| Title:
|
All-at-once preconditioning in PDE-constrained optimization (English) |
| Author:
|
Rees, Tyrone |
| Author:
|
Stoll, Martin |
| Author:
|
Wathen, Andy |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 |
| Volume:
|
46 |
| Issue:
|
2 |
| Year:
|
2010 |
| Pages:
|
341-360 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The optimization of functions subject to partial differential equations (PDE) plays an important role in many areas of science and industry. In this paper we introduce the basic concepts of PDE-constrained optimization and show how the all-at-once approach will lead to linear systems in saddle point form. We will discuss implementation details and different boundary conditions. We then show how these system can be solved efficiently and discuss methods and preconditioners also in the case when bound constraints for the control are introduced. Numerical results will illustrate the competitiveness of our techniques. (English) |
| Keyword:
|
optimal control |
| Keyword:
|
preconditioning |
| Keyword:
|
partial differential equations |
| MSC:
|
49J20 |
| MSC:
|
49M25 |
| MSC:
|
65F08 |
| MSC:
|
65F10 |
| MSC:
|
65F50 |
| MSC:
|
65K10 |
| MSC:
|
65N22 |
| MSC:
|
76D07 |
| idZBL:
|
Zbl 1195.65083 |
| idMR:
|
MR2663605 |
| . |
| Date available:
|
2010-09-13T16:43:16Z |
| Last updated:
|
2013-07-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140748 |
| . |
| Reference:
|
[1] Bangerth, W., Hartmann, R., Kanschat, G.: deal.II—a general-purpose object-oriented finite element library.ACM Trans. Math. Software 33 (2007), 24, 27. MR 2404402, 10.1145/1268776.1268779 |
| Reference:
|
[2] Benzi, M., Golub, G. H., Liesen, J.: Numerical solution of saddle point problems.Acta Numer. 14 (2005), 1–137. Zbl 1115.65034, MR 2168342, 10.1017/S0962492904000212 |
| Reference:
|
[3] Bergounioux, M., Ito, K., Kunisch, K.: Primal-dual strategy for constrained optimal control problems.SIAM J. Control Optim. 37 (1999), 1176–1194 (electronic). Zbl 0937.49017, MR 1691937, 10.1137/S0363012997328609 |
| Reference:
|
[4] Bertsekas, D. P.: Projected Newton methods for optimization problems with simple constraints.SIAM J. Control Optim. 20 (1982), 221–246. Zbl 0507.49018, MR 0646950, 10.1137/0320018 |
| Reference:
|
[5] Bramble, J. H., Pasciak, J. E.: A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems.Math. Comp 50 (1988), 1–17. Zbl 0649.65061, MR 0917816, 10.1090/S0025-5718-1988-0917816-8 |
| Reference:
|
[6] Collis, S. S., Heinkenschloss, M.: Analysis of the Streamline Upwind/Petrov Galerkin Method Applied to the Solution of Optimal Control Problems.Tech. Rep. TR02–01, Department of Computational and Applied Mathematics, Rice University, Houston, TX 77005–1892, 2002. |
| Reference:
|
[7] Elman, H. C.: Multigrid and Krylov subspace methods for the discrete Stokes equations.In: Seventh Copper Mountain Conference on Multigrid Methods (N. D. Melson, T. A. Manteuffel, S. F. McCormick, and C. C. Douglas, eds.), Vol. CP 3339, Hampton 1996, NASA, pp. 283–299. Zbl 0865.76078 |
| Reference:
|
[8] Elman, H. C., Silvester, D. J., Wathen, A. J.: Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics.Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2005. Zbl 1083.76001, MR 2155549 |
| Reference:
|
[9] Engel, M., Griebel, M.: A multigrid method for constrained optimal control problems: SFB-Preprint 406, Sonderforschungsbereich 611, Rheinische Friedrich-Wilhelms-Universität Bonn, 2008.Submitted. |
| Reference:
|
[10] Gee, M., Siefert, C., Hu, J., Tuminaro, R., Sala, M.: ML 5.0 smoothed aggregation user’s guide.Tech. Rep. SAND2006-2649, Sandia National Laboratories, 2006. |
| Reference:
|
[11] Gill, P. E., Murray, W., Wright, M. H.: Practical Optimization.Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1981. Zbl 0503.90062, MR 0634376 |
