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.
@article{bwmeta1.element.bwnjournal-article-zmv26i3p333bwm,
author = {J\'anos Kar\'atson},
title = {Gradient method for non-injective operators in Hilbert space with application to Neumann problems},
journal = {Applicationes Mathematicae},
volume = {26},
year = {1999},
pages = {333-346},
zbl = {1002.46046},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-zmv26i3p333bwm}
}
Karátson, János. Gradient method for non-injective operators in Hilbert space with application to Neumann problems. Applicationes Mathematicae, Tome 26 (1999) pp. 333-346. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-zmv26i3p333bwm/
[000] [1] J. Céa, Lectures on Optimization. Theory and Algorithms, Springer, 1978.
[001] [2] J. W. Daniel, The conjugate gradient method for linear and nonlinear operator equations, SIAM J. Numer. Anal. 4 (1967), 10-26. | Zbl 0154.40302
[002] [3] Yu. V. Egorov and M. A. Shubin, Partial Differential Equations I, Encyclopaedia Math. Sci., Springer, 1992.
[003] [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. | Zbl 0987.65121
[004] [5] H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Akademie-Verlag, Berlin, 1974. | Zbl 0289.47029
[005] [6] L. V. Kantorovich, On an effective method of solving extremal problems for quadratic functionals, Dokl. Akad. Nauk SSSR 48 (1945), 455-460.
[006] [7] L. V. Kantorovich and G. P. Akilov, Functional Analysis, Pergamon Press, 1982. | Zbl 0484.46003
[007] [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. | Zbl 0899.46061
[008] [9] J. Nečas, Introduction to the Theory of Nonlinear Elliptic Equations, Wiley, 1986.
[009] [10] V. S. Vladimirov, A Collection of Problems on the Equations of Mathematical Physics, Mir, Moscow, 1986. | Zbl 0607.35001