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 and Continuous Dynamical Systems, 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), 139-148.

[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., (). 

[3]

W. Auzinger, R. Frank and G. Kirlinger, An extension of $B$-convergence for Runge-Kutta methods, Appl. Num. Math., 9, (1992), 91-109.

[4]

W. Auzinger, R. Frank and G. Kirlinger, Modern convergence theory for stiff initial value problems, J. Comput. Appl. Math., 45 (1993), 5-16. doi: 10.1016/0378-3782(93)90046-W.

[5]

G. D. Byrne and A. C. Hindmarsh, Stift ODE solvers: A review of current and coming attractions, J. Comput. Phys., 70 (1987), 1-62.

[6]

G. Dahlquist, A special stability problem for linear multistep methods, BIT, 3 (1963), 27-43. doi: 10.1007/BF01963532.

[7]

J. H. Ferziger and M. Peric, "Computational Methods for Fluid Dynamics," Springer-Verlag, Berlin, Germany, 1980.

[8]

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

[9]

E. Hairer and G. Wanner, "Solving Ordinary Differential Equations II: Stiff and Differential Algebraic Problems," Springer-Verlag, Berlin, Germany, 1991.

[10]

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

[11]

A. Lew, J. E. Marsden, M. Ortiz and M. West, Variational time integrators, Internat. J. Numer. Methods Engrg., 60 (2004), 153-212.

[12]

J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer., 10 (2001), 357-514.

[13]

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

[14]

P. Pedregal, A variational approach to dynamical systems, and its numerical simulation, Numer. Funct. Anal. Opt., 31 (2010), 1532-2467. doi: 10.1080/01630563.2010.497237.

[15]

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

show all references

References:
[1]

S. Amat and P. Pedregal, A variational approach to implicit ODEs and differential inclusions, ESAIM-COCV, 15 (2009), 139-148.

[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., (). 

[3]

W. Auzinger, R. Frank and G. Kirlinger, An extension of $B$-convergence for Runge-Kutta methods, Appl. Num. Math., 9, (1992), 91-109.

[4]

W. Auzinger, R. Frank and G. Kirlinger, Modern convergence theory for stiff initial value problems, J. Comput. Appl. Math., 45 (1993), 5-16. doi: 10.1016/0378-3782(93)90046-W.

[5]

G. D. Byrne and A. C. Hindmarsh, Stift ODE solvers: A review of current and coming attractions, J. Comput. Phys., 70 (1987), 1-62.

[6]

G. Dahlquist, A special stability problem for linear multistep methods, BIT, 3 (1963), 27-43. doi: 10.1007/BF01963532.

[7]

J. H. Ferziger and M. Peric, "Computational Methods for Fluid Dynamics," Springer-Verlag, Berlin, Germany, 1980.

[8]

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

[9]

E. Hairer and G. Wanner, "Solving Ordinary Differential Equations II: Stiff and Differential Algebraic Problems," Springer-Verlag, Berlin, Germany, 1991.

[10]

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

[11]

A. Lew, J. E. Marsden, M. Ortiz and M. West, Variational time integrators, Internat. J. Numer. Methods Engrg., 60 (2004), 153-212.

[12]

J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer., 10 (2001), 357-514.

[13]

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

[14]

P. Pedregal, A variational approach to dynamical systems, and its numerical simulation, Numer. Funct. Anal. Opt., 31 (2010), 1532-2467. doi: 10.1080/01630563.2010.497237.

