Gradient method for non-injective operators in Hilbert space with application to Neumann problems

János Karátson

Applicationes Mathematicae (1999)

  • Volume: 26, Issue: 3, page 333-346
  • ISSN: 1233-7234

Abstract

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The gradient method is developed for non-injective non-linear operators in Hilbert space that satisfy a translation invariance condition. The focus is on a class of non-differentiable operators. Linear convergence in norm is obtained. The method can be applied to quasilinear elliptic boundary value problems with Neumann boundary conditions.

How to cite

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Karátson, János. "Gradient method for non-injective operators in Hilbert space with application to Neumann problems." Applicationes Mathematicae 26.3 (1999): 333-346. <http://eudml.org/doc/219243>.

@article{Karátson1999,
abstract = {The gradient method is developed for non-injective non-linear operators in Hilbert space that satisfy a translation invariance condition. The focus is on a class of non-differentiable operators. Linear convergence in norm is obtained. The method can be applied to quasilinear elliptic boundary value problems with Neumann boundary conditions.},
author = {Karátson, János},
journal = {Applicationes Mathematicae},
keywords = {Neumann boundary value problems; non-injective non-linear operator; gradient method; Hilbert space; noninjective nonlinear operators; translation invariance; nondifferentiable operators; linear convergence in norm; quasilinear elliptic boundary value problems; Neumann boundary conditions},
language = {eng},
number = {3},
pages = {333-346},
title = {Gradient method for non-injective operators in Hilbert space with application to Neumann problems},
url = {http://eudml.org/doc/219243},
volume = {26},
year = {1999},
}

TY - JOUR
AU - Karátson, János
TI - Gradient method for non-injective operators in Hilbert space with application to Neumann problems
JO - Applicationes Mathematicae
PY - 1999
VL - 26
IS - 3
SP - 333
EP - 346
AB - The gradient method is developed for non-injective non-linear operators in Hilbert space that satisfy a translation invariance condition. The focus is on a class of non-differentiable operators. Linear convergence in norm is obtained. The method can be applied to quasilinear elliptic boundary value problems with Neumann boundary conditions.
LA - eng
KW - Neumann boundary value problems; non-injective non-linear operator; gradient method; Hilbert space; noninjective nonlinear operators; translation invariance; nondifferentiable operators; linear convergence in norm; quasilinear elliptic boundary value problems; Neumann boundary conditions
UR - http://eudml.org/doc/219243
ER -

References

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  1. [1] J. Céa, Lectures on Optimization. Theory and Algorithms, Springer, 1978. 
  2. [2] J. W. Daniel, The conjugate gradient method for linear and nonlinear operator equations, SIAM J. Numer. Anal. 4 (1967), 10-26. Zbl0154.40302
  3. [3] Yu. V. Egorov and M. A. Shubin, Partial Differential Equations I, Encyclopaedia Math. Sci., Springer, 1992. 
  4. [4] I. Faragó and J. Karátson, The gradient-finite element method for elliptic problems, in: Conference on Numerical Mathematics and Computational Mechanics, University of Miskolc, 1998. Zbl0987.65121
  5. [5] H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Akademie-Verlag, Berlin, 1974. Zbl0289.47029
  6. [6] L. V. Kantorovich, On an effective method of solving extremal problems for quadratic functionals, Dokl. Akad. Nauk SSSR 48 (1945), 455-460. 
  7. [7] L. V. Kantorovich and G. P. Akilov, Functional Analysis, Pergamon Press, 1982. Zbl0484.46003
  8. [8] J. Karátson, The gradient method for non-differentiable operators in product Hilbert spaces and applications to elliptic systems of quasilinear differential equations, J. Appl. Anal. 3 (1997), 225-237. Zbl0899.46061
  9. [9] J. Nečas, Introduction to the Theory of Nonlinear Elliptic Equations, Wiley, 1986. 
  10. [10] V. S. Vladimirov, A Collection of Problems on the Equations of Mathematical Physics, Mir, Moscow, 1986. Zbl0607.35001

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