Mesh-independence and preconditioning for solving parabolic control problems with mixed control-state constraints
ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 3, pp. 626-652.

Optimal control problems for the heat equation with pointwise bilateral control-state constraints are considered. A locally superlinearly convergent numerical solution algorithm is proposed and its mesh independence is established. Further, for the efficient numerical solution reduced space and Schur complement based preconditioners are proposed which take into account the active and inactive set structure of the problem. The paper ends by numerical tests illustrating our theoretical findings and comparing the efficiency of the proposed preconditioners.

DOI : https://doi.org/10.1051/cocv:2008042
Classification : 49K20,  65K05
Mots clés : bilateral control-state constraints, heat equation, mesh independence, optimal control, PDE-constrained optimization, semismooth Newton method
@article{COCV_2009__15_3_626_0,
     author = {Hinterm\"uller, Michael and Kopacka, Ian and Volkwein, Stefan},
     title = {Mesh-independence and preconditioning for solving parabolic control problems with mixed control-state constraints},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {626--652},
     publisher = {EDP-Sciences},
     volume = {15},
     number = {3},
     year = {2009},
     doi = {10.1051/cocv:2008042},
     zbl = {1167.49027},
     mrnumber = {2542576},
     language = {en},
     url = {www.numdam.org/item/COCV_2009__15_3_626_0/}
}
Hintermüller, Michael; Kopacka, Ian; Volkwein, Stefan. Mesh-independence and preconditioning for solving parabolic control problems with mixed control-state constraints. ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 3, pp. 626-652. doi : 10.1051/cocv:2008042. http://www.numdam.org/item/COCV_2009__15_3_626_0/

[1] R.A. Adams, Sobolev Spaces, Pure and Applied Mathematics 65. Academic Press, New York-London (1975). | MR 450957 | Zbl 0314.46030

[2] 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. | MR 1627643 | Zbl 0909.49015

[3] 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. | MR 1941049 | Zbl 0992.49022

[4] G. Biros and O. Ghattas, Parallel Lagrange-Newton-Krylov-Schur methods for PDE-constrained optimization. I. The Krylov-Schur solver. SIAM J. Sci. Comput. 27 (2005) 687-713. | MR 2202240 | Zbl 1091.65061

[5] R. Dautray and J.-L. Lions, Evolution Problems I, Mathematical Analysis and Numerical Methods for Science and Technology 5. Springer-Verlag, Berlin (1992). | MR 1156075

[6] L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics 19. American Mathematical Society, Providence, Rhode Island (1998). | MR 1625845 | Zbl 0902.35002

[7] C. Geiger and C. Kanzow, Theorie und Numerik restringierter Optimierungsaufgaben. Springer-Verlag, Berlin (2002). | Zbl 1003.90044

[8] W. Hackbusch, Optimal H p,p/2 error estimates for a parabolic Galerkin method. SIAM J. Numer. Anal. 18 (1981) 681-692. | MR 622703 | Zbl 0483.65063

[9] M. Hintermüller, 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. | MR 2378147 | Zbl 1154.65057

[10] M. Hintermüller and M. Hinze, A SQP-semismooth Newton-type algorithm applied to control of the instationary Navier-Stokes system subject to control constraints. SIAM J. Opt. 16 (2006) 1177-1200. | MR 2219138 | Zbl 1131.90073

[11] M. Hintermüller and M. Ulbrich, A mesh-independence result for semismooth Newton methods. Math. Program. Ser. B 101 (2004) 151-184. | MR 2085262 | Zbl 1079.65065

[12] M. Hintermüller, K. Ito and K. Kunisch, The primal-dual active set strategy as a semi-smooth Newton method. SIAM J. Opt. 13 (2003) 865-888. | MR 1972219 | Zbl 1080.90074

[13] M. Hintermüller, S. Volkwein and F. Diwoky, Fast solution techniques in constrained optimal boundary control of the semilinear heat equation. Internat. Ser. Numer. Math. 155 (2007) 119-147. | MR 2328613 | Zbl 1239.49039

[14] J.-L. Lions, Optimal control of systems governed by partial differential equations. Springer-Verlag, Berlin (1971). | MR 271512 | Zbl 0203.09001

[15] K. Malanowski, Convergence of approximations versus regularity of solutions for convex, control-constrained optimal control problems. Appl. Math. Optim. 8 (1981) 69-95. | MR 646505 | Zbl 0479.49017

[16] 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). | MR 273810 | Zbl 0241.65046

[17] K. Rektorys, The Method of Discretization in Time and Partial Differential Equations, Mathematics and Applications 4. D. Reichel Publishing Company, Boston-Dordrecht-London (1982). | MR 689712 | Zbl 0505.65029

[18] R. Temam, Navier-Stokes Equations, Studies in Mathematics and its Applications. North-Holland, Amsterdam (1979). | MR 603444 | Zbl 0426.35003

[19] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators. North-Holland Publishing Company, Amsterdam (1978). | MR 503903 | Zbl 0387.46032

[20] F. Tröltzsch, Regular Lagrange multipliers for control problems with mixed pointwise control-state constraints. SIAM J. Opt. 15 (2005) 616-634. | MR 2144184 | Zbl 1083.49018

[21] F. Tröltzsch, Optimale Steuerung partieller Differentialgleichungen. Vieweg Verlag, Wiesbaden (2005). | Zbl 1142.49001