| Reference:
|
[12] Golub, G. H., Varga, R. S.: Chebyshev semi-iterative methods.successive over-relaxation iterative methods, and second order Richardson iterative methods. I. Numer. Math. 3 (1961), 147–156. Zbl 0099.10903, MR 0145678, 10.1007/BF01386013 |
| Reference:
|
[13] Golub, G. H., Varga, R. S.: Chebyshev semi-iterative methods, successive over-relaxation iterative methods, and second order Richardson iterative methods.II. Numer. Math. 3 (1961), 157–168. MR 0145679, 10.1007/BF01386014 |
| Reference:
|
[14] Greenbaum, A.: Iterative Methods for Solving Linear Systems.Vol. 17 of Frontiers in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia 1997. Zbl 0883.65022, MR 1474725 |
| Reference:
|
[15] Hestenes, M. R., Stiefel, E.: Methods of conjugate gradients for solving linear systems.J. Res. Nat. Bur. Stand 49 (1952), 409–436 (1953)???. Zbl 0048.09901, MR 0060307, 10.6028/jres.049.044 |
| Reference:
|
[16] Hintermüller, M., Ito, K., Kunisch, K.: The primal-dual active set strategy as a semismooth Newton method.SIAM J. Optim. 13 (2002), 865–888. MR 1972219, 10.1137/S1052623401383558 |
| Reference:
|
[17] Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints.Mathematical Modelling: Theory and Applications. Springer-Verlag, New York 2009. Zbl 1167.49001, MR 2516528 |
| Reference:
|
[18] Ito, K., Kunisch, K.: Lagrange multiplier approach to variational problems and applications.Vol. 15 of Advances in Design and Control, Society for Industrial and Applied Mathematics (SIAM), Philadelphia 2008. Zbl 1156.49002, MR 2441683 |
| Reference:
|
[19] Murphy, M. F., Golub, G. H., Wathen, A. J.: A note on preconditioning for indefinite linear systems.SIAM J. Sci. Comput. 21 (2000), 1969–1972. Zbl 0959.65063, MR 1762024, 10.1137/S1064827599355153 |
| Reference:
|
[20] Nocedal, J., Wright, S. J.: Numerical Optimization.(Springer Series in Operations Research.) Springer-Verlag, New York 1999. Zbl 1104.65059, MR 1713114 |
| Reference:
|
[21] Paige, C. C., Saunders, M. A.: Solutions of sparse indefinite systems of linear equations.SIAM J. Numer. Anal. 12 (1975), 617–629. MR 0383715, 10.1137/0712047 |
| Reference:
|
[22] Peisker, P., Braess, D.: A conjugate gradient method and a multigrid algorithm for Morley’s finite element approximation of the biharmonic equation.Numer. Math. 50 (1987), 567–586. Zbl 0595.65113, MR 0880336, 10.1007/BF01408577 |
| Reference:
|
[23] Rees, T., Dollar, H. S., Wathen, A. J.: Optimal solvers for PDE-constrained optimization.SIAM J. Sci. Comput. 32 (2010), 271–298. MR 2599778, 10.1137/080727154 |
| Reference:
|
[24] Rees, T., Stoll, M.: Block triangular preconditioners for PDE constrained optimization.Numer. Linear Algebra Appl., (2009), to appear. MR 2759604 |
| Reference:
|
[25] Saad, Y.: Iterative Methods for Sparse Linear Systems.Society for Industrial and Applied Mathematics, Philadelphia 2003. Zbl 1031.65046, MR 1990645 |
| Reference:
|
[26] Stoll, M.: Solving Linear Systems Using the Adjoint.PhD. Thesis, University of Oxford 2009. |
| Reference:
|
[27] Stoll, M., Wathen, A.: Preconditioning for active set and projected gradient methods as semi-smooth Newton methods for PDE-constrained optimization with control constraints.Submitted. |
| Reference:
|
[28] Tröltzsch, F.: Optimale Steuerung partieller Differentialgleichungen: Theorie, Verfahren und Anwendungen.Vieweg Verlag, Wiesbaden 2005. |
| Reference:
|
[29] Wathen, A. J.: Realistic eigenvalue bounds for the Galerkin mass matrix.IMA J. Numer. Anal. 7 (1987), 449–457. Zbl 0648.65076, MR 0968517, 10.1093/imanum/7.4.449 |
| Reference:
|
[30] Wathen, A. J., Rees, T.: Chebyshev semi-iteration in preconditioning for problems including the mass matrix.Electron. Trans. Numer. Anal. 34 (2008–2009), 125–135. Zbl 1189.65065, MR 2597805 |
| . |
