A leapfrog semi-smooth Newton-multigrid method for semilinear parabolic optimal control problems

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• 作者：Jun Liu ; Mingqing Xiao
• 关键词：Parabolic PDE control ; Leapfrog scheme ; Multigrid method
• 刊名：Computational Optimization and Applications
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：63
• 期：1
• 页码：69-95
• 全文大小：672 KB
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• 作者单位：Jun Liu (1)
  Mingqing Xiao (2)
  
  1. Department of Mathematics and Statistical Sciences, Jackson State University, Jackson, MS, 39217, USA
  2. Department of Mathematics, Southern Illinois University, Carbondale, IL, 62901, USA
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Optimization
  Operations Research and Mathematical Programming
  Operation Research and Decision Theory
  Statistics
  Convex and Discrete Geometry
  
• 出版者：Springer Netherlands
• ISSN：1573-2894

文摘

A new semi-smooth Newton multigrid algorithm is proposed for solving the discretized first order necessary optimality systems that characterizing the optimal solutions of a class of two dimensional semi-linear parabolic PDE optimal control problems with control constraints. A new computational scheme (leapfrog scheme) in time associated with the standard five-point stencil in space is established to achieve the second-order finite difference discretization. The convergence (or unconditional stability) of the proposed scheme is proved when assuming time-periodic solutions. Moreover, the derived well-structured discretized Jacobian matrices greatly facilitate the development of effective smoother in our multigrid algorithm. Numerical simulations are provided to illustrate the effectiveness of the proposed method, which validates the second-order accuracy in solution approximations and the optimal linear complexity of computing time.