A new semi-smooth Newton multigrid method for control-constrained semi-linear elliptic PDE problems

Jun Liu () and Mingqing Xiao ()

Journal of Global Optimization, 2016, vol. 64, issue 3, 468 pages

Abstract: In this paper a new multigrid algorithm is proposed to accelerate the convergence of the semi-smooth Newton method that is applied to the first order necessary optimality systems arising from a class of semi-linear control-constrained elliptic optimal control problems. Under admissible assumptions on the nonlinearity, the discretized Jacobian matrix is proved to have an uniformly bounded inverse with respect to mesh size. Different from current available approaches, a new numerical implementation that leads to a robust multigrid solver is employed to coarsen the grid operator. Numerical simulations are provided to illustrate the efficiency of the proposed method, which shows to be computationally more efficient than the full-approximation-storage multigrid in current literature. In particular, our proposed approach achieves a mesh-independent convergence and its performance is highly robust with respect to the regularization parameter. Copyright Springer Science+Business Media New York 2016

Keywords: Semi-smooth Newton method; Linear multigrid; Collective Jacobi smoother; Semi-linear elliptic equation; Full-approximation storage multigrid (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1007/s10898-014-0206-y

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