Second-order approximation and fast multigrid solution of parabolic bilinear optimization problems

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• 作者：Alfio Borzì ; Sergio González Andrade
• 关键词：Multigrid methods ; Newton methods ; Finite differences ; Parabolic partial differential equations ; Bilinear control ; ptimal control theory ; 49K20 ; 49J20 ; 65M55 ; 65C20
• 刊名：Advances in Computational Mathematics
• 出版年：2015
• 出版时间：April 2015
• 年：2015
• 卷：41
• 期：2
• 页码：457-488
• 全文大小：3,118 KB
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• 作者单位：Alfio Borzì (1)
  Sergio González Andrade (2)
  
  1. Institut für Mathematik, Universit?t Würzburg, Emil-Fischer-Strasse 30, 97074, Würzburg, Germany
  2. Research Center on Mathematical Modelling MODEMAT, Escuela Politécnica Nacional, Ladrón de Guevara E11-253, Quito, Ecuador
  
• 刊物类别：Computer Science
• 刊物主题：Numeric Computing
  Calculus of Variations and Optimal Control
  Mathematics
  Algebra
  Theory of Computation
  
• 出版者：Springer U.S.
• ISSN：1572-9044

文摘

An accurate and fast solution scheme for parabolic bilinear optimization problems is presented. Parabolic models where the control plays the role of a reaction coefficient and the objective is to track a desired trajectory are formulated and investigated. Existence and uniqueness of optimal solution are proved. A space-time discretization is proposed and second-order accuracy for the optimal solution is discussed. The resulting optimality system is solved with a nonlinear multigrid strategy that uses a local semismooth Newton step as smoothing scheme. Results of numerical experiments validate the theoretical accuracy estimates and demonstrate the ability of the multigrid scheme to solve the given optimization problems with mesh-independent efficiency.