Estimating the control error in discretized PDE-constrained optimization

Journal of Numerical Mathematics

Volumn 14, Issue 3, 2006, Pages 163-185

  Becker, R ^a  

^a UNIVERSITÉ DE PAU ET DES PAYS DE L'ADOUR   (France)

Author keywords

Adaptive methods; Finite elements; Optimal control

Indexed keywords

ADAPTIVE METHODS; SECOND-ORDER OPTIMIZATION ITERATION;

ADAPTIVE ALGORITHMS; CONSTRAINT THEORY; DISCRETE TIME CONTROL SYSTEMS; FINITE ELEMENT METHOD; ITERATIVE METHODS; OPTIMAL CONTROL SYSTEMS; OPTIMIZATION; PARTIAL DIFFERENTIAL EQUATIONS;

ERROR ANALYSIS;

EID: 33846242194     PISSN: 15702820     EISSN: None     Source Type: Journal    
DOI: 10.1163/156939506778658302     Document Type: Article

References (20)

1. I. Babuška and W. C. Rheinboldt, Error estimates for adaptive finite element computations. SIAM J. Numer Anal. (1978) 15, 736-754.
2. R. Becker, H. Kapp, and R. Rannacher, Adaptive finite element methods for optimal control of partial differential equations: Basic concepts. SIAM J. Control Optimization (2000) 39, No. 1, 113-132.
3. R. Becker and B. Vexler, A posteriori error estimation for finite element discretizations of parameter identification problems. Numer. Math. (2004) 96, No. 3, 435-459.
4. P. Binev, W. Dahmen, and R. DeVore, Adaptive finite element methods with convergence rates. Numerische Mathematik (2004).
5. J. F. Bonnans, Second order analysis for control constrained optimal control problems of semi-linear elliptic systems. Appl. Math. Optimization (1998) 38, No. 3, 303 -325.
6. E. Casas and F. Tröltzsch, Error estimates for linear-quadratic elliptic control problems. In: Analysis and Optimization of Differential Systems (Eds. V. Barbu, I. Lasiecka, D. Tiba, and C. Varsan), IFIP TC7/WG 7.2 International Working Conference, Constanta, Romania, September 10-14, 2002. Boston, MA: Kluwer Academic Publishers, 2003. pp. 89-100.
7. Z. Chen and R. H. Nochetto, Residual type a posteriori error estimates for elliptic obstacle problems. Numer. Math. (2000) 84, No. 4, 527-548.
8. W. Dahmen and A. Kunoth, Adaptive wavelet methods for linear-quadratic elliptic control prob lems: convergence rates. SIAM J. Control Optimization (2005) 43, No. 5, 1640-1675.
9. K. Eriksson, D. Estep, P. Hansbo, and C. Johnson, Introduction to adaptive methods for differential equations. In: Acta Numerica (Ed. A. Iserles), Cambridge University Press, 1995, pp. 105-158.
10. V. Girault and P.-A. Raviart, Finite Elements for the Navier Stokes Equations. Springer, Berlin, 1986.
11. A. Haraux, How to differentiate the projection on a convex set in Hilbert space. Some applications to variational inequalities. J. Math. Soc. Japan (1977) 29, 615 -631.
12. R. Li, W. Liu, H. Ma, and T. Tang, Adaptive finite element approximation for distributed elliptic optimal control problems. SIAM J. Control Optimization (2002) 41, No. 5, 1321-1349.
13. W. Liu and N. Yan, A posteriori error estimates for convex boundary control problems. SIAM J. Numer. Anal. (2001) 39, No. 1, 73 -99.
14. K. Malanowski, Convergence of approximations vs. regularity of solutions for convex, control-constrained optimal-control problems. Appl. Math. Optimization (1981) 8, 69-95.
15. H. Maurer and J. Zowe, First and second-order necessary and sufficient optimality conditions for infinite-dimensional programming problems. Math. Program. (1979) 16, 98-110.
16. R. H. Nochetto, K. G. Siebert, and A. Veeser, Fully localized a posteriori error estimators and barrier sets for contact problems. SIAM J. Numer. Anal. (2005) 42, No. 5, 2118-2135.
17. E. Polak, Optimization. Algorithms and consistent approximations. Appl. Math. Sci. (1997), No. 124.
18. A. Rösch, Error estimates for parabolic optimal control problems with control constraints. ZAA (2004) 23, 353-376.
19. A. Veeser, Efficient and reliable a posteriori error estimators for elliptic obstacle problems. SIAM J. Numer. Anal. (2001) 39, No. 1, 146-167.
20. R. Verfürth, A Review of Error Estimation and Adaptive Mesh-Refinement Techniques. John Wiley/Teubner, New York-Stuttgart, 1996.