3 (1963), 27.  doi: 10.1007/BF01963532.  Google Scholar

[7]

J. H. Ferziger and M. Peric, "Computational Methods for Fluid Dynamics,", Springer-Verlag, (1980).   Google Scholar

[8]

E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations,", Springer Series in Computational Mathematics, (2006).   Google Scholar

[9]

E. Hairer and G. Wanner, "Solving Ordinary Differential Equations II: Stiff and Differential Algebraic Problems,", Springer-Verlag, (1991).   Google Scholar

[10]

J. D. Lambert, "Numerical Methods for Ordinary Differntial Systems: The Initial Value Problem,", John Wiley and Sons Ltd. 1991., (1991).   Google Scholar

[11]

A. Lew, J. E. Marsden, M. Ortiz and M. West, Variational time integrators,, Internat. J. Numer. Methods Engrg., 60 (2004), 153.   Google Scholar

[12]

J. E. Marsden and M. West, Discrete mechanics and variational integrators,, Acta Numer., 10 (2001), 357.   Google Scholar

[13]

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

[14]

P. Pedregal, A variational approach to dynamical systems, and its numerical simulation,, Numer. Funct. Anal. Opt., 31 (2010), 1532.  doi: 10.1080/01630563.2010.497237.  Google Scholar

[15]

J. Stoer and R. Bulirsch, "Introduction to Numerical Analysis,", Springer-Verlag. Second edition, (1993).   Google Scholar

show all references

References:
[1]

S. Amat and P. Pedregal, A variational approach to implicit ODEs and differential inclusions,, ESAIM-COCV, 15 (2009), 139.   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]

W. Auzinger, R. Frank and G. Kirlinger, An extension of $B$-convergence for Runge-Kutta methods,, Appl. Num. Math., 9 (1992), 91.   Google Scholar

[4]

W. Auzinger, R. Frank and G. Kirlinger, Modern convergence theory for stiff initial value problems,, J. Comput. Appl. Math., 45 (1993), 5.  doi: 10.1016/0378-3782(93)90046-W.  Google Scholar

[5]

G. D. Byrne and A. C. Hindmarsh, Stift ODE solvers: A review of current and coming attractions,, J. Comput. Phys., 70 (1987), 1.   Google Scholar

[6]

G. Dahlquist, A special stability problem for linear multistep methods,, BIT, 3 (1963), 27.  doi: 10.1007/BF01963532.  Google Scholar

[7]

J. H. Ferziger and M. Peric, "Computational Methods for Fluid Dynamics,", Springer-Verlag, (1980).   Google Scholar

[8]

E. Hairer, C. Lubich and G. Wanner, "Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations,", Springer Series in Computational Mathematics, (2006).   Google Scholar

[9]

E. Hairer and G. Wanner, "Solving Ordinary Differential Equations II: Stiff and Differential Algebraic Problems,", Springer-Verlag, (1991).   Google Scholar

[10]

J. D. Lambert, "Numerical Methods for Ordinary Differntial Systems: The Initial Value Problem,", John Wiley and Sons Ltd. 1991., (1991).   Google Scholar

[11]

A. Lew, J. E. Marsden, M. Ortiz and M. West, Variational time integrators,, Internat. J. Numer. Methods Engrg., 60 (2004), 153.   Google Scholar

[12]

J. E. Marsden and M. West, Discrete mechanics and variational integrators,, Acta Numer., 10 (2001), 357.   Google Scholar

[13]

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

[14]

P. Pedregal, A variational approach to dynamical systems, and its numerical simulation,, Numer. Funct. Anal. Opt., 31 (2010), 1532.  doi: 10.1080/01630563.2010.497237.  Google Scholar

[15]

J. Stoer and R. Bulirsch, "Introduction to Numerical Analysis,", Springer-Verlag. Second edition, (1993).   Google Scholar

[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

Xianming Liu, Guangyue Han. A Wong-Zakai approximation of stochastic differential equations driven by a general semimartingale. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2499-2508. doi: 10.3934/dcdsb.2020192

[7]

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

[8]

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

[9]

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

[10]

Valery Y. Glizer. Novel Conditions of Euclidean space controllability for singularly perturbed systems with input delay. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 307-320. doi: 10.3934/naco.2020027

[11]

Hong Seng Sim, Wah June Leong, Chuei Yee Chen, Siti Nur Iqmal Ibrahim. Multi-step spectral gradient methods with modified weak secant relation for large scale unconstrained optimization. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 377-387. doi: 10.3934/naco.2018024

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (47)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]