# 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

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topCarlsson, 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

top- G. Allaire, Shape optimization by the homogenization method, Applied Mathematical Sciences146. Springer-Verlag, New York (2002). Zbl0990.35001
- 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
- G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi, Mathématiques & Applications17 (Berlin) [Mathematics & Applications]. Springer-Verlag, Paris (1994). Zbl0819.35002
- M.P. Bendsøe and O. Sigmund, Topology optimization, Theory, methods and applications. Springer-Verlag, Berlin (2003). Zbl1059.74001
- L. Borcea, Electrical impedance tomography. Inverse Problems18 (2002) R99–R136. Zbl1031.35147
- S.C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics15. Springer-Verlag, New York (1994).
- P. Cannarsa and H. Frankowska, Some characterizations of optimal trajectories in control theory. SIAM J. Control Optim.29 (1991) 1322–1347. Zbl0744.49011
- P. Cannarsa and H. Frankowska, Value function and optimality conditions for semilinear control problems. Appl. Math. Optim.26 (1992) 139–169. Zbl0765.49001
- 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
- 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
- J. Carlsson, Symplectic reconstruction of data for heat and wave equations. Preprint (2008) . URIhttp://arxiv.org/abs/0809.3621
- 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
- M. Cheney and D. Isaacson, Distinguishability in impedance imaging. IEEE Trans. Biomed. Eng.39 (1992) 852–860.
- F.H. Clarke, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series in Mathematics. John Wiley and Sons, Inc. (1983). Zbl0582.49001
- 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
- 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
- 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
- B. Dacorogna, Direct methods in the calculus of variations, Appl. Math. Sci.78. Springer-Verlag, Berlin (1989). Zbl0703.49001
- H.W. Engl, M. Hanke and A. Neubauer, Regularization of inverse problems, Mathematics and its Applications375. Kluwer Academic Publishers Group, Dordrecht (1996). Zbl0859.65054
- L.C. Evans, Partial differential equations, Graduate Studies in Mathematics19. American Mathematical Society, Providence, USA (1998). Zbl0902.35002
- H. Frankowska, Contingent cones to reachable sets of control systems. SIAM J. Control Optim.27 (1989) 170–198. Zbl0671.49030
- 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
- 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
- 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
- R.V. Kohn and A. McKenney, Numerical implementation of a variational method for electrical impedance tomography. Inverse Problems6 (1990) 389–414. Zbl0718.65089
- R.V. Kohn and G. Strang, Optimal design and relaxation of variational problems. I. Comm. Pure Appl. Math.39 (1986) 113–137. Zbl0609.49008
- R.V. Kohn and G. Strang, Optimal design and relaxation of variational problems. II. Comm. Pure Appl. Math.39 (1986) 139–182. Zbl0621.49008
- R.V. Kohn and G. Strang, Optimal design and relaxation of variational problems. III. Comm. Pure Appl. Math.39 (1986) 353–377. Zbl0694.49004
- 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
- O. Pironneau, Optimal shape design for elliptic systems, Springer Series in Computational Physics. Springer-Verlag, New York (1984).
- R.T. Rockafellar, Convex analysis, Princeton Mathematical Series28. Princeton University Press, Princeton, USA (1970).
- M. Sandberg, Convergence rates for numerical approximations of an optimally controlled Ginzburg-Landau equation. Preprint (2008) . URIhttp://arxiv.org/abs/0809.1834
- M. Sandberg and A. Szepessy, Convergence rates of symplectic Pontryagin approximations in optimal control theory. ESAIM: M2AN40 (2006) 149–173. Zbl1091.49027
- D. Tataru, Viscosity solutions for Hamilton-Jacobi equations with unbounded nonlinear term: a simplified approach. J. Diff. Eq.111 (1994) 123–146. Zbl0810.34060
- 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.
- 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
- A. Wexler, B. Fry and M.R. Neuman, Impedance-computed tomography algorithm and system. Appl. Opt.24 (1985) 3985–3992.

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