Hostname: page-component-6898cfcb55-cbk24 Total loading time: 0 Render date: 2026-11-11T07:06:29.375Z Has data issue: false hasContentIssue false
  • English
  • Français

Mesh-independence and preconditioning for solving parabolic control problems with mixed control-state constraints

Published online by Cambridge University Press:  19 July 2008

Michael Hintermüller,
Ian Kopacka and
Stefan Volkwein
Michael Hintermüller
Affiliation:
University of Sussex Department of Mathematics Mantell Building Falmer, Brighton BN1 9RF, UK. michael.hintermueller@uni-graz.at
Ian Kopacka
Affiliation:
Karl-Franzens-University of Graz Department of Mathematics and Scientific Computing Heinrichstrasse 36, 8010 Graz, Austria. ian.kopacka@uni-graz.at; stefan.volkwein@uni-graz.at
Stefan Volkwein
Affiliation:
Karl-Franzens-University of Graz Department of Mathematics and Scientific Computing Heinrichstrasse 36, 8010 Graz, Austria. ian.kopacka@uni-graz.at; stefan.volkwein@uni-graz.at
Get access

Abstract

Optimal control problems for the heat equation with pointwisebilateral control-state constraints are considered. A locallysuperlinearly convergent numerical solution algorithm is proposedand its mesh independence is established. Further, for theefficient numerical solution reduced space and Schur complementbased preconditioners are proposed which take into account theactive and inactive set structure of the problem. The paper endsby numerical tests illustrating our theoretical findings andcomparing the efficiency of the proposed preconditioners.

Keywords

Bilateral control-state constraintsheat equationmeshindependenceoptimal controlPDE-constrained optimizationsemismooth Newton method

Information

Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 15 , Issue 3 , July 2009, pp. 626 - 652
Copyright
© EDP Sciences, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

R.A. Adams, Sobolev Spaces, Pure and Applied Mathematics 65. Academic Press, New York-London (1975).
A. Battermann and M. Heinkenschloss, Preconditioners for Karush-Kuhn-Tucker matrices arising in the optimal control of distributed systems, in Control and estimation of distributed parameter systems (Vorau, 1996), Internat. Ser. Numer. Math. 126 (1998) 15–32.
A. Battermann and E.W. Sachs, Block preconditioners for KKT systems in PDE-governed optimal control problems, in Fast solution of discretized optimization problems (Berlin, 2000), Internat. Ser. Numer. Math. 138 (2001) 1–18.
Biros, G. and Ghattas, O., Parallel Lagrange-Newton-Krylov-Schur methods for PDE-constrained optimization. I. The Krylov-Schur solver. SIAM J. Sci. Comput. 27 (2005) 687–713. CrossRef
R. Dautray and J.-L. Lions, Evolution Problems I, Mathematical Analysis and Numerical Methods for Science and Technology 5. Springer-Verlag, Berlin (1992).
L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics 19. American Mathematical Society, Providence, Rhode Island (1998).
C. Geiger and C. Kanzow, Theorie und Numerik restringierter Optimierungsaufgaben. Springer-Verlag, Berlin (2002).
Hackbusch, W., Optimal $H^{p,p/2}$ error estimates for a parabolic Galerkin method. SIAM J. Numer. Anal. 18 (1981) 681–692. CrossRef
Hintermüller, M., Mesh-independence and fast local convergence of a primal-dual active-set method for mixed control-state constrained elliptic control problems. ANZIAM Journal 49 (2007) 1–38. CrossRef
Hintermüller, M. and Hinze, M., SQP-semismooth Newton-type, A algorithm applied to control of the instationary Navier-Stokes system subject to control constraints. SIAM J. Opt. 16 (2006) 1177–1200. CrossRef
Hintermüller, M. and Ulbrich, M., A mesh-independence result for semismooth Newton methods. Math. Program. Ser. B 101 (2004) 151–184. CrossRef
Hintermüller, M., Ito, K. and Kunisch, K., The primal-dual active set strategy as a semi-smooth Newton method. SIAM J. Opt. 13 (2003) 865–888. CrossRef
Hintermüller, M., Volkwein, S. and Diwoky, F., Fast solution techniques in constrained optimal boundary control of the semilinear heat equation. Internat. Ser. Numer. Math. 155 (2007) 119–147.
J.-L. Lions, Optimal control of systems governed by partial differential equations. Springer-Verlag, Berlin (1971).
Malanowski, K., Convergence of approximations versus regularity of solutions for convex, control-constrained optimal control problems. Appl. Math. Optim. 8 (1981) 69–95. CrossRef
J.M. Ortega and W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in several Variables, Computer Science and Applied Mathematics. Academic Press, New York (1970).
K. Rektorys, The Method of Discretization in Time and Partial Differential Equations, Mathematics and Applications 4. D. Reichel Publishing Company, Boston-Dordrecht-London (1982).
R. Temam, Navier-Stokes Equations, Studies in Mathematics and its Applications. North-Holland, Amsterdam (1979).
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators. North-Holland Publishing Company, Amsterdam (1978).
Tröltzsch, F., Regular Lagrange multipliers for control problems with mixed pointwise control-state constraints. SIAM J. Opt. 15 (2005) 616–634. CrossRef
F. Tröltzsch, Optimale Steuerung partieller Differentialgleichungen. Vieweg Verlag, Wiesbaden (2005).