# Symplectic Pontryagin approximations for optimal design

• Volume: 43, Issue: 1, page 3-32
• ISSN: 0764-583X

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## Abstract

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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 ${\mathrm{L}}^{2}$ projection, i.e. not on the difference of the exact and approximate solutions to the hamiltonian systems.

## How to cite

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Carlsson, Jesper, Sandberg, Mattias, and Szepessy, Anders. "Symplectic Pontryagin approximations for optimal design." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 43.1 (2009): 3-32. <http://eudml.org/doc/245361>.

@article{Carlsson2009,
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 $\mathrm \{L\}^\{2\}$ 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 - Modélisation Mathématique et Analyse Numérique},
keywords = {topology optimization; inverse problems; Hamilton-Jacobi; regularization; error estimates; impedance tomography; convexification; homogenization; optimal design problems; Pontryagin method; Newton method},
language = {eng},
number = {1},
pages = {3-32},
publisher = {EDP-Sciences},
title = {Symplectic Pontryagin approximations for optimal design},
url = {http://eudml.org/doc/245361},
volume = {43},
year = {2009},
}

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 - Modélisation Mathématique et Analyse Numérique
PY - 2009
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 $\mathrm {L}^{2}$ 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; optimal design problems; Pontryagin method; Newton method
UR - http://eudml.org/doc/245361
ER -

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