An adaptive wavelet collocation method for solving optimal control of elliptic variational inequalities of the obstacle type

This paper presents a fast computational technique based on the wavelet collocation method for the numerical solution of an optimal control problem governed by elliptic variational inequalities of obstacle type. In this problem, the solution divides the domain into contact and noncontact sets. The b...

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Bibliographic Details
Published in:Computers & mathematics with applications (1987) 2018-01, Vol.75 (2), p.470-485
Main Authors: Khaksar-e Oshagh, M., Shamsi, M.
Format: Article
Language:English
Subjects:
Adaptive computational grids
Adaptive control
Adaptive mesh generation
Algorithms
Barriers
Collocation methods
Computational mathematics
Fast wavelet transform
Free boundaries
Free boundary problems
Inequalities
Mathematical analysis
Mesh generation
Numerical analysis
Optimal control
Optimal control of the obstacle problem
Optimization
Studies
Wavelet collocation method
Wavelet transforms
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Summary:This paper presents a fast computational technique based on the wavelet collocation method for the numerical solution of an optimal control problem governed by elliptic variational inequalities of obstacle type. In this problem, the solution divides the domain into contact and noncontact sets. The boundary between the contact and noncontact sets is a free boundary, which is not known a priori and the solution is not smooth on it. Accordingly, a very fine grid is needed in order to obtain a solution with a reasonable accuracy. In this paper, our aim is to propose an adaptive scheme in order to generate an appropriate and economic irregular dyadic mesh for finding the optimal control and state functions. The irregular mesh will be generated such that its density around the free boundary is higher than in other places and high-resolution computations are focused on these zones. To this aim, we use an adaptive wavelet collocation method and take advantage of the fast wavelet transform of compact-supported interpolating wavelets to develop a multi-level algorithm, which generates an adaptive computational grid. Using this adaptive grid takes less CPU time than using a full regular mesh. At each step of the algorithm, the active set method is used for solving the optimality system of the obstacle problem on the adapted mesh. Finally, the numerical examples are presented to show the validity and efficiency of the technique.
ISSN:0898-1221
1873-7668
DOI:10.1016/j.camwa.2017.09.026