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Pontryagin approximations for optimal design
KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
2006 (English)Licentiate thesis, comprehensive summary (Other scientific)
Abstract [en]

This thesis concerns the approximation of optimally controlled partial differential equations for applications in optimal design and reconstruction. Such optimal control problems are often ill-posed and need to be regularized to obtain good approximations. We here use the theory of the corresponding Hamilton-Jacobi-Bellman equations to construct regularizations and derive error estimates for optimal design problems. The constructed Pontryagin method is a simple and general method where the first, analytical, step is to regularize the Hamiltonian. Next its stationary Hamiltonian system, a nonlinear partial differential equation, is computed efficiently with the Newton method using a sparse Jacobian. 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. In the thesis we present solutions to applications such as optimal design and reconstruction of conducting materials and elastic structures.

Place, publisher, year, edition, pages
Stockholm: KTH , 2006. , 16 p.
Series
Trita-CSC-A, ISSN 1653-5723 ; 2006:11
Keyword [en]
Topology Optimization, Inverse Problems, Hamilton-Jacobi, Regularization, Error Estimates, Impedance Tomography
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-4089ISBN: 91-7178-417-9 (print)OAI: oai:DiVA.org:kth-4089DiVA: diva2:10677
Presentation
2006-09-08, D41, Huvudbyggnaden, Lindstedtsvägen 17 1tr, Stockholm, 10:00
Supervisors
Note
QC 20101110Available from: 2006-08-29 Created: 2006-08-29 Last updated: 2010-11-10Bibliographically approved
List of papers
1. Symplectic Pontryagin Approximations for Optimal Design
Open this publication in new window or tab >>Symplectic Pontryagin Approximations for Optimal Design
2009 (English)In: Mathematical Modelling and Numerical Analysis, ISSN 0764-583X, Vol. 43, no 1, 3-32 p.Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
EDP Sciences, 2009
Keyword
Topology optimization; inverse problems; Hamilton-Jacobi; regularization; error estimates; impedance tomography; convexification; homogenization
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-6071 (URN)10.1051/m2an/2008038 (DOI)000262735400002 ()2-s2.0-59049090036 (Scopus ID)
Note
QC 20100712. Uppdaterad från accepted till published (20100712).Available from: 2008-10-21 Created: 2008-10-21 Last updated: 2011-12-20Bibliographically approved
2. Pontryagin Approximations for Optimal Design of Elastic Structures
Open this publication in new window or tab >>Pontryagin Approximations for Optimal Design of Elastic Structures
(English)Manuscript (Other academic)
Abstract [en]

This article presents a numerical method for approximation of some optimal control problems for partial differential equations. The method uses regularization derived from consistency with the corresponding Hamilton-Jacobi-Bellman equations in infinite dimension. In particular, optimal designs of elastic structures such as distributing a limited amount of material to minimize its compliance, or to detect interior material distributions from surface measurements, are computed. The derived Pontryagin based method presented here is simple to use with standard PDE-software using Newton iterations with a sparse Hessian.

National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-9384 (URN)
Note
QC 20100712Available from: 2008-10-28 Created: 2008-10-28 Last updated: 2010-11-10Bibliographically approved

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Citation style
  • apa
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  • Other locale
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Output format
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