Symplectic Pontryagin approximations for optimal design

Jesper Carlsson; Mattias Sandberg; Anders Szepessy

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2009)

  • Volume: 43, Issue: 1, page 3-32
  • ISSN: 0764-583X

Abstract

top
The powerful Hamilton-Jacobi theory is used for constructing regularizations and error estimates for optimal design problems. The constructed Pontryagin method is a simple and general method for optimal design and reconstruction: the first, analytical, step is to regularize the hamiltonian; next the solution to its stationary hamiltonian system, a nonlinear partial differential equation, is computed with the Newton method. The method is efficient for designs where the hamiltonian function can be explicitly formulated and when the jacobian is sparse, but becomes impractical otherwise (e.g. for non local control constraints). An error estimate for the difference between exact and approximate objective functions is derived, depending only on the difference of the hamiltonian and its finite dimensional regularization along the solution path and its L 2 projection, i.e. not on the difference of the exact and approximate solutions to the hamiltonian systems.

How to cite

top

Carlsson, Jesper, Sandberg, Mattias, and Szepessy, Anders. "Symplectic Pontryagin approximations for optimal design." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 43.1 (2009): 3-32. <http://eudml.org/doc/245361>.

@article{Carlsson2009,
abstract = {The powerful Hamilton-Jacobi theory is used for constructing regularizations and error estimates for optimal design problems. The constructed Pontryagin method is a simple and general method for optimal design and reconstruction: the first, analytical, step is to regularize the hamiltonian; next the solution to its stationary hamiltonian system, a nonlinear partial differential equation, is computed with the Newton method. The method is efficient for designs where the hamiltonian function can be explicitly formulated and when the jacobian is sparse, but becomes impractical otherwise (e.g. for non local control constraints). An error estimate for the difference between exact and approximate objective functions is derived, depending only on the difference of the hamiltonian and its finite dimensional regularization along the solution path and its $\mathrm \{L\}^\{2\}$ projection, i.e. not on the difference of the exact and approximate solutions to the hamiltonian systems.},
author = {Carlsson, Jesper, Sandberg, Mattias, Szepessy, Anders},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {topology optimization; inverse problems; Hamilton-Jacobi; regularization; error estimates; impedance tomography; convexification; homogenization; optimal design problems; Pontryagin method; Newton method},
language = {eng},
number = {1},
pages = {3-32},
publisher = {EDP-Sciences},
title = {Symplectic Pontryagin approximations for optimal design},
url = {http://eudml.org/doc/245361},
volume = {43},
year = {2009},
}

TY - JOUR
AU - Carlsson, Jesper
AU - Sandberg, Mattias
AU - Szepessy, Anders
TI - Symplectic Pontryagin approximations for optimal design
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2009
PB - EDP-Sciences
VL - 43
IS - 1
SP - 3
EP - 32
AB - The powerful Hamilton-Jacobi theory is used for constructing regularizations and error estimates for optimal design problems. The constructed Pontryagin method is a simple and general method for optimal design and reconstruction: the first, analytical, step is to regularize the hamiltonian; next the solution to its stationary hamiltonian system, a nonlinear partial differential equation, is computed with the Newton method. The method is efficient for designs where the hamiltonian function can be explicitly formulated and when the jacobian is sparse, but becomes impractical otherwise (e.g. for non local control constraints). An error estimate for the difference between exact and approximate objective functions is derived, depending only on the difference of the hamiltonian and its finite dimensional regularization along the solution path and its $\mathrm {L}^{2}$ projection, i.e. not on the difference of the exact and approximate solutions to the hamiltonian systems.
LA - eng
KW - topology optimization; inverse problems; Hamilton-Jacobi; regularization; error estimates; impedance tomography; convexification; homogenization; optimal design problems; Pontryagin method; Newton method
UR - http://eudml.org/doc/245361
ER -

References

top
  1. [1] G. Allaire, Shape optimization by the homogenization method, Applied Mathematical Sciences 146. Springer-Verlag, New York (2002). Zbl0990.35001MR1859696
  2. [2] M. Bardi and I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations, Systems & Control: Foundations & Applications. Birkhäuser Boston Inc., Boston, USA (1997). With appendices by M. Falcone and P. Soravia. Zbl0890.49011MR1484411
  3. [3] G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi, Mathématiques & Applications 17 (Berlin) [Mathematics & Applications]. Springer-Verlag, Paris (1994). Zbl0819.35002MR1613876
  4. [4] M.P. Bendsøe and O. Sigmund, Topology optimization, Theory, methods and applications. Springer-Verlag, Berlin (2003). Zbl1059.74001MR2008524
  5. [5] L. Borcea, Electrical impedance tomography. Inverse Problems 18 (2002) R99–R136. Zbl1031.35147MR1955896
  6. [6] S.C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics 15. Springer-Verlag, New York (1994). Zbl0804.65101MR1278258
  7. [7] P. Cannarsa and H. Frankowska, Some characterizations of optimal trajectories in control theory. SIAM J. Control Optim. 29 (1991) 1322–1347. Zbl0744.49011MR1132185
  8. [8] P. Cannarsa and H. Frankowska, Value function and optimality conditions for semilinear control problems. Appl. Math. Optim. 26 (1992) 139–169. Zbl0765.49001MR1166210
  9. [9] P. Cannarsa and H. Frankowska, Value function and optimality condition for semilinear co problems. II. Parabolic case. Appl. Math. Optim. 33 (1996) 1–33. Zbl0862.49021MR1359346
  10. [10] P. Cannarsa and C. Sinestrari, Semiconcave functions, Hamilton-Jacobi equations, and optimal control, Progress in Nonlinear Differential Equations and their Applications 58. Birkhäuser Boston Inc., Boston, USA (2004). Zbl1095.49003MR2041617
  11. [11] J. Carlsson, Symplectic reconstruction of data for heat and wave equations. Preprint (2008) http://arxiv.org/abs/0809.3621. 
  12. [12] J. Céa, S. Garreau, P. Guillaume and M. Masmoudi, The shape and topological optimizations connection. Comput. Methods Appl. Mech. Engrg. 188 (2000) 713–726. IV WCCM, Part II (Buenos Aires, 1998). Zbl0972.74057MR1784106
  13. [13] M. Cheney and D. Isaacson, Distinguishability in impedance imaging. IEEE Trans. Biomed. Eng. 39 (1992) 852–860. 
  14. [14] F.H. Clarke, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series in Mathematics. John Wiley and Sons, Inc. (1983). Zbl0582.49001MR709590
  15. [15] M.G. Crandall, L.C. Evans and P.-L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 282 (1984) 487–502. Zbl0543.35011MR732102
  16. [16] M.G. Crandall, H. Ishii and P.-L. Lions, User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 (1992) 1–67. Zbl0755.35015MR1118699
  17. [17] M. Crouzeix and V. Thomée, The stability in L p and W p 1 of the L 2 -projection onto finite element function spaces. Math. Comp. 48(178) (1987) 521–532. Zbl0637.41034MR878688
  18. [18] B. Dacorogna, Direct methods in the calculus of variations, Appl. Math. Sci. 78. Springer-Verlag, Berlin (1989). Zbl0703.49001MR990890
  19. [19] H.W. Engl, M. Hanke and A. Neubauer, Regularization of inverse problems, Mathematics and its Applications 375. Kluwer Academic Publishers Group, Dordrecht (1996). Zbl0859.65054MR1408680
  20. [20] L.C. Evans, Partial differential equations, Graduate Studies in Mathematics 19. American Mathematical Society, Providence, USA (1998). Zbl0902.35002
  21. [21] H. Frankowska, Contingent cones to reachable sets of control systems. SIAM J. Control Optim. 27 (1989) 170–198. Zbl0671.49030MR980229
  22. [22] J. Goodman, R.V. Kohn and L. Reyna, Numerical study of a relaxed variational problem from optimal design. Comput. Methods Appl. Mech. Engrg. 57 (1986) 107–127. Zbl0591.73119MR859964
  23. [23] E. Hairer, C. Lubich and G. Wanner, Geometric numerical integration, Structure-preserving algorithms for ordinary differential equations, Springer Series in Computational Mathematics 31. Springer-Verlag, Berlin (2002). Zbl0994.65135MR1904823
  24. [24] B. Kawohl, J. Stará and G. Wittum, Analysis and numerical studies of a problem of shape design. Arch. Rational Mech. Anal. 114 (1991) 349–363. Zbl0726.65071MR1100800
  25. [25] R.V. Kohn and A. McKenney, Numerical implementation of a variational method for electrical impedance tomography. Inverse Problems 6 (1990) 389–414. Zbl0718.65089MR1057033
  26. [26] R.V. Kohn and G. Strang, Optimal design and relaxation of variational problems. I. Comm. Pure Appl. Math. 39 (1986) 113–137. Zbl0609.49008MR820342
  27. [27] R.V. Kohn and G. Strang, Optimal design and relaxation of variational problems. II. Comm. Pure Appl. Math. 39 (1986) 139–182. Zbl0621.49008MR820067
  28. [28] R.V. Kohn and G. Strang, Optimal design and relaxation of variational problems. III. Comm. Pure Appl. Math. 39 (1986) 353–377. Zbl0694.49004MR829845
  29. [29] F. Natterer, The mathematics of computerized tomography, Classics in Applied Mathematics 32. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, USA (2001). Reprint of the 1986 original. Zbl0973.92020MR1847845
  30. [30] O. Pironneau, Optimal shape design for elliptic systems, Springer Series in Computational Physics. Springer-Verlag, New York (1984). Zbl0534.49001MR725856
  31. [31] R.T. Rockafellar, Convex analysis, Princeton Mathematical Series 28. Princeton University Press, Princeton, USA (1970). Zbl0193.18401MR274683
  32. [32] M. Sandberg, Convergence rates for numerical approximations of an optimally controlled Ginzburg-Landau equation. Preprint (2008) http://arxiv.org/abs/0809.1834. 
  33. [33] M. Sandberg and A. Szepessy, Convergence rates of symplectic Pontryagin approximations in optimal control theory. ESAIM: M2AN 40 (2006) 149–173. Zbl1091.49027MR2223508
  34. [34] D. Tataru, Viscosity solutions for Hamilton-Jacobi equations with unbounded nonlinear term: a simplified approach. J. Diff. Eq. 111 (1994) 123–146. Zbl0810.34060MR1280618
  35. [35] A.N. Tikhonov and V.Y. Arsenin, Solutions of ill-posed problems, in Scripta Series in Mathematics, V.H. Winston & Sons, Washington, D.C., John Wiley & Sons, New York (1977). Translated from the Russian, Preface by translation editor F. John. Zbl0354.65028MR455365
  36. [36] C.R. Vogel, Computational methods for inverse problems, Frontiers in Applied Mathematics 23. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2002). With a foreword by H.T. Banks. Zbl1008.65103MR1928831
  37. [37] A. Wexler, B. Fry and M.R. Neuman, Impedance-computed tomography algorithm and system. Appl. Opt. 24 (1985) 3985–3992. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.