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August  2021, 14(8): 3017-3025. doi: 10.3934/dcdss.2020465

On the convergence to equilibria of a sequence defined by an implicit scheme

Thierry Horsin 1,, and  Mohamed Ali Jendoubi 2,3,

1. 

Laboratoire M2N, EA7340, CNAM, 292 rue Saint-Martin, 75003, Paris, France
2. 

Université de Carthage, Institut Préparatoire aux Etudes Scientifiques et Techniques, B.P. 51 2070 La Marsa, Tunisie
3. 

Laboratoire équations aux dérivées partielles, Faculté des sciences de Tunis, Université Tunis El Manar, Campus universitaire El-Manar, 2092 El-Manar, Tunisie

* Corresponding author: Thierry Horsin

Dedicated to the memory of Ezzeddine ZAHROUNI
Both authors wishes to thank Morgan Pierre for fruitful comments. They are also grateful to the referees for their very useful comments. The first author wishes to thanks the organizers of ICAAM 2019 in Hammamet, Tunisia, where this work was initiated. The second author wishes to thanks CNAM, France where this work was partially completed.

Received  September 2020 Revised  October 2020 Published  August 2021 Early access  November 2020

We present numerical implicit schemes based on a geometric approach of the study of the convergence of solutions of gradient-like systems given in [3]. Depending on the globality of the induced metric, we can prove the convergence of these algorithms.

Keywords: Implicit scheme, asymptotic analysis, riemannian geometry, gradient systems, dynamical systems.
Mathematics Subject Classification: Primary: 34D05, 65L07; Secondary: 34C40.
Citation: Thierry Horsin, Mohamed Ali Jendoubi. On the convergence to equilibria of a sequence defined by an implicit scheme. Discrete & Continuous Dynamical Systems - S, 2021, 14 (8) : 3017-3025. doi: 10.3934/dcdss.2020465
References:
[1]

P.-A. Absil, R. Mahony and B. Andrews, Convergence of the iterates of descent methods for analytic cost functions, SIAM J. Optim., 16 (2005), 531-547.  doi: 10.1137/040605266.  Google Scholar

[2]

H. Attouch and J. Bolte, On the convergence of the proximal algorithm for nonsmooth functions involving analytic features, Math. Program., 116 (2009), 5-16.  doi: 10.1007/s10107-007-0133-5.  Google Scholar

[3]

T. Barta, R. Chill and E. Fašangová, Every ordinary differential equation with a strict Lyapunov function is a gradient system, Monatsh. Math., 166 (2012), 57-72.  doi: 10.1007/s00605-011-0322-4.  Google Scholar

[4]

J. Bolte, A. Daniilidis, O. Ley and L. Mazet, Characterizations of Lojasiewicz inequalities and applications, Trans. Amer. Math. Soc, 362 (6) (2010), 3319-3363.  doi: 10.1090/S0002-9947-09-05048-X.  Google Scholar

[5]

R. Chill, A. Haraux and M. A. Jendoubi, Applications of the Lojasiewicz-Simon gradient inequality to gradient-like evolution equations, Anal. Appl., 7 (2009), 351-372.  doi: 10.1142/S0219530509001438.  Google Scholar

[6]

A. Haraux and M. A. Jendoubi, The Convergence Problem for Dissipative Autonomous Systems. Classical Methods and Recent Advances, SpringerBriefs in Mathematics. Cham : Springer. 2015 doi: 10.1007/978-3-319-23407-6.  Google Scholar

[7]

J. X. Hong, Isometric Embedding of Riemannian Manifolds in Euclidean Spaces, AMS, 2006. Google Scholar

[8]

M. A. Jendoubi, Convergence des solutions globales et bornées de quelques problèmes d'évolution avec nonlinéarité analytique, in Progress in Partial Differential Equations. Papers from the 3rd European conference on elliptic and parabolic problems, Pont-à-Mousson, France, June 1997. Vol. 1. (eds Amann, H. (ed.) et al.) Harlow: Longman. Pitman Res. Notes Math. Ser., 383 (1998), 181–190.  Google Scholar

[9]

W. Klingenberg, Riemannian Geometry, De Gruyter Studies in Mathematics, 1, Berlin: Walter de Gruyter & Co. 1982.  Google Scholar

[10]

S. Lojasiewicz, Ensembles semi-analytiques, Preprint, I.H.E.S, Bures-sur-Yvette, 1965. Google Scholar

[11]

S. Lojasiewicz, Une proprièté topologique des sous ensembles analytiques réels, in Les Équations aux Dérivées Partielles, Colloques internationaux du C.N.R.S, 117. 1963.  Google Scholar

[12]

