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Title: A new multigrid method for unconstrained parabolic optimal control problems
Authors: Li, BY 
Liu, J
Xiao, MQ
Keywords: Parabolic optimal control
Leapfrog scheme
Finite difference
Multigrid method
Issue Date: 2017
Publisher: North-Holland
Source: Journal of computational and applied mathematics, 2017, v. 326, p. 358-373 How to cite?
Journal: Journal of computational and applied mathematics 
Abstract: A second-order leapfrog finite difference scheme in time is proposed and developed for solving the first-order necessary optimality system of the distributed parabolic optimal control problems. Different from available approaches, the proposed leapfrog scheme for the two-point boundary optimality system is shown to be unconditionally stable and provides a second-order accuracy, though the classical leapfrog scheme usually is unstable. Moreover the proposed leapfrog scheme provides a feasible structure that leads to an effective implementation of a fast solver under the multigrid framework. A detailed mathematical proof for the stability of the proposed scheme is provided in terms of a new norm that is more suitable and stronger to characterize the convergence than the L-2 norm often used in literature. Numerical experiments show that the proposed scheme significantly outperforms the widely used second-order backward time differentiation approach and the resultant fast solver demonstrates a mesh-independent convergence as well as a linear time complexity.
ISSN: 0377-0427
EISSN: 1879-1778
DOI: 10.1016/
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