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Optimal Control of Partial Differential Equations in Optimal Design
KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA. (Numerical Analysis)
2008 (English)Doctoral thesis, comprehensive summary (Other scientific)
Abstract [en]

This thesis concerns the approximation of optimally controlled partial differential equations for inverse problems in optimal design. Important examples of such problems are optimal material design and parameter reconstruction. In optimal material design the goal is to construct a material that meets some optimality criterion, e.g. to design a beam, with fixed weight, that is as stiff as possible. Parameter reconstrucion concerns, for example, the problem to find the interior structure of a material from surface displacement measurements resulting from applied external forces.

Optimal control problems, particularly for partial differential equations, 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 Hamiltonian system 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.

Another treated issue is the relevance of input data for parameter reconstruction problems, where the goal is to determine a spacially distributed coefficient of a partial differential equation from partial observations of the solution. It is here shown that the choice of input data, that generates the partial observations, affects the reconstruction, and that it is possible to formulate meaningful optimality criteria for the input data that enhances the quality of the reconstructed coefficient.

In the thesis we present solutions to various applications in optimal material design and reconstruction.

Abstract [sv]

Denna avhandling handlar om approximation av optimalt styrda partiella differentialekvationer för inversa problem inom optimal design. Viktiga exempel på sådana problem är optimal materialdesign och parameterskattning. Inom materialdesign är målet att konstruera ett material som uppfyller vissa optimalitetsvillkor, t.ex. att konstruera en så styv balk som möjligt under en given vikt, medan ett exempel på parameterskattning är att hitta den inre strukturen hos ett material genom att applicera ytkrafter och mäta de resulterande förskjutningarna.

Problem inom optimal styrning, speciellt för styrning av partiella differentialekvationer,är ofta illa ställa och måste regulariseras för att kunna lösas numeriskt. Teorin för Hamilton-Jacobi-Bellmans ekvationer används här för att konstruera regulariseringar och ge feluppskattningar till problem inom optimaldesign. Den konstruerade Pontryaginmetoden är en enkel och generell metod där det första analytiska steget är att regularisera Hamiltonianen. I nästa steg löses det Hamiltonska systemet effektivt med Newtons metod och en gles Jacobian. Vi härleder även en feluppskattning för skillnaden mellan den exakta och den approximerade målfunktionen. Denna uppskattning beror endast på skillnaden mellan den sanna och den regulariserade, ändligt dimensionella, Hamiltonianen, båda utvärderade längst lösningsbanan och dessL²-projektion. Felet beror alltså ej på skillnaden mellan den exakta och denapproximativa lösningen till det Hamiltonska systemet.

Ett annat fall som behandlas är frågan hur indata ska väljas för parameterskattningsproblem. För sådana problem är målet vanligen att bestämma en rumsligt beroende koefficient till en partiell differentialekvation, givet ofullständiga mätningar av lösningen. Här visas att valet av indata, som genererarde ofullständiga mätningarna, påverkar parameterskattningen, och att det är möjligt att formulera meningsfulla optimalitetsvillkor för indata som ökar kvaliteten på parameterskattningen. I avhandlingen presenteras lösningar för diverse tillämpningar inom optimal materialdesign och parameterskattning.

Place, publisher, year, edition, pages
Stockholm: KTH , 2008. , viii, 20 p.
Series
Trita-CSC-A, ISSN 1653-5723 ; 2008:15
Keyword [en]
Optimal design
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-9293ISBN: 978-91-7415-149-7 (print)OAI: oai:DiVA.org:kth-9293DiVA: diva2:54461
Public defence
2008-11-07, F3, Lindstedtsvägen 26, Stockholm, 10:00 (English)
Opponent
Supervisors
Note
QC 20100712Available from: 2008-10-27 Created: 2008-10-16 Last updated: 2010-07-12Bibliographically 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
3. Symplectic Reconstruction of Data for Heat and Wave Equations
Open this publication in new window or tab >>Symplectic Reconstruction of Data for Heat and Wave Equations
(English)Manuscript (Other academic)
Abstract [en]

This report concerns the inverse problem of estimating a spacially dependent coefficient of a partial differential equation from observations of the solution at the boundary. Such a problem can be formulated as an optimal control problem with the coefficient as the control variable and the solution as state variable. The heat or the wave equation is here considered as state equation. It is well known that such inverse problems are ill-posed and need to be regularized. The powerful Hamilton-Jacobi theory is used to construct a simple and general method where the first step is to analytically regularize the Hamiltonian; next its Hamiltonian system, a system of nonlinear partial differential equations, is solved with the Newton method and a sparse Jacobian.

National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-9385 (URN)
Note
QC 20100712Available from: 2008-10-28 Created: 2008-10-28 Last updated: 2010-07-12Bibliographically approved
4. Inverse reconstruction from optimal input data
Open this publication in new window or tab >>Inverse reconstruction from optimal input data
(English)Manuscript (Other academic)
Abstract [en]

This report concerns the problem to find optimal input data for an inverse reconstruction problem. In a classical parameter reconstruction problem the goal is to determine a spacially distributed (and optionally time dependent) coefficient of a partial differential equation from observed data. Here, the spacially dependent wave speed coefficient of the acoustic wave equation is sought, given observations of the solution on the boundary. The reconstruction of the coefficient is highly dependent on input data, e.g. if Neumann boundary values serve as input data it is in general not possible to determine the coefficient for all possible input data. It is shown that it is possible to formulate meaningful optimality criteria for the input data that enhances quality of the reconstructed coefficient. Both the problem of estimating the coefficient and the problem of finding optimal input data are ill-posed inverse problems and need to be regularized.

National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-9378 (URN)
Note
QC 20100712Available from: 2008-10-27 Created: 2008-10-27 Last updated: 2010-07-12Bibliographically approved

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