Transforming smoothers for PDE constrained optimization problems

Computing and Visualization in Science

Volumn 11, Issue 4-6, 2008, Pages 207-219

  Schulz, Volker ^a   Wittum, Gabriel ^b  

^a UNIVERSITY OF TRIER   (Germany)

^b UNIVERSITY OF HEIDELBERG   (Germany)

Author keywords

Large scale optimization; Multigrid methods; Optimal control problems; Quadratic programming; Saddlepoint problems; Transforming smoothers

Indexed keywords

BESSEL FUNCTIONS; CONSTRAINT THEORY; CONVERGENCE OF NUMERICAL METHODS; DIFFERENCE EQUATIONS; DIFFERENTIAL EQUATIONS; DIFFERENTIATION (CALCULUS); LEAD; NUMERICAL ANALYSIS; OPTIMIZATION; PARTIAL DIFFERENTIAL EQUATIONS;

CONVERGENCE PROPERTIES; MODEL PROBLEMS; MULTI GRID; MULTIGRID (MG) METHOD; NUMERICAL INVESTIGATIONS; OPTIMIZ ATION PROBLEMS; PARTIAL DIFFERENTIAL; PDE-CONSTRAINED OPTIMIZATION; SADDLE POINT PROBLEMS (SPP); SADDLE POINTS; SPRINGER (CO); VARIATIONAL PROBLEMS;

CONSTRAINED OPTIMIZATION;

EID: 47849129989     PISSN: 14329360     EISSN: 14330369     Source Type: Journal    
DOI: 10.1007/s00791-008-0094-0     Document Type: Article

References (26)

1. Antil, H., Hoppe, R.H.W., Linsenmann, C.: Path-following primal-dual interior point methods for shape optimization of stationary flow problems. J. Numer. Math. 11, 81-100 (2007)
2. Ascher, U., Haber, E.: A multigrid method for distributed parameter estimation problems. ETNA 18, 1-18 (2003)
3. Borzi, A., Kunisch, K., Kwak, D.Y.: Accuracy and convergence properties of the finite difference multigrid solution of an optimal control optimality system. SIAM J. Control Opt. 41(5), 1477-1497 (2004)
4. Borzi, A., Schulz, V.: Multigrid methods for PDE optimization. SIAM Rev. (2007) (to appear)
5. Braess, D., Sarazin, R.: An efficient smoother for the Stokes problem. Appl. Numer. Math. 23, 3-19 (1996)
6. Dreyer, Th., Maar, B., Schulz, V.: Multigrid Optimization in Applications. J. Comput. Appl. Math. 120, 67-84 (2000)
7. Hackbusch, W.: On the fast solving of parabolic boundary control problems. SIAM J. Control Optim. 17, 231-244 (1979)
8. Hackbusch, W.: Fast solution of elliptic control problems. J. Optim. Theory Appl. 31, 565-581 (1980)
9. Hackbusch, W.: Numerical solution of linear and nonlinear parabolic control problems. In: Auslender, A., Oettli, W., Stoer, J.(eds) Optimization and Optimal Control, pp. 179-185. Springer, New York (1981)
10. Hackbusch, W.: Multigrid Methods and Applications. Springer, New York (1985)
11. Hackbusch, W., Trottenberg, U. (eds.): Multigrid Methods. Springer, New York (1982)
12. Hackbusch, W., Will, Th.: A numerical method for a parabolic bang-bang problem. In: Hoffmann, K.-H., Krabs, W.(eds) Optimal Control of Partial Differential Equations., pp. 20-45. Birkhäuser, Basel (1982)
13. Hackbusch, W., Will, Th.: A numerical method for a parabolic bang-bang problem. Control Cyberne. 12, 99-116 (1983)
14. Hoppe, R.H.W., Linsenmann, C., Petrova, S.I.: Primal-dual Newton methods in structural optimization. Comp. Visual. Sci. 9, 71-87 (2006)
15. Hoppe, R.H.W., Petrova, S.I.: Primal-dual Newton interior point methods in shape and topology optimization. Numer. Linear Algebra Appl. 11, 413-429 (2004)
16. Hoppe, R.H.W., Petrova, S.I., Schulz, V.H.: A primal-dual Newton-type interior-point method for topology optimization. J. Optim. Theory Appl. 114(3), 545-571 (2002)
17. Kelley, C.T., Sachs, E.W.: Fast algorithms for compact fixed point problems with inexact function evaluations. SIAM J. Sci. Stat. Comput. 12(4), 725-742 (1991)
18. Maar, B.: Nicht-Newtonsche Fluide. Mehrgitterverfahren für Bingham-Strömungen. Ph.D. thesis, Universität Stuttgart (1998)
19. Maar, B., Schulz, V.: Interior point multigrid methods for topology optimization. Struct. Optim. 19(3), 214-224 (2000)
20. Nash, S.: A multigrid approach to discretized optimization problems. Optim. Methods Softw. 14, 99-116 (2000)
21. Schulz, V., Bardossy, A., Helmig, R.: Conditional statistical inverse modeling in groundwater flow by multigrid methods. Comput. Geosci. 3, 49-68 (1999)
22. Schulz, V.H.: Solving discretized optimization problems by partially reduced SQP methods. Comput. Visual. Sci. 1(2), 83-96 (1998)
23. Schulz, V.H., Dreyer, Th., Speer, Th., Bock, H.G.: Optimum shape design of turbine blades. In: Kleinschmidt, P., Bachem, A., Derigs, U., Fischer, D., Leopold-Wildburger, U., Möhring, R.(eds) Operations Research Proceedings 1995, pp. 190-195. Springer, Heidelberg (1996)
24. Schulz, V.H., Wittum, G.: Multigrid optimization methods for stationary parameter identification problems in groundwater flow. In: Hackbusch W., Wittum G. (eds.) Multigrid Methods V, Lecture Notes in Computational Science and Engineering, vol. 3, pp. 276-288. Springer, New York (1998)
25. Wittum, G.: Multi-grid methods for the Stokes and Navier-Stokes equations. Numer. Math. 54, 543-563 (1989)
26. Wittum, G.: On the convergence of multigrid methods with transforming smoothers. theory with application to the Navier-Stokes equations. Numer. Math. 57, 15-38 (1989)