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Inverse reconstruction from optimal input data
KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
(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: urn:nbn:se:kth:diva-9378OAI: oai:DiVA.org:kth-9378DiVA: diva2:113752
Note
QC 20100712Available from: 2008-10-27 Created: 2008-10-27 Last updated: 2010-07-12Bibliographically approved
In thesis
1. Optimal Control of Partial Differential Equations in Optimal Design
Open this publication in new window or tab >>Optimal Control of Partial Differential Equations in Optimal Design
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
Optimal design
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-9293 (URN)978-91-7415-149-7 (ISBN)
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

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