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Global and uniform convergence of subspace correction methods for some convex optimization problems


Authors: Xue-Cheng Tai and Jinchao Xu
Journal: Math. Comp. 71 (2002), 105-124
MSC (2000): Primary 65N22, 65N55, 65K10, 65J15
Published electronically: May 11, 2001
MathSciNet review: 1862990
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Abstract:

This paper gives some global and uniform convergence estimates for a class of subspace correction (based on space decomposition) iterative methods applied to some unconstrained convex optimization problems. Some multigrid and domain decomposition methods are also discussed as special examples for solving some nonlinear elliptic boundary value problems.


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  • 1. O. Axelsson and W. Layton, A two-level discretization of nonlinear boundary value problems, SIAM J. Numer. Anal. 33 (1996), no. 6, 2359–2374. MR 1427468, 10.1137/S0036142993247104
  • 2. Randolph E. Bank and Donald J. Rose, Analysis of a multilevel iterative method for nonlinear finite element equations, Math. Comp. 39 (1982), no. 160, 453–465. MR 669639, 10.1090/S0025-5718-1982-0669639-X
  • 3. James H. Bramble, Joseph E. Pasciak, Jun Ping Wang, and Jinchao Xu, Convergence estimates for product iterative methods with applications to domain decomposition, Math. Comp. 57 (1991), no. 195, 1–21. MR 1090464, 10.1090/S0025-5718-1991-1090464-8
  • 4. Xiao-Chuan Cai and Maksymilian Dryja, Domain decomposition methods for monotone nonlinear elliptic problems, Domain decomposition methods in scientific and engineering computing (University Park, PA, 1993) Contemp. Math., vol. 180, Amer. Math. Soc., Providence, RI, 1994, pp. 21–27. MR 1312374, 10.1090/conm/180/01953
  • 5. Jean Céa, Optimisation: Théorie et algorithmes, Dunod, Paris, 1971 (French). Méthodes Mathématiques de l’Informatique, 2. MR 0298892
  • 6. T. F. Chan and I. Sharapov.
    Subspace correction multilevel methods for elliptic eigenvalue problems.
    In P. Bjørstad, M. Espedal, and D. Keyes, editors, Domain decomposition methods in science and engineering, 9th international conference, Bergen, Norway., pages 311-317. DDM.org, 1998.
    Available online at http://www.ddm.org/DD9/.
  • 7. Philippe G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR 0520174
  • 8. P. Deuflhard and M. Weiser.
    Global inexact Newton multilevel FEM for nonlinear elliptic problems.
    In W. Hackbusch and G. Wittum, editors, Multigrid methods V, Lecture Notes in Computational Science and Engineering, Stuttgart, 1996, Springer. CMP 2000:06
  • 9. Jim Douglas Jr., Todd Dupont, and Lars Wahlbin, The stability in 𝐿^{𝑞} of the 𝐿²-projection into finite element function spaces, Numer. Math. 23 (1974/75), 193–197. MR 0383789
  • 10. M. Dryja and W. Hackbusch, On the nonlinear domain decomposition method, BIT 37 (1997), no. 2, 296–311. MR 1450962, 10.1007/BF02510214
  • 11. Maksymilian Dryja and Olof B. Widlund, Towards a unified theory of domain decomposition algorithms for elliptic problems, Third International Symposium on Domain Decomposition Methods for Partial Differential Equations (Houston, TX, 1989) SIAM, Philadelphia, PA, 1990, pp. 3–21. MR 1064335
  • 12. Ivar Ekeland and Roger Temam, Convex analysis and variational problems, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1976. Translated from the French; Studies in Mathematics and its Applications, Vol. 1. MR 0463994
  • 13. E. Gelman and J. Mandel, On multilevel iterative methods for optimization problems, Math. Programming 48 (1990), no. 1, (Ser. B), 1–17. MR 1049769, 10.1007/BF01582249
  • 14. R. Glowinski and A. Marrocco, Sur l’approximation, par éléments finis d’ordre un, et la résolution, par pénalisation-dualité, d’une classe de problèmes de Dirichlet non linéaires, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge Anal. Numér. 9 (1975), no. R-2, 41–76 (French, with Loose English summary). MR 0388811
  • 15. M. Griebel and P. Oswald, On the abstract theory of additive and multiplicative Schwarz algorithms, Numer. Math. 70 (1995), no. 2, 163–180. MR 1324736, 10.1007/s002110050115
  • 16. W. Hackbusch and A. Reusken, Analysis of a damped nonlinear multilevel method, Numer. Math. 55 (1989), no. 2, 225–246. MR 987387, 10.1007/BF01406516
  • 17. Ralf Kornhuber, Adaptive monotone multigrid methods for nonlinear variational problems, Advances in Numerical Mathematics, B. G. Teubner, Stuttgart, 1997. MR 1469497
  • 18. R. Kornhuber.
    Globally convergent multigrid methods for porous medium type equations.
    Preprint, 1999.
  • 19. P.-L. Lions, On the Schwarz alternating method. I, First International Symposium on Domain Decomposition Methods for Partial Differential Equations (Paris, 1987) SIAM, Philadelphia, PA, 1988, pp. 1–42. MR 972510
  • 20. Jan Mandel, Étude algébrique d’une méthode multigrille pour quelques problèmes de frontière libre, C. R. Acad. Sci. Paris Sér. I Math. 298 (1984), no. 18, 469–472 (French, with English summary). MR 750748
  • 21. Stephen F. McCormick, Multilevel projection methods for partial differential equations, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 62, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1146209
  • 22. S. Oualibouch and N. el Mansouri.
