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 projection, i.e. not on the difference of the exact and approximate solutions to the hamiltonian systems.
Mots clés : topology optimization, inverse problems, Hamilton-Jacobi, regularization, error estimates, impedance tomography, convexification, homogenization
@article{M2AN_2009__43_1_3_0,
author = {Carlsson, Jesper and Sandberg, Mattias and Szepessy, Anders},
title = {Symplectic {Pontryagin} approximations for optimal design},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {3--32},
publisher = {EDP-Sciences},
volume = {43},
number = {1},
year = {2009},
doi = {10.1051/m2an/2008038},
mrnumber = {2494792},
zbl = {1159.65068},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2008038/}
}
TY - JOUR AU - Carlsson, Jesper AU - Sandberg, Mattias AU - Szepessy, Anders TI - Symplectic Pontryagin approximations for optimal design JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2009 SP - 3 EP - 32 VL - 43 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2008038/ DO - 10.1051/m2an/2008038 LA - en ID - M2AN_2009__43_1_3_0 ER -
%0 Journal Article %A Carlsson, Jesper %A Sandberg, Mattias %A Szepessy, Anders %T Symplectic Pontryagin approximations for optimal design %J ESAIM: Modélisation mathématique et analyse numérique %D 2009 %P 3-32 %V 43 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2008038/ %R 10.1051/m2an/2008038 %G en %F M2AN_2009__43_1_3_0
Carlsson, Jesper; Sandberg, Mattias; Szepessy, Anders. Symplectic Pontryagin approximations for optimal design. ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 1, pp. 3-32. doi : 10.1051/m2an/2008038. http://www.numdam.org/articles/10.1051/m2an/2008038/
[1] , Shape optimization by the homogenization method, Applied Mathematical Sciences 146. Springer-Verlag, New York (2002). | MR | Zbl
[2] and , 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. | MR | Zbl
[3] , Solutions de viscosité des équations de Hamilton-Jacobi, Mathématiques & Applications 17 (Berlin) [Mathematics & Applications]. Springer-Verlag, Paris (1994). | MR | Zbl
[4] and , Topology optimization, Theory, methods and applications. Springer-Verlag, Berlin (2003). | MR | Zbl
[5] , Electrical impedance tomography. Inverse Problems 18 (2002) R99-R136. | MR | Zbl
[6] and , The mathematical theory of finite element methods, Texts in Applied Mathematics 15. Springer-Verlag, New York (1994). | MR | Zbl
[7] and , Some characterizations of optimal trajectories in control theory. SIAM J. Control Optim. 29 (1991) 1322-1347. | MR | Zbl
[8] and , Value function and optimality conditions for semilinear control problems. Appl. Math. Optim. 26 (1992) 139-169. | MR | Zbl
[9] and , Value function and optimality condition for semilinear co problems. II. Parabolic case. Appl. Math. Optim. 33 (1996) 1-33. | MR | Zbl
[10] and , Semiconcave functions, Hamilton-Jacobi equations, and optimal control, Progress in Nonlinear Differential Equations and their Applications 58. Birkhäuser Boston Inc., Boston, USA (2004). | MR | Zbl
[11] , Symplectic reconstruction of data for heat and wave equations. Preprint (2008) http://arxiv.org/abs/0809.3621.
[12] , , and , The shape and topological optimizations connection. Comput. Methods Appl. Mech. Engrg. 188 (2000) 713-726. IV WCCM, Part II (Buenos Aires, 1998). | MR | Zbl
[13] and , Distinguishability in impedance imaging. IEEE Trans. Biomed. Eng. 39 (1992) 852-860.
[14] , Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series in Mathematics. John Wiley and Sons, Inc. (1983). | MR | Zbl
[15] , and , Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 282 (1984) 487-502. | MR | Zbl
[16] , and , User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 (1992) 1-67. | MR | Zbl
[17] and , The stability in and of the -projection onto finite element function spaces. Math. Comp. 48(178) (1987) 521-532. | MR | Zbl
[18] , Direct methods in the calculus of variations, Appl. Math. Sci. 78. Springer-Verlag, Berlin (1989). | MR | Zbl
[19] , and , Regularization of inverse problems, Mathematics and its Applications 375. Kluwer Academic Publishers Group, Dordrecht (1996). | MR | Zbl
[20] , Partial differential equations, Graduate Studies in Mathematics 19. American Mathematical Society, Providence, USA (1998). | Zbl
[21] , Contingent cones to reachable sets of control systems. SIAM J. Control Optim. 27 (1989) 170-198. | MR | Zbl
[22] , and , Numerical study of a relaxed variational problem from optimal design. Comput. Methods Appl. Mech. Engrg. 57 (1986) 107-127. | MR | Zbl
[23] , and , Geometric numerical integration, Structure-preserving algorithms for ordinary differential equations, Springer Series in Computational Mathematics 31. Springer-Verlag, Berlin (2002). | MR | Zbl
[24] , and , Analysis and numerical studies of a problem of shape design. Arch. Rational Mech. Anal. 114 (1991) 349-363. | MR | Zbl
[25] and , Numerical implementation of a variational method for electrical impedance tomography. Inverse Problems 6 (1990) 389-414. | MR | Zbl
[26] and , Optimal design and relaxation of variational problems. I. Comm. Pure Appl. Math. 39 (1986) 113-137. | MR | Zbl
[27] and , Optimal design and relaxation of variational problems. II. Comm. Pure Appl. Math. 39 (1986) 139-182. | MR | Zbl
[28] and , Optimal design and relaxation of variational problems. III. Comm. Pure Appl. Math. 39 (1986) 353-377. | MR | Zbl
[29] , 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. | MR | Zbl
[30] , Optimal shape design for elliptic systems, Springer Series in Computational Physics. Springer-Verlag, New York (1984). | MR | Zbl
[31] , Convex analysis, Princeton Mathematical Series 28. Princeton University Press, Princeton, USA (1970). | MR | Zbl
[32] , Convergence rates for numerical approximations of an optimally controlled Ginzburg-Landau equation. Preprint (2008) http://arxiv.org/abs/0809.1834.
[33] and , Convergence rates of symplectic Pontryagin approximations in optimal control theory. ESAIM: M2AN 40 (2006) 149-173. | Numdam | MR | Zbl
[34] , Viscosity solutions for Hamilton-Jacobi equations with unbounded nonlinear term: a simplified approach. J. Diff. Eq. 111 (1994) 123-146. | MR | Zbl
[35] and , 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. | MR | Zbl
[36] , 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. | MR | Zbl
[37] , and , Impedance-computed tomography algorithm and system. Appl. Opt. 24 (1985) 3985-3992.
Cité par Sources :
