Semi–Smooth Newton Methods for Variational Inequalities of the First Kind

Kazufumi Ito; Karl Kunisch

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 37, Issue: 1, page 41-62
  • ISSN: 0764-583X

Abstract

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Semi–smooth Newton methods are analyzed for a class of variational inequalities in infinite dimensions. It is shown that they are equivalent to certain active set strategies. Global and local super-linear convergence are proved. To overcome the phenomenon of finite speed of propagation of discretized problems a penalty version is used as the basis for a continuation procedure to speed up convergence. The choice of the penalty parameter can be made on the basis of an L∞ estimate for the penalized solutions. Unilateral as well as bilateral problems are considered.

How to cite

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Ito, Kazufumi, and Kunisch, Karl. "Semi–Smooth Newton Methods for Variational Inequalities of the First Kind." ESAIM: Mathematical Modelling and Numerical Analysis 37.1 (2010): 41-62. <http://eudml.org/doc/194155>.

@article{Ito2010,
abstract = { Semi–smooth Newton methods are analyzed for a class of variational inequalities in infinite dimensions. It is shown that they are equivalent to certain active set strategies. Global and local super-linear convergence are proved. To overcome the phenomenon of finite speed of propagation of discretized problems a penalty version is used as the basis for a continuation procedure to speed up convergence. The choice of the penalty parameter can be made on the basis of an L∞ estimate for the penalized solutions. Unilateral as well as bilateral problems are considered. },
author = {Ito, Kazufumi, Kunisch, Karl},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Semi-smooth Newton methods; contact problems; variational inequalities; bilateral constraints; superlinear convergence.; semi-smooth Newton methods; superlinear convergence},
language = {eng},
month = {3},
number = {1},
pages = {41-62},
publisher = {EDP Sciences},
title = {Semi–Smooth Newton Methods for Variational Inequalities of the First Kind},
url = {http://eudml.org/doc/194155},
volume = {37},
year = {2010},
}

TY - JOUR
AU - Ito, Kazufumi
AU - Kunisch, Karl
TI - Semi–Smooth Newton Methods for Variational Inequalities of the First Kind
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 37
IS - 1
SP - 41
EP - 62
AB - Semi–smooth Newton methods are analyzed for a class of variational inequalities in infinite dimensions. It is shown that they are equivalent to certain active set strategies. Global and local super-linear convergence are proved. To overcome the phenomenon of finite speed of propagation of discretized problems a penalty version is used as the basis for a continuation procedure to speed up convergence. The choice of the penalty parameter can be made on the basis of an L∞ estimate for the penalized solutions. Unilateral as well as bilateral problems are considered.
LA - eng
KW - Semi-smooth Newton methods; contact problems; variational inequalities; bilateral constraints; superlinear convergence.; semi-smooth Newton methods; superlinear convergence
UR - http://eudml.org/doc/194155
ER -

References

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  2. M. Bergounioux, M. Haddou, M. Hintermüller and K. Kunisch, A comparison of a Moreau-Yosida based active strategy and interior point methods for constrained optimal control problems. SIAM J. Optim.11 (2000) 495-521.  
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  7. M. Hintermüller, K. Ito and K. Kunisch, The primal-dual active set strategy as semi-smooth Newton method. SIAM J. Optim. (to appear).  
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  9. R. Hoppe and R. Kornhuber, Adaptive multigrid methods for obstacle problems. SIAM J. Numer. Anal.31 (1994) 301-323.  
  10. K. Ito and K. Kunisch, Augmented Lagrangian methods for nonsmooth convex optimization in Hilbert spaces. Nonlinear Anal.41 (2000) 573-589.  
  11. K. Ito and K. Kunisch, Optimal control of elliptic variational inequalities. Appl. Math. Optim.41 (2000) 343-364.  
  12. D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980).  
  13. D.M. Troianello, Elliptic Differential Equations and Obstacle Problems. Plenum Press, New York (1987).  
  14. M. Ulbrich, Semi-smooth Newton methods for operator equations in function space. SIAM J. Optim. (to appear).  

Citations in EuDML Documents

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  1. Juan Carlos De Los Reyes, Sergio González, Path following methods for steady laminar Bingham flow in cylindrical pipes
  2. Juan Carlos De Los Reyes, Sergio González, Path following methods for steady laminar Bingham flow in cylindrical pipes
  3. Kazufumi Ito, Karl Kunisch, Semi-smooth Newton methods for the Signorini problem
  4. Karl Kunisch, Georg Stadler, Generalized Newton methods for the 2D-Signorini contact problem with friction in function space
  5. Luise Blank, Martin Butz, Harald Garcke, Solving the Cahn-Hilliard variational inequality with a semi-smooth Newton method
  6. Luise Blank, Martin Butz, Harald Garcke, Solving the Cahn-Hilliard variational inequality with a semi-smooth Newton method
  7. Karl Kunisch, Georg Stadler, Generalized Newton methods for the 2D-Signorini contact problem with friction in function space
  8. Karl Kunisch, Daniel Wachsmuth, Sufficient optimality conditions and semi-smooth newton methods for optimal control of stationary variational inequalities
  9. Karl Kunisch, Daniel Wachsmuth, Sufficient optimality conditions and semi-smooth newton methods for optimal control of stationary variational inequalities

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