    Proximal domain decomposition algorithms and applications to elliptic problems.
    In R. Glowinski, J. Periaux, Z-C. Shi, and O. Widlund, editors, Domain decomposition methods in sciences and engineering, 8th international conference, Beijing , China, pages 91-98. Wiley, 1997.
  • 23. R. Rannacher.
    On the convergence of the Newton-Raphson method for strongly nonlinear finite element equations.
    In P. Wriggers and W. Wagner, editors, Nonlinear computational mechanics, pages 11-30. Springer-Verlag, 1991.
  • 24. Arnold Reusken, Convergence of the multilevel full approximation scheme including the 𝑉-cycle, Numer. Math. 53 (1988), no. 6, 663–686. MR 955979, 10.1007/BF01397135
  • 25. Recent advances in matrix theory, Proceedings of an Advanced Seminar Conducted by the Mathematics Research Center, United States Army, at the University of Wisconsin, Madison, October 1 4-16, vol. 1963, The University of Wisconsin Press, Madison, Wis., 1964. MR 0164974
  • 26. I. Sharapov.
    Multilevel subspace correction for large scale optimization problems.
    Technical Report CAM-97-31, University of California at Los Angeles, 1997.
  • 27. Barry F. Smith, Petter E. Bjørstad, and William D. Gropp, Domain decomposition, Cambridge University Press, Cambridge, 1996. Parallel multilevel methods for elliptic partial differential equations. MR 1410757
  • 28. X.-C. Tai.
    Parallel function decomposition and space decomposition methods with applications to optimisation, splitting and domain decomposition.
    Report No. 231-1992, Institut für Mathematik, TechnischeUniversität Graz, 1992.
    Available online at http://www.mi.uib.no/~tai.
  • 29. X.-C. Tai.
    Parallel function and space decomposition methods.
    In P. Neittaanmäki, editor, Finite element methods, fifty years of the Courant element, Lecture notes in pure and applied mathematics, volume 164, pages 421-432. Marcel Dekker inc., 1994.
    Available online at http://www.mi.uib.no/~tai. CMP 95:03
  • 30. Xue Cheng Tai, Domain decomposition for linear and nonlinear elliptic problems via function or space decomposition, Domain decomposition methods in scientific and engineering computing (University Park, PA, 1993) Contemp. Math., vol. 180, Amer. Math. Soc., Providence, RI, 1994, pp. 355–360. MR 1312410, 10.1090/conm/180/01992
  • 31. X.-C. Tai.
    Parallel function and space decomposition methods - Part I. function decomposition.
    Beijing Mathematics, 1, part 2:104-134, 1995.
    Available online at http://www.mi.uib.no/~tai.
  • 32. X.-C. Tai.
    Parallel function decomposition and space decomposition methods: Part II. space decomposition.
    Beijing Mathematics, 1, part 2:135-152, 1995.
    Available online at http://www.mi.uib.no/~tai.
  • 33. Xue-Cheng Tai and Magne Espedal, Rate of convergence of some space decomposition methods for linear and nonlinear problems, SIAM J. Numer. Anal. 35 (1998), no. 4, 1558–1570. MR 1626026, 10.1137/S0036142996297461
  • 34. X.-C. Tai and J. Xu.
    Global convergence of subspace correction methods for convex optimization problems.
    Report no. 114, Department of Mathematics, University of Bergen, Norway, 1998.
    Available online at http://www.mi.uib.no/~tai.
  • 35. Olof B. Widlund, Some Schwarz methods for symmetric and nonsymmetric elliptic problems, Fifth International Symposium on Domain Decomposition Methods for Partial Differential Equations (Norfolk, VA, 1991) SIAM, Philadelphia, PA, 1992, pp. 19–36. MR 1189560
  • 36. J. Xu.
    Theory of Multilevel Methods.
    PhD thesis, Cornell University, 1989.
  • 37. Jinchao Xu, Iterative methods by space decomposition and subspace correction, SIAM Rev. 34 (1992), no. 4, 581–613. MR 1193013, 10.1137/1034116
  • 38. Jinchao Xu, A novel two-grid method for semilinear elliptic equations, SIAM J. Sci. Comput. 15 (1994), no. 1, 231–237. MR 1257166, 10.1137/0915016
  • 39. Jinchao Xu, Two-grid discretization techniques for linear and nonlinear PDEs, SIAM J. Numer. Anal. 33 (1996), no. 5, 1759–1777. MR 1411848, 10.1137/S0036142992232949
  • 40. Jun Zou, A new fast solver—monotone MG method (MMG), J. Comput. Math. 5 (1987), no. 4, 325–335. MR 958612

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Additional Information

Xue-Cheng Tai
Affiliation: Department of Mathematics, University of Bergen, Johannes Brunsgate 12, 5007, Bergen, Norway
Email: tai@mi.uib.no

Jinchao Xu
Affiliation: Center for Computational Mathematics and Applications and Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802
Email: xu@math.psu.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-01-01311-4
Keywords: Parallel, multigrid, domain decomposition, nonlinear, elliptic equation, space decomposition, convex optimization
Received by editor(s): March 10, 1998
Received by editor(s) in revised form: September 15, 1999, and March 24, 2000
Published electronically: May 11, 2001
Additional Notes: The work of the first author was partially supported by the Norwegian Research Council Strategic Institute Program within Inverse Problems at RF–Rogaland Research.
The work of the second author was partially supported by NSF DMS-9706949 NSF ACI-9800244, NASA NAG2-1236 through Pennsylvania State University.
Article copyright: © Copyright 2001 American Mathematical Society