Newton and conjugate gradient for harmonic maps from the disc into the sphere

Morgan Pierre

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 10, Issue: 1, page 142-167
  • ISSN: 1292-8119

Abstract

top
We compute numerically the minimizers of the Dirichlet energy E ( u ) = 1 2 ∫ B 2 | ∇ u | 2 d x among maps u : B 2 → S 2 from the unit disc into the unit sphere that satisfy a boundary condition and a degree condition. We use a Sobolev gradient algorithm for the minimization and we prove that its continuous version preserves the degree. For the discretization of the problem we use continuous P1 finite elements. We propose an original mesh-refining strategy needed to preserve the degree with the discrete version of the algorithm (which is a preconditioned projected gradient). In order to improve the convergence, we generalize to manifolds the classical Newton and conjugate gradient algorithms. We give a proof of the quadratic convergence of the Newton algorithm for manifolds in a general setting.

How to cite

top

Pierre, Morgan. "Newton and conjugate gradient for harmonic maps from the disc into the sphere." ESAIM: Control, Optimisation and Calculus of Variations 10.1 (2010): 142-167. <http://eudml.org/doc/90717>.

@article{Pierre2010,
abstract = { We compute numerically the minimizers of the Dirichlet energy $$E(u)=\frac\{1\}\{2\}\int\_\{B^2\}|\nabla u|^2 \{\rm d\}x$$ among maps $u:B^2\to S^2$ from the unit disc into the unit sphere that satisfy a boundary condition and a degree condition. We use a Sobolev gradient algorithm for the minimization and we prove that its continuous version preserves the degree. For the discretization of the problem we use continuous P1 finite elements. We propose an original mesh-refining strategy needed to preserve the degree with the discrete version of the algorithm (which is a preconditioned projected gradient). In order to improve the convergence, we generalize to manifolds the classical Newton and conjugate gradient algorithms. We give a proof of the quadratic convergence of the Newton algorithm for manifolds in a general setting. },
author = {Pierre, Morgan},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Harmonic maps; finite elements; mesh-refinement; Sobolev gradient; Newton algorithm; conjugate gradient.},
language = {eng},
month = {3},
number = {1},
pages = {142-167},
publisher = {EDP Sciences},
title = {Newton and conjugate gradient for harmonic maps from the disc into the sphere},
url = {http://eudml.org/doc/90717},
volume = {10},
year = {2010},
}

TY - JOUR
AU - Pierre, Morgan
TI - Newton and conjugate gradient for harmonic maps from the disc into the sphere
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 10
IS - 1
SP - 142
EP - 167
AB - We compute numerically the minimizers of the Dirichlet energy $$E(u)=\frac{1}{2}\int_{B^2}|\nabla u|^2 {\rm d}x$$ among maps $u:B^2\to S^2$ from the unit disc into the unit sphere that satisfy a boundary condition and a degree condition. We use a Sobolev gradient algorithm for the minimization and we prove that its continuous version preserves the degree. For the discretization of the problem we use continuous P1 finite elements. We propose an original mesh-refining strategy needed to preserve the degree with the discrete version of the algorithm (which is a preconditioned projected gradient). In order to improve the convergence, we generalize to manifolds the classical Newton and conjugate gradient algorithms. We give a proof of the quadratic convergence of the Newton algorithm for manifolds in a general setting.
LA - eng
KW - Harmonic maps; finite elements; mesh-refinement; Sobolev gradient; Newton algorithm; conjugate gradient.
UR - http://eudml.org/doc/90717
ER -

