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Symplectic Pontryagin Approximations for Optimal Design
KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
Univ Oslo, CMA.ORCID iD: 0000-0003-2669-359X
KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
2009 (English)In: Mathematical Modelling and Numerical Analysis, ISSN 0764-583X, E-ISSN 1290-3841, 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. Vol. 43, no 1, 3-32 p.
Keyword [en]
Topology optimization; inverse problems; Hamilton-Jacobi; regularization; error estimates; impedance tomography; convexification; homogenization
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-6071DOI: 10.1051/m2an/2008038ISI: 000262735400002Scopus ID: 2-s2.0-59049090036OAI: oai:DiVA.org:kth-6071DiVA: diva2:113548
Note
QC 20100712. Uppdaterad från accepted till published (20100712).Available from: 2008-10-21 Created: 2008-10-21 Last updated: 2017-12-14Bibliographically 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
2. Approximation of Optimally Controlled Ordinary and Partial Differential Equations
Open this publication in new window or tab >>Approximation of Optimally Controlled Ordinary and Partial Differential Equations
2006 (English)Doctoral thesis, comprehensive summary (Other scientific)
Abstract [en]

In this thesis, which consists of four papers, approximation of optimal control problems is studied. In Paper I the Symplectic Pontryagin method for approximation of optimally controlled ordinary differential equations is presented. The method consists of a Symplectic Euler time stepping scheme for a Hamiltonian system with a regularized Hamiltonian. Under some assumptions it is shown that the approximate value function associated with this scheme converges to the original value function with a linear rate.

In Paper II the ideas from Paper I are extended to approximation of an optimally controlled partial differential equation, a one-dimensional Ginzburg-Landau equation. The approximation is performed in two steps. In the first step a value function associated with a finite element spatial discretization is shown to converge quadratically in the mesh size to the original value function. In the second step a Symplectic Euler discretization in time is shown to converge with a linear rate. The behavior of optimal solutions is shown by numerical examples.

In Paper III the same approximation method as in Paper II is applied to three other problems; the optimal design of an electric conductor, the design of an elastic domain, and the problem of reconstructing the interior of an object from measured electrical surface currents. Since these problems are time-independent the Hamilton-Jacobi theory can not be used. In order to be able to obtain error bounds the problems are therefore transferred to a setting where time plays a role. Computer experiments with the Symplectic Pontryagin method is performed for all three problems.

Common to the three first papers is that the convergence proofs use that the approximate value functions solve Hamilton-Jacobi equations consistent with the original Hamilton-Jacobi equations.

Paper IV concerns convergence of attainable sets for non-