Symplectic Pontryagin approximations for optimal design
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 43 (2009) no. 1, pp. 3-32.

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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 L 2 projection, i.e. not on the difference of the exact and approximate solutions to the hamiltonian systems.

DOI : 10.1051/m2an/2008038
Classification : 65N21, 49L25
Mots-clés : topology optimization, inverse problems, Hamilton-Jacobi, regularization, error estimates, impedance tomography, convexification, homogenization 
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     title = {Symplectic {Pontryagin} approximations for optimal design},
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Carlsson, Jesper; Sandberg, Mattias; Szepessy, Anders. Symplectic Pontryagin approximations for optimal design. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 43 (2009) no. 1, pp. 3-32. doi : 10.1051/m2an/2008038. https://geodesic-test.mathdoc.fr/articles/10.1051/m2an/2008038/

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