References

top
  1. F. Alouges, A new algorithm for computing liquid crystal stable configurations: the harmonic mapping case. SIAM J. Numer. Anal.34 (1997) 1708–1726.  
  2. F. Alouges and B.D. Coleman, Numerical bifurcation of equilibria of nematic crystals between non-co-axial cylinders. Math. Models Methods Appl. Sci.11 (2001) 459–473.  
  3. D. Braess, Finite elements, in Theory, fast solvers, and applications in solid mechanics. Translated from the 1992 German edition by Larry L. Schumaker. Cambridge University Press, Cambridge, 2nd edn. (2001).  
  4. H. Brézis, Analyse fonctionnelle. Masson (1996).  
  5. H. Brézis and J.-M. Coron, Large solutions for harmonic maps in two dimensions. Comm. Math. Phys.92 (1983) 203–215.  
  6. K.-C. Chang, W.-Y. Ding and R. Ye, Finite-time blow up of the heat flow of harmonic maps from surfaces. J. Differ. Geom.36 (1992) 507–515.  
  7. P.G. Ciarlet, Introduction à l'analyse numérique matricielle et à l'optimisation. Masson (1988).  
  8. P.-G. De Gennes and J. Prost, The physics of liquid crystals. Clarendon Press, Oxford (1993).  
  9. R. Fletcher and C.M. Reeves, Function minimization by conjugate gradients. Comput. J.7 (1994) 149–154.  
  10. M. Giaquinta, G. Modica and J. Souček, Cartesian currents in the calculus of variations. I. Springer-Verlag, Berlin (1998).  
  11. M. Giaquinta, G. Modica and J. Souček, Cartesian currents in the calculus of variations. II. Springer-Verlag, Berlin (1998).  
  12. R.M. Hardt, Singularities of harmonic maps. Bull. Amer. Math. Soc. (N.S.)34 (1997) 15–34.  
  13. E. Hebey, Introduction à l'analyse non linéaire sur les variétés. Diderot Editeur Arts et Sciences (1987).  
  14. F. Hélein, Régularité des applications faiblement harmoniques entre une surface et une sphère. C. R. Acad. Sci. Paris Sér. I Math.311 (1990) 519–524.  
  15. F. Hélein, Symétries dans les problèmes variationnels et applications harmoniques. Istituti Editoriali e Poligrafici Internazionali, Pisa-Roma (1998).  
  16. J. Jost, Harmonic mappings betwenn surfaces. Springer-verlag, Lecture Notes in Math.1062 (1984).  
  17. W.P.A Klingenberg, Riemannian Geometry. Walter de Gruyter (1995).  
  18. E. Kuwert, Minimizing the energy of maps from a surface into a 2-sphere with prescribed degree and boundary values. Manuscripta Math.83 (1994) 31–38.  
  19. L. Lemaire, Applications harmoniques de surfaces riemanniennes. J. Differ. Geom.13 (1978) 51–78.  
  20. A. Lichnewsky, Une méthode de gradient conjugué sur des variétés : application à certains problèmes de valeurs propres non linéaires. Numer. Funct. Anal. Optim.1 (1979) 515–560.  
  21. P.L. Lions, The concentration-compactness principle in the calculus of variations. The limit case, part 2. Rev. Mat. Iberoamericana1 (1985) 45–121.  
  22. C.B. Morrey, Multiple integrals in the calculus of variations. Springer, New York (1966).  
  23. J.W. Neuberger, Sobolev gradients and boundary conditions for partial differential equations, in Recent developments in optimization theory and nonlinear analysis (Jerusalem, 1995), Amer. Math. Soc., Providence, RI. Contemp. Math.204 (1997) 171–181  
  24. E. Polak, Optimization, Appl. Math. Sci.124 (1997).  
  25. J. Qing, Remark on the Dirichlet problem for harmonic maps from the disc into the 2-sphere. Proc. R. Soc. Edinb.122A (1992) 63–67.  
  26. J. Qing, Boundary regularity of weakly harmonic maps from surfaces. J. Funct. Anal.114 (1993) 63–67.  
  27. R. Schoen and K. Uhlenbeck, Boundary regularity and the Dirichlet problem for harmonic maps. J. Dif. Geom.18 (1983) 253–268.  
  28. J.R. Shewchuk, Triangle: engineering a 2d quality mesh generator and delaunay triangulator. .  URIhttp://www-2.cs.cmu.edu/quake/triangle.html
  29. J.R. Shewchuk, An introduction to the conjugate gradient method without the agonizing pain. (1994).  URIhttp://www-2.cs.cmu.edu/jrs/jrspapers.html#cg
  30. A. Soyeur, The Dirichlet problem for harmonic maps from the disc into the 2-sphere. Proc. R. Soc. Edinb.113A (1989) 229–234.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.