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Journal of Numerical Mathematics

Editor-in-Chief: Hoppe, Ronald H. W. / Kuznetsov, Yuri

Managing Editor: Olshanskii, Maxim

Editorial Board: Benzi, Michele / Brenner, Susanne C. / Carstensen, Carsten / Dryja, M. / Feistauer, Miloslav / Glowinski, R. / Lazarov, Raytcho / Nataf, Frédéric / Neittaanmaki, P. / Bonito, Andrea / Quarteroni, Alfio / Guzman, Johnny / Rannacher, Rolf / Repin, Sergey I. / Shi, Zhong-ci / Tyrtyshnikov, Eugene E. / Zou, Jun / Simoncini, Valeria / Reusken, Arnold

IMPACT FACTOR 2018: 3.107

CiteScore 2018: 2.43

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Source Normalized Impact per Paper (SNIP) 2018: 1.618

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Volume 19, Issue 3


A full multigrid solution of control-constrained Cauchy–Riemann optimal control problems

M. M. Butt
  • *Institut für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universität Graz, Heinrichstr. 36, 8010 Graz, Austria and Department of Mathematics, GC University, Katchery Road, Lahore 54000, Pakistan.
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/ A. Borzì
  • Institut für Mathematik, Universität Würzburg, Hubland Campus Nord, Emil-Fischer-Str. 30, 97074 Würzburg, Germany.
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Published Online: 2011-11-15 | DOI: https://doi.org/10.1515/jnum.2011.009


The formulation as well as full-multigrid solution of control-constrained Cauchy–Riemann optimal control problems are presented. A constrained distributed control mechanism through divergence and curl sources is considered with mixed boundary conditions. The corresponding optimal solutions are obtained solving a Cauchy–Riemann optimality system consisting of four first-order partial differential equations and two inequality constraints. For the solution of the optimality system, staggered grids and a full-multigrid scheme are considered. The proposed full-multigrid method is based on a coarsening strategy by a factor of three that results in a nested hierarchy of staggered grids. The smoothing procedure consists of a distributed-Gauss–Seidel scheme for the state and adjoint equations and a projected gradient step for the controls. Numerical results validate the effectiveness of the proposed approach.

Keywords:: Cauchy–Riemann equations; constrained optimal control problems; staggered grids; finite differences; full multigrid method

About the article

Received: 2011-03-24

Revised: 2011-08-02

Published Online: 2011-11-15

Published in Print: 2011-10-01

Citation Information: Journal of Numerical Mathematics, Volume 19, Issue 3, Pages 189–214, ISSN (Online) 1569-3953, ISSN (Print) 1570-2820, DOI: https://doi.org/10.1515/jnum.2011.009.

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