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.
@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: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {43},
year = {2009},
pages = {3-32},
doi = {10.1051/m2an/2008038},
mrnumber = {2494792},
zbl = {1159.65068},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2009__43_1_3_0}
}
Carlsson, Jesper; Sandberg, Mattias; Szepessy, Anders. Symplectic Pontryagin approximations for optimal design. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 43 (2009) pp. 3-32. doi : 10.1051/m2an/2008038. http://gdmltest.u-ga.fr/item/M2AN_2009__43_1_3_0/
[1] , Shape optimization by the homogenization method, Applied Mathematical Sciences 146. Springer-Verlag, New York (2002). | MR 1859696 | Zbl 0990.35001
[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 1484411 | Zbl 0890.49011
[3] , Solutions de viscosité des équations de Hamilton-Jacobi, Mathématiques & Applications 17 (Berlin) [Mathematics & Applications]. Springer-Verlag, Paris (1994). | MR 1613876 | Zbl 0819.35002
[4] and , Topology optimization, Theory, methods and applications. Springer-Verlag, Berlin (2003). | MR 2008524 | Zbl 1059.74001
[5] , Electrical impedance tomography. Inverse Problems 18 (2002) R99-R136. | MR 1955896 | Zbl 1031.35147
[6] and , The mathematical theory of finite element methods, Texts in Applied Mathematics 15. Springer-Verlag, New York (1994). | MR 1278258 | Zbl 0804.65101
[7] and , Some characterizations of optimal trajectories in control theory. SIAM J. Control Optim. 29 (1991) 1322-1347. | MR 1132185 | Zbl 0744.49011
[8] and , Value function and optimality conditions for semilinear control problems. Appl. Math. Optim. 26 (1992) 139-169. | MR 1166210 | Zbl 0765.49001
[9] and , Value function and optimality condition for semilinear co problems. II. Parabolic case. Appl. Math. Optim. 33 (1996) 1-33. | MR 1359346 | Zbl 0862.49021
[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 2041617 | Zbl 1095.49003
[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 1784106 | Zbl 0972.74057
[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 709590 | Zbl 0582.49001
[15] , and , Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 282 (1984) 487-502. | MR 732102 | Zbl 0543.35011
[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 1118699 | Zbl 0755.35015
[17] and , The stability in and of the -projection onto finite element function spaces. Math. Comp. 48(178) (1987) 521-532. | MR 878688 | Zbl 0637.41034
[18] , Direct methods in the calculus of variations, Appl. Math. Sci. 78. Springer-Verlag, Berlin (1989). | MR 990890 | Zbl 0703.49001
[19] , and , Regularization of inverse problems, Mathematics and its Applications 375. Kluwer Academic Publishers Group, Dordrecht (1996). | MR 1408680 | Zbl 0859.65054
[20] , Partial differential equations, Graduate Studies in Mathematics 19. American Mathematical Society, Providence, USA (1998). | Zbl 0902.35002
[21] , Contingent cones to reachable sets of control systems. SIAM J. Control Optim. 27 (1989) 170-198. | MR 980229 | Zbl 0671.49030
[22] , and , Numerical study of a relaxed variational problem from optimal design. Comput. Methods Appl. Mech. Engrg. 57 (1986) 107-127. | MR 859964 | Zbl 0591.73119
[23] , and , Geometric numerical integration, Structure-preserving algorithms for ordinary differential equations, Springer Series in Computational Mathematics 31. Springer-Verlag, Berlin (2002). | MR 1904823 | Zbl 0994.65135
[24] , and , Analysis and numerical studies of a problem of shape design. Arch. Rational Mech. Anal. 114 (1991) 349-363. | MR 1100800 | Zbl 0726.65071
[25] and , Numerical implementation of a variational method for electrical impedance tomography. Inverse Problems 6 (1990) 389-414. | MR 1057033 | Zbl 0718.65089
[26] and , Optimal design and relaxation of variational problems. I. Comm. Pure Appl. Math. 39 (1986) 113-137. | MR 820342 | Zbl 0609.49008
[27] and , Optimal design and relaxation of variational problems. II. Comm. Pure Appl. Math. 39 (1986) 139-182. | MR 820067 | Zbl 0621.49008
[28] and , Optimal design and relaxation of variational problems. III. Comm. Pure Appl. Math. 39 (1986) 353-377. | MR 829845 | Zbl 0694.49004
[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 1847845 | Zbl 0973.92020
[30] , Optimal shape design for elliptic systems, Springer Series in Computational Physics. Springer-Verlag, New York (1984). | MR 725856 | Zbl 0534.49001
[31] , Convex analysis, Princeton Mathematical Series 28. Princeton University Press, Princeton, USA (1970). | MR 274683 | Zbl 0193.18401
[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 2223508 | Zbl 1091.49027
[34] , Viscosity solutions for Hamilton-Jacobi equations with unbounded nonlinear term: a simplified approach. J. Diff. Eq. 111 (1994) 123-146. | MR 1280618 | Zbl 0810.34060
[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 455365 | Zbl 0354.65028
[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 1928831 | Zbl 1008.65103
[37] , and , Impedance-computed tomography algorithm and system. Appl. Opt. 24 (1985) 3985-3992.