A Modified Augemented Lagrangian Method for a Class of Nonlinear Ill-Posed Problems ()
Abstract
A class of nonlinear problems with real parameters is
defined. Generally, in this class of problems, when the parametric values are
very large, the problems become ill-posed and numerical difficulties are
encountered when trying to solve these problems. In this paper, the nonlinear
problems are reformulated to overcome the numerical difficulties associated
with large parametric values. A novel iterative algorithm, which is suitable
for large scale problems and can be easily parallelized, is proposed to solve
the reformulated problems. Numerical tests indicate that the proposed algorithm
gives stable solutions. Convergence properties of the proposed algorithm are
investigated. In the limiting case, when the corresponding constraint is
exactly satisfied, the proposed method is equivalent to the standard augmented
Lagrangian method.
Share and Cite:
M. Shariff, "A Modified Augemented Lagrangian Method for a Class of Nonlinear Ill-Posed Problems,"
Open Journal of Applied Sciences, Vol. 3 No. 1B, 2013, pp. 70-73. doi:
10.4236/ojapps.2013.31B1014.
Conflicts of Interest
The authors declare no conflicts of interest.
References
[1]
|
M. H. B. M. Shariff and D. F. Parker, “An Extension of Key’s Principle to Nonlinear Elasticity,” Journal of Engngineering Mathematics, Vol. 37, No. 1-3, 2000, pp. 171-190. doi:10.1023/A:1004734311626
|
[2]
|
D. P. Bertsekas, “Constrained Optimization and Lagrangian Multiplier Methods,” Athena Scientific, Massachusetts, USA, 1981.
|
[3]
|
M. H. B. M. Shariff, “ A modified augmented Lagrangian method for a class of constrained problems,” Journal of Computational and Applied Mathematics, Vol. 151, No. 2, 2003, pp. 257-270. doi:10.1016/S0377-0427(02)00813-0
|
[4]
|
R. W. Ogden, “Volume Changes Associated with the Deformation of Rubber-like Solids,” Journal of Mechanics and Physics Solids, Vol. 24, No. 6, 1976, pp. 323-338.
doi:10.1016/0022-5096(76)90007-7
|
[5]
|
P. J. Blatz, “Finite Elastic Deformation of a Plane Strain Wedge-Shaped Radial Crack in a Compressible Cylinder,” Second-Order Effects in Elasticity, Plasticity and Fluid Dynamics, Pergamon Press, Oxford, 1964, pp. 145-161.
|
[6]
|
M. H. B. M. Shariff, “Parallel Finite Element Software for Nonlinear Stress Problems,” Parallel Computing and Transputer Applications, IOS Press, Amsterdam, 1992, pp. 1261-1270.
|