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Symplectic Pontryagin approximations for optimal design

Published online by Cambridge University Press:  16 October 2008

Jesper Carlsson ,
Mattias Sandberg  and
Anders Szepessy
Jesper Carlsson
Affiliation:
Department of Numerical Analysis, Kungl. Tekniska Högskolan, 100 44 Stockholm, Sweden. jesperc@kth.se
Mattias Sandberg
Affiliation:
CMA, University of Oslo, P.O. Box 1053 Blindern, 0316 Oslo, Norway. mattias.sandberg@cma.uio.no
Anders Szepessy
Affiliation:
Department of Numerical Analysis, Kungl. Tekniska Högskolan, 100 44 Stockholm, Sweden. jesperc@kth.se Department of Mathematics, Kungl. Tekniska Högskolan, 100 44 Stockholm, Sweden. szepessy@kth.se
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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.

Keywords

Topology optimizationinverse problemsHamilton-Jacobiregularizationerror estimatesimpedance tomographyconvexificationhomogenization.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 1 , January 2009 , pp. 3 - 32
Copyright
© EDP Sciences, SMAI, 2008

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References

G. Allaire, Shape optimization by the homogenization method, Applied Mathematical Sciences 146. Springer-Verlag, New York (2002).
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.
G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi, Mathématiques & Applications 17 (Berlin) [Mathematics & Applications]. Springer-Verlag, Paris (1994).
M.P. Bendsøe and O. Sigmund, Topology optimization, Theory, methods and applications. Springer-Verlag, Berlin (2003).
Borcea, L., Electrical impedance tomography. Inverse Problems 18 (2002) R99–R136. CrossRef
S.C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics 15. Springer-Verlag, New York (1994).
Cannarsa, P. and Frankowska, H., Some characterizations of optimal trajectories in control theory. SIAM J. Control Optim. 29 (1991) 1322–1347. CrossRef
Cannarsa, P. and Frankowska, H., Value function and optimality conditions for semilinear control problems. Appl. Math. Optim. 26 (1992) 139–169. CrossRef
Cannarsa, P. and Frankowska, H., Value function and optimality condition for semilinear co problems. II. Parabolic case. Appl. Math. Optim. 33 (1996) 1–33. CrossRef
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).
J. Carlsson, Symplectic reconstruction of data for heat and wave equations. Preprint (2008) http://arxiv.org/abs/0809.3621.
Céa, J., Garreau, S., Guillaume, P. and Masmoudi, M., The shape and topological optimizations connection. Comput. Methods Appl. Mech. Engrg. 188 (2000) 713–726. IV WCCM, Part II (Buenos Aires, 1998). CrossRef
Cheney, M. and Isaacson, D., Distinguishability in impedance imaging. IEEE Trans. Biomed. Eng. 39 (1992) 852–860. CrossRef
F.H. Clarke, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series in Mathematics. John Wiley and Sons, Inc. (1983).
Crandall, M.G., Evans, L.C. and Lions, P.-L., Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 282 (1984) 487–502. CrossRef
Crandall, M.G., Ishii, H. and Lions, P.-L., User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 (1992) 1–67. CrossRef
M. Crouzeix and V. Thomée, The stability in Lp and $W\sp 1\sb p$ of the L2 -projection onto finite element function spaces. Math. Comp. 48(178) (1987) 521–532.
B. Dacorogna, Direct methods in the calculus of variations, Appl. Math. Sci. 78. Springer-Verlag, Berlin (1989).
H.W. Engl, M. Hanke and A. Neubauer, Regularization of inverse problems, Mathematics and its Applications 375. Kluwer Academic Publishers Group, Dordrecht (1996).
L.C. Evans, Partial differential equations, Graduate Studies in Mathematics 19. American Mathematical Society, Providence, USA (1998).
Frankowska, H., Contingent cones to reachable sets of control systems. SIAM J. Control Optim. 27 (1989) 170–198. CrossRef
Goodman, J., Kohn, R.V. and Reyna, L., Numerical study of a relaxed variational problem from optimal design. Comput. Methods Appl. Mech. Engrg. 57 (1986) 107–127. CrossRef
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).
Kawohl, B., Stará, J. and Wittum, G., Analysis and numerical studies of a problem of shape design. Arch. Rational Mech. Anal. 114 (1991) 349–363. CrossRef
Kohn, R.V. and McKenney, A., Numerical implementation of a variational method for electrical impedance tomography. Inverse Problems 6 (1990) 389–414. CrossRef
Kohn, R.V. and Strang, G., Optimal design and relaxation of variational problems. I. Comm. Pure Appl. Math. 39 (1986) 113–137. CrossRef
Kohn, R.V. and Strang, G., Optimal design and relaxation of variational problems. II. Comm. Pure Appl. Math. 39 (1986) 139–182. CrossRef
Kohn, R.V. and Strang, G., Optimal design and relaxation of variational problems. III. Comm. Pure Appl. Math. 39 (1986) 353–377. CrossRef
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.
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 Series 28. Princeton University Press, Princeton, USA (1970).
M. Sandberg, Convergence rates for numerical approximations of an optimally controlled Ginzburg-Landau equation. Preprint (2008) http://arxiv.org/abs/0809.1834.
Sandberg, M. and Szepessy, A., Convergence rates of symplectic Pontryagin approximations in optimal control theory. ESAIM: M2AN 40 (2006) 149–173. CrossRef
Tataru, D., Viscosity solutions for Hamilton-Jacobi equations with unbounded nonlinear term: a simplified approach. J. Diff. Eq. 111 (1994) 123–146. CrossRef
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 Mathematics 23. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2002). With a foreword by H.T. Banks.
Wexler, A., Fry, B. and Neuman, M.R., Impedance-computed tomography algorithm and system. Appl. Opt. 24 (1985) 3985–3992. CrossRef