B. Merlet and M. Pierre, Convergence to equilibrium for the backward Euler scheme and applications, Comm. Pure and Appl. Anal., 9 (2010), 665-702.  doi: 10.3934/cpaa.2010.9.685.  Google Scholar

[13]

B. Merlet and T. H. Nguyen, Convergence to equilibrium for the backward euler scheme and applicationsconvergence to equilibrium for discretizations of gradient-like flows on Riemannian manifolds, Differential Integral Equations 26 (2013), 571–602. https://projecteuclid.org/euclid.die/1363266079 Google Scholar

[14]

J. Nash, The imbedding problem for Riemannian manifolds, Annals of Maths, 63 (1956), 20-63.  doi: 10.2307/1969989.  Google Scholar

[15]

J. H. C. Whitehead, Convex regions in the geometry of paths, Quart. J. Math., Oxford 3 (1932), 33–42, . doi: 10.1093/qmath/os-3.1.33.  Google Scholar

show all references

References:
[1]

P.-A. Absil, R. Mahony and B. Andrews, Convergence of the iterates of descent methods for analytic cost functions, SIAM J. Optim., 16 (2005), 531-547.  doi: 10.1137/040605266.  Google Scholar

[2]

H. Attouch and J. Bolte, On the convergence of the proximal algorithm for nonsmooth functions involving analytic features, Math. Program., 116 (2009), 5-16.  doi: 10.1007/s10107-007-0133-5.  Google Scholar

[3]

T. Barta, R. Chill and E. Fašangová, Every ordinary differential equation with a strict Lyapunov function is a gradient system, Monatsh. Math., 166 (2012), 57-72.  doi: 10.1007/s00605-011-0322-4.  Google Scholar

[4]

J. Bolte, A. Daniilidis, O. Ley and L. Mazet, Characterizations of Lojasiewicz inequalities and applications, Trans. Amer. Math. Soc, 362 (6) (2010), 3319-3363.  doi: 10.1090/S0002-9947-09-05048-X.  Google Scholar

[5]

R. Chill, A. Haraux and M. A. Jendoubi, Applications of the Lojasiewicz-Simon gradient inequality to gradient-like evolution equations, Anal. Appl., 7 (2009), 351-372.  doi: 10.1142/S0219530509001438.  Google Scholar

[6]

A. Haraux and M. A. Jendoubi, The Convergence Problem for Dissipative Autonomous Systems. Classical Methods and Recent Advances, SpringerBriefs in Mathematics. Cham : Springer. 2015 doi: 10.1007/978-3-319-23407-6.  Google Scholar

[7]

J. X. Hong, Isometric Embedding of Riemannian Manifolds in Euclidean Spaces, AMS, 2006. Google Scholar

[8]

M. A. Jendoubi, Convergence des solutions globales et bornées de quelques problèmes d'évolution avec nonlinéarité analytique, in Progress in Partial Differential Equations. Papers from the 3rd European conference on elliptic and parabolic problems, Pont-à-Mousson, France, June 1997. Vol. 1. (eds Amann, H. (ed.) et al.) Harlow: Longman. Pitman Res. Notes Math. Ser., 383 (1998), 181–190.  Google Scholar

[9]

W. Klingenberg, Riemannian Geometry, De Gruyter Studies in Mathematics, 1, Berlin: Walter de Gruyter & Co. 1982.  Google Scholar

[10]

S. Lojasiewicz, Ensembles semi-analytiques, Preprint, I.H.E.S, Bures-sur-Yvette, 1965. Google Scholar

[11]

S. Lojasiewicz, Une proprièté topologique des sous ensembles analytiques réels, in Les Équations aux Dérivées Partielles, Colloques internationaux du C.N.R.S, 117. 1963.  Google Scholar

[12]

B. Merlet and M. Pierre, Convergence to equilibrium for the backward Euler scheme and applications, Comm. Pure and Appl. Anal., 9 (2010), 665-702.  doi: 10.3934/cpaa.2010.9.685.  Google Scholar

[13]

B. Merlet and T. H. Nguyen, Convergence to equilibrium for the backward euler scheme and applicationsconvergence to equilibrium for discretizations of gradient-like flows on Riemannian manifolds, Differential Integral Equations 26 (2013), 571–602. https://projecteuclid.org/euclid.die/1363266079 Google Scholar

[14]

J. Nash, The imbedding problem for Riemannian manifolds, Annals of Maths, 63 (1956), 20-63.  doi: 10.2307/1969989.  Google Scholar

[15]

J. H. C. Whitehead, Convex regions in the geometry of paths, Quart. J. Math., Oxford 3 (1932), 33–42, . doi: 10.1093/qmath/os-3.1.33.  Google Scholar

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