Maximum time step for the BDF3 scheme applied to gradient flows

Abstract

For backward differentiation formulae (BDF) applied to gradient flows of semiconvex functions, quadratic stability implies the existence of a Lyapunov functional. We compute the maximum time step which can be derived from quadratic stability for the 3-step BDF method (BDF3). Applications to the asymptotic behaviour of sequences generated by the BDF3 scheme are given.

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References

  1. 1.

    Absil, P.A., Mahony, R., Andrews, B.: Convergence of the iterates of descent methods for analytic cost functions. SIAM J. Optim. 16(2), 531–547 (2005)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Alaa, N.E., Pierre, M.: Convergence to equilibrium for discretized gradient-like systems with analytic features. IMA J. Numer. Anal. 33(4), 1291–1321 (2013)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Attouch, H., Bolte, J.: On the convergence of the proximal algorithm for nonsmooth functions involving analytic features. Math. Program. 116(1–2, Ser. B), 5–16 (2009)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Attouch, H., Bolte, J., Redont, P., Soubeyran, A.: Proximal alternating minimization and projection methods for nonconvex problems: an approach based on the Kurdyka-Łojasiewicz inequality. Math. Oper. Res. 35(2), 438–457 (2010)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Attouch, H., Bolte, J., Svaiter, B.F.: Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods. Math. Program. 137(1–2, Ser. A), 91–129 (2013)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Boţ, R.I., Csetnek, E.R., László, S.C.: An inertial forward-backward algorithm for the minimization of the sum of two nonconvex functions. EURO J. Comput. Optim. 4(1), 3–25 (2016)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Bolte, J., Daniilidis, A., Lewis, A.: The Łojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems. SIAM J. Optim. 17(4), 1205–1223 (2006)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Bolte, J., Daniilidis, A., Lewis, A., Shiota, M.: Clarke subgradients of stratifiable functions. SIAM J. Optim. 18(2), 556–572 (2007)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Bolte, J., Daniilidis, A., Ley, O., Mazet, L.: Characterizations of Łojasiewicz inequalities: subgradient flows, talweg, convexity. Trans. Am. Math. Soc. 362(6), 3319–3363 (2010)

    Article  Google Scholar 

  10. 10.

    Bouchriti, A., Pierre, M., Alaa, N.E.: Gradient stability of high-order BDF methods and some applications. J. Differ. Equ. Appl. 26(1), 74–103 (2020)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Bouchriti, A., Pierre, M., Alaa, N.E.: Remarks on the asymptotic behaviour of scalar auxiliary variable (SAV) schemes for gradient-like flows. J. Appl. Anal. Comput. 10(5), 2198–2219 (2020)

    MathSciNet  Google Scholar 

  12. 12.

    Brezis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York (1973)

  13. 13.

    de Carvalho Bento, G., da Cruz Neto, J.A.X., Soubeyran, A., de Sousa Júnior, V.L.: Dual descent methods as tension reduction systems. J. Optim. Theory Appl. 171(1), 209–227 (2016)

  14. 14.

    Curtiss, C.F., Hirschfelder, J.O.: Integration of stiff equations. Proc. Nat. Acad. Sci. USA 38, 235–243 (1952)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Giaquinta, M., Modica, G., Souček, J.: Cartesian currents in the calculus of variations. II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 38. Springer-Verlag, Berlin (1998)

  16. 16.

    Hairer, E., Nørsett, S.P., Wanner, G.: Solving ordinary differential equations. I, Springer Series in Computational Mathematics, vol. 8, second edn. Springer, Berlin (1993)

  17. 17.

    Haraux, A., Jendoubi, M.A.: The convergence problem for dissipative autonomous systems. SpringerBriefs in Mathematics. Springer, Cham; BCAM Basque Center for Applied Mathematics, Bilbao (2015)

  18. 18.

    Łojasiewicz, S.: Ensembles semi-analytiques. I.H.E.S. Notes (1965)

  19. 19.

    Merlet, B., Pierre, M.: Convergence to equilibrium for the backward Euler scheme and applications. Commun. Pure Appl. Anal. 9(3), 685–702 (2010)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Rockafellar, R.T., Wets, R.J.B.: Variational Analysis, Grundlehren der Mathematischen Wissenschaften, vol. 317. Springer, Berlin (1998)

    Google Scholar 

  21. 21.

    Stuart, A.M., Humphries, A.R.: Dynamical Systems and Numerical Analysis, Cambridge Monographs on Applied and Computational Mathematics, vol. 2. Cambridge University Press, Cambridge (1996)

    Google Scholar 

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Acknowledgements

The author is thankful to Frédéric Bosio, Anass Bouchriti and Nour Eddine Alaa for helpful discussions. The author also wishes to thank the anonymous referee for his valuable comments.

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Correspondence to Morgan Pierre.

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Pierre, M. Maximum time step for the BDF3 scheme applied to gradient flows. Calcolo 58, 3 (2021). https://doi.org/10.1007/s10092-020-00393-3

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Keywords

  • Gradient system
  • BDF method
  • Semiconvex function
  • Kurdyka–Łojasiewicz property
  • Multivalued dynamical system

Mathematics Subject Classification

  • 65P40
  • 65L04