Mesh-independence and preconditioning for solving parabolic control problems with mixed control-state constraints
Michael Hintermüller; Ian Kopacka; Stefan Volkwein
ESAIM: Control, Optimisation and Calculus of Variations (2009)
- Volume: 15, Issue: 3, page 626-652
- ISSN: 1292-8119
Access Full Article
topAbstract
topHow to cite
topReferences
top- [1] R.A. Adams, Sobolev Spaces, Pure and Applied Mathematics 65. Academic Press, New York-London (1975). Zbl0314.46030MR450957
- [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. Zbl0909.49015MR1627643
- [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. Zbl0992.49022MR1941049
- [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. Zbl1091.65061MR2202240
- [5] R. Dautray and J.-L. Lions, Evolution Problems I, Mathematical Analysis and Numerical Methods for Science and Technology 5. Springer-Verlag, Berlin (1992). Zbl0755.35001MR1156075
- [6] L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics 19. American Mathematical Society, Providence, Rhode Island (1998). Zbl0902.35002MR1625845
- [7] C. Geiger and C. Kanzow, Theorie und Numerik restringierter Optimierungsaufgaben. Springer-Verlag, Berlin (2002). Zbl1003.90044
- [8] W. Hackbusch, Optimal error estimates for a parabolic Galerkin method. SIAM J. Numer. Anal. 18 (1981) 681–692. Zbl0483.65063MR622703
- [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. Zbl1154.65057MR2378147
- [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. Zbl1131.90073MR2219138
- [11] M. Hintermüller and M. Ulbrich, A mesh-independence result for semismooth Newton methods. Math. Program. Ser. B 101 (2004) 151–184. Zbl1079.65065MR2085262
- [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. Zbl1080.90074MR1972219
- [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. Zbl1239.49039MR2328613
- [14] J.-L. Lions, Optimal control of systems governed by partial differential equations. Springer-Verlag, Berlin (1971). Zbl0203.09001MR271512
- [15] K. Malanowski, Convergence of approximations versus regularity of solutions for convex, control-constrained optimal control problems. Appl. Math. Optim. 8 (1981) 69–95. Zbl0479.49017MR646505
- [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). Zbl0241.65046MR273810
- [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). Zbl0505.65029MR689712
- [18] R. Temam, Navier-Stokes Equations, Studies in Mathematics and its Applications. North-Holland, Amsterdam (1979). Zbl0426.35003MR603444
- [19] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators. North-Holland Publishing Company, Amsterdam (1978). Zbl0387.46032MR503903
- [20] F. Tröltzsch, Regular Lagrange multipliers for control problems with mixed pointwise control-state constraints. SIAM J. Opt. 15 (2005) 616–634. Zbl1083.49018MR2144184
- [21] F. Tröltzsch, Optimale Steuerung partieller Differentialgleichungen. Vieweg Verlag, Wiesbaden (2005). Zbl1142.49001