[15]

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

[1]

Ricardo Almeida. Optimality conditions for fractional variational problems with free terminal time. Discrete and Continuous Dynamical Systems - S, 2018, 11 (1) : 1-19. doi: 10.3934/dcdss.2018001

[2]

Monika Dryl, Delfim F. M. Torres. Necessary optimality conditions for infinite horizon variational problems on time scales. Numerical Algebra, Control and Optimization, 2013, 3 (1) : 145-160. doi: 10.3934/naco.2013.3.145

[3]

Stepan Sorokin, Maxim Staritsyn. Feedback necessary optimality conditions for a class of terminally constrained state-linear variational problems inspired by impulsive control. Numerical Algebra, Control and Optimization, 2017, 7 (2) : 201-210. doi: 10.3934/naco.2017014

[4]

Laurent Pfeiffer. Optimality conditions in variational form for non-linear constrained stochastic control problems. Mathematical Control and Related Fields, 2020, 10 (3) : 493-526. doi: 10.3934/mcrf.2020008

[5]

Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control and Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017

[6]

Ariadna Farrés, Àngel Jorba. On the high order approximation of the centre manifold for ODEs. Discrete and Continuous Dynamical Systems - B, 2010, 14 (3) : 977-1000. doi: 10.3934/dcdsb.2010.14.977

[7]

Richard A. Norton, David I. McLaren, G. R. W. Quispel, Ari Stern, Antonella Zanna. Projection methods and discrete gradient methods for preserving first integrals of ODEs. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 2079-2098. doi: 10.3934/dcds.2015.35.2079

[8]

B. Bonnard, J.-B. Caillau, E. Trélat. Second order optimality conditions with applications. Conference Publications, 2007, 2007 (Special) : 145-154. doi: 10.3934/proc.2007.2007.145

[9]

Mansoureh Alavi Hejazi, Soghra Nobakhtian. Optimality conditions for multiobjective fractional programming, via convexificators. Journal of Industrial and Management Optimization, 2020, 16 (2) : 623-631. doi: 10.3934/jimo.2018170

[10]

Luong V. Nguyen. A note on optimality conditions for optimal exit time problems. Mathematical Control and Related Fields, 2015, 5 (2) : 291-303. doi: 10.3934/mcrf.2015.5.291

[11]

Piernicola Bettiol, Nathalie Khalil. Necessary optimality conditions for average cost minimization problems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2093-2124. doi: 10.3934/dcdsb.2019086

[12]

Geng-Hua Li, Sheng-Jie Li. Unified optimality conditions for set-valued optimizations. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1101-1116. doi: 10.3934/jimo.2018087

[13]

Ying Gao, Xinmin Yang, Kok Lay Teo. Optimality conditions for approximate solutions of vector optimization problems. Journal of Industrial and Management Optimization, 2011, 7 (2) : 483-496. doi: 10.3934/jimo.2011.7.483

[14]

Adela Capătă. Optimality conditions for vector equilibrium problems and their applications. Journal of Industrial and Management Optimization, 2013, 9 (3) : 659-669. doi: 10.3934/jimo.2013.9.659

[15]

Qiu-Sheng Qiu. Optimality conditions for vector equilibrium problems with constraints. Journal of Industrial and Management Optimization, 2009, 5 (4) : 783-790. doi: 10.3934/jimo.2009.5.783

[16]

Majid E. Abbasov. Generalized exhausters: Existence, construction, optimality conditions. Journal of Industrial and Management Optimization, 2015, 11 (1) : 217-230. doi: 10.3934/jimo.2015.11.217

[17]

Shahlar F. Maharramov. Necessary optimality conditions for switching control problems. Journal of Industrial and Management Optimization, 2010, 6 (1) : 47-55. doi: 10.3934/jimo.2010.6.47

[18]

Gang Li, Yinghong Xu, Zhenhua Qin. Optimality conditions for composite DC infinite programming problems. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022064

[19]

Gennaro Infante. Eigenvalues and positive solutions of odes involving integral boundary conditions. Conference Publications, 2005, 2005 (Special) : 436-442. doi: 10.3934/proc.2005.2005.436

[20]

Lori Badea. Multigrid methods for some quasi-variational inequalities. Discrete and Continuous Dynamical Systems - S, 2013, 6 (6) : 1457-1471. doi: 10.3934/dcdss.2013.6.1457

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (78)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]