Symplectic Pontryagin approximations for optimal design

Jesper Carlsson; Mattias Sandberg; Anders Szepessy

ESAIM: Mathematical Modelling and Numerical Analysis (2008)

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

Abstract

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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 L2 projection, i.e. not on the difference of the exact and approximate solutions to the Hamiltonian systems.

How to cite

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Carlsson, Jesper, Sandberg, Mattias, and Szepessy, Anders. "Symplectic Pontryagin approximations for optimal design." ESAIM: Mathematical Modelling and Numerical Analysis 43.1 (2008): 3-32. <http://eudml.org/doc/194446>.

@article{Carlsson2008,
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 L2 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},
keywords = {Topology optimization; inverse problems; Hamilton-Jacobi; regularization; error estimates; impedance tomography; convexification; homogenization.; topology optimization; regularization; homogenization; optimal design problems; Pontryagin method; Newton method},
language = {eng},
month = {10},
number = {1},
pages = {3-32},
publisher = {EDP Sciences},
title = {Symplectic Pontryagin approximations for optimal design},
url = {http://eudml.org/doc/194446},
volume = {43},
year = {2008},
}

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
DA - 2008/10//
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 L2 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.; topology optimization; regularization; homogenization; optimal design problems; Pontryagin method; Newton method
UR - http://eudml.org/doc/194446
ER -

References

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  1. G. Allaire, Shape optimization by the homogenization method, Applied Mathematical Sciences146. Springer-Verlag, New York (2002).  Zbl0990.35001
  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.  Zbl0890.49011
  3. G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi, Mathématiques & Applications17 (Berlin) [Mathematics & Applications]. Springer-Verlag, Paris (1994).  Zbl0819.35002
  4. M.P. Bendsøe and O. Sigmund, Topology optimization, Theory, methods and applications. Springer-Verlag, Berlin (2003).  Zbl1059.74001
  5. L. Borcea, Electrical impedance tomography. Inverse Problems18 (2002) R99–R136.  Zbl1031.35147
  6. S.C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics15. Springer-Verlag, New York (1994).  
  7. P. Cannarsa and H. Frankowska, Some characterizations of optimal trajectories in control theory. SIAM J. Control Optim.29 (1991) 1322–1347.  Zbl0744.49011
  8. P. Cannarsa and H. Frankowska, Value function and optimality conditions for semilinear control problems. Appl. Math. Optim.26 (1992) 139–169.  Zbl0765.49001
  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.  Zbl0862.49021
  10. P. Cannarsa and C. Sinestrari, Semiconcave functions, Hamilton-Jacobi equations, and optimal control, Progress in Nonlinear Differential Equations and their Applications58. Birkhäuser Boston Inc., Boston, USA (2004).  Zbl1095.49003
  11. J. Carlsson, Symplectic reconstruction of data for heat and wave equations. Preprint (2008) .  URIhttp://arxiv.org/abs/0809.3621
  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).  Zbl0972.74057
  13. M. Cheney and D. Isaacson, Distinguishability in impedance imaging. IEEE Trans. Biomed. Eng.39 (1992) 852–860.  
  14. F.H. Clarke, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series in Mathematics. John Wiley and Sons, Inc. (1983).  Zbl0582.49001
  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.  Zbl0543.35011
  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.  Zbl0755.35015
  17. M. Crouzeix and V. Thomée, The stability in Lp and W p 1 of the L2-projection onto finite element function spaces. Math. Comp.48(178) (1987) 521–532.  Zbl0637.41034
  18. B. Dacorogna, Direct methods in the calculus of variations, Appl. Math. Sci.78. Springer-Verlag, Berlin (1989).  Zbl0703.49001
  19. H.W. Engl, M. Hanke and A. Neubauer, Regularization of inverse problems, Mathematics and its Applications375. Kluwer Academic Publishers Group, Dordrecht (1996).  Zbl0859.65054
  20. L.C. Evans, Partial differential equations, Graduate Studies in Mathematics19. American Mathematical Society, Providence, USA (1998).  Zbl0902.35002
  21. H. Frankowska, Contingent cones to reachable sets of control systems. SIAM J. Control Optim.27 (1989) 170–198.  Zbl0671.49030
  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.  Zbl0591.73119
  23. E. Hairer, C. Lubich and G. Wanner, Geometric numerical integration, Structure-preserving algorithms for ordinary differential equations, Springer Series in Computational Mathematics31. Springer-Verlag, Berlin (2002).  Zbl0994.65135
  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.  Zbl0726.65071
  25. R.V. Kohn and A. McKenney, Numerical implementation of a variational method for electrical impedance tomography. Inverse Problems6 (1990) 389–414.  Zbl0718.65089
  26. R.V. Kohn and G. Strang, Optimal design and relaxation of variational problems. I. Comm. Pure Appl. Math.39 (1986) 113–137.  Zbl0609.49008
  27. R.V. Kohn and G. Strang, Optimal design and relaxation of variational problems. II. Comm. Pure Appl. Math.39 (1986) 139–182.  Zbl0621.49008
  28. R.V. Kohn and G. Strang, Optimal design and relaxation of variational problems. III. Comm. Pure Appl. Math.39 (1986) 353–377.  Zbl0694.49004
  29. F. Natterer, The mathematics of computerized tomography, Classics in Applied Mathematics32. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, USA (2001). Reprint of the 1986 original.  Zbl0973.92020
  30. O. Pironneau, Optimal shape design for elliptic systems, Springer Series in Computational Physics. Springer-Verlag, New York (1984).  
  31. R.T. Rockafellar, Convex analysis, Princeton Mathematical Series28. Princeton University Press, Princeton, USA (1970).  
  32. M. Sandberg, Convergence rates for numerical approximations of an optimally controlled Ginzburg-Landau equation. Preprint (2008) .  URIhttp://arxiv.org/abs/0809.1834
  33. M. Sandberg and A. Szepessy, Convergence rates of symplectic Pontryagin approximations in optimal control theory. ESAIM: M2AN40 (2006) 149–173.  Zbl1091.49027
  34. D. Tataru, Viscosity solutions for Hamilton-Jacobi equations with unbounded nonlinear term: a simplified approach. J. Diff. Eq.111 (1994) 123–146.  Zbl0810.34060
  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.  
  36. C.R. Vogel, Computational methods for inverse problems, Frontiers in Applied Mathematics23. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2002). With a foreword by H.T. Banks.  Zbl1008.65103
  37. A. Wexler, B. Fry and M.R. Neuman, Impedance-computed tomography algorithm and system. Appl. Opt.24 (1985) 3985–3992.  

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