Adaptive space-time finite element methods for parabolic optimization problems

SIAM Journal on Control and Optimization

Volumn 46, Issue 1, 2007, Pages 116-142

  Meidner, Dominik ^a   Vexler, Boris ^b  

^a UNIVERSITY OF HEIDELBERG   (Germany)

^b JOHANN RADON INSTITUTE FOR COMPUTATIONAL AND APPLIED MATHEMATICS   (Austria)

Author keywords

A posteriori error estimation; Mesh refinement; Optimal control; Parabolic equations; Parameter identification

Indexed keywords

ADAPTIVE ALGORITHMS; ERROR ANALYSIS; FINITE ELEMENT METHOD; MESH GENERATION;

A POSTERIORI ERROR ESTIMATION; OPTIMAL CONTROL; PARABOLIC EQUATIONS; SPACE DISCRETIZATIONS;

IDENTIFICATION (CONTROL SYSTEMS);

EID: 39449084428     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/060648994     Document Type: Article

References (29)

1. R. BECKER, H. KAPP, AND R. RANNACHER, Adaptive finite element methods for optimal control of partial differential equations: Basic concepts, SIAM J. Control Optim., 39 (2000), pp. 113-132.
2. R. BECKER, D. MEIDNER, AND B. VEXLER, Efficient numerical solution of parabolic optimization problems by finite element methods, Optim. Methods Softw., to appear.
3. R. BECKER AND R. RANNACHER, An Optimal Control Approach to A-Posteriori Error Estimation, Acta Numer. 2001, Arieh Iserles, ed., Cambridge University Press, London, 2001, pp. 1-102.
4. R. BECKER AND B. VEXLER, A posteriori error estimation for finite element discretizations of parameter identification problems, Numer. Math., 96 (2004), pp. 435-459.
5. R. BECKER AND B. VEXLER, Mesh refinement and numerical sensitivity analysis for parameter calibration of partial differential equations, J. Comput. Phys., 206 (2005), pp. 95-110.
6. P. G. CIARLET, The Finite Element Method for Elliptic Problems, Classics Appl. Math. 40, SIAM, Philadelphia, 2002.
7. A. R. CONN, N. GOULD, AND PH. L. TOINT, Trust-Region Methods, MPS/SIAM Ser. Optim., SIAM, Philadelphia, 2000.
8. R. DAUTRAY AND J.-L. LIONS, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 5, Springer-Verlag, Berlin, 1992.
9. K. EHIKSSON, D. ESTEP, P. HANSBO, AND C. JOHNSON, Introduction to Adaptive Methods for Differential Equations, Acta Numer. 1995, Arieh Iserles, ed., Cambridge University Press, London, 1995, pp. 105-158.
10. K. ERIKSSON, D. ESTEP, P. HANSBO, AND C. JOHNSON, Computational differential equations, Cambridge University Press, Cambridge, 1996.
11. K. ERIKSSON, C. JOHNSON, AND V. THOMÉE, Time discretization of parabolic problems by the discontinuous Galerkin method, RAIRO Modelisation Math. Anal. Numer., 19 (1985), pp. 611-643.
12. K. ERIKSSON AND C. JOHNSON, Adaptive finite element methods for parabolic problems V: Long-time integration, SIAM J. Numer. Anal., 32 (1995), pp. 1750-1763.
13. A. V. FURSIKOV, Optimal Control of Distributed Systems: Theory and Applications, Transl. Math. Monogr. 187, AMS, Providence, 1999.
14. A. GRIEWANK AND A. WALTHER, Revolve: An implementation of checkpointing for the reverse or adjoint mode of computational differentiation, ACM Trans. Math. Software, 26 (2000), pp. 19-45.
15. A. GRIEWANK, Achieving logarithmic growth of temporal and spatial complexity in reverse automatic differentiation, Optim. Methods Softw., 1 (1992), pp. 35-54.
16. M. HINZE, A variational discretization concept in control constrained optimization: The linear-quadratic case, Comput. Optim. Appl., 30 (2005), pp. 45-61.
17. J. LANG, Adaptive Multilevel Solution of Nonlinear Parabolic PDE Systems. Theory, Algorithm, and Applications, Lecture Notes in Earth Sci. 16, Springer-Verlag, Berlin, 1999.
18. J.-L. LIONS, Optimal Control of Systems Governed by Partial Differential Equations, Grundlehren Math. Wiss. 170, Springer-Verlag, Berlin, 1971.
19. W. LIU, H. MA, T. TANG, AND N. YAN, A posteriori error estimates for discontinuous galerkin time-stepping method for optimal control problems governed by parabolic equations, SIAM J. Numer. Anal., 42 (2004), pp. 1032-1061.
20. W. LIU AND N. YAN, A posteriori error estimates for distributed convex optimal control problems, Adv. Comput. Math, 15 (2001), pp. 285-309.
21. W. LIU AND N. YAN, A posteriori error estimates for control problems governed by nonlinear elliptic equations, Appl. Numer. Math., 47 (2003), pp. 173-187.
22. C. MEYER AND A. RÖSCH, Superconvergence properties of optimal control problems, SIAM J. Control Optim., 43 (2004), pp. 970-985.
23. J. NOCEDAL AND S. J. WRIGHT, Numerical Optimization, Springer Ser. Oper. Res., Springer-Verlag, New York, 1999.
24. M. PICASSO, Anisotropic A Posteriori Error Estimates for an Optimal Control Problem Governed by the Heat Equation, Internat. J. Numer. Methods PDE, 22 (2006), pp. 1314-1336.
25. R. RANNACHER AND B. VEXLER, A priori error estimates for the finite element discretization of elliptic parameter identification problems with pointwise measurements, SIAM J. Control Optim., 44 (2005), pp. 1844-1863.
26. M. SCHMICH AND B. VEXLER, Adaptivity with dynamic meshes for space-time finite element discretizations of parabolic equations, SIAM J. Sci. Comput., submitted.
27. F. TRÖLTZSCH, Optimale Steuerung Partieller Differentialgleichungen, Friedr. Vieweg &. Sohn Verlag, Wiesbaden, 2005.
28. R. VERFÜRTH, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley/Teubner, New York/Stuttgart, 1996.
29. B. VEXLER, Adaptive Finite Elements for Parameter Identification Problems, Ph.D. thesis, Institut für Angewandte Mathematik, Universität Heidelberg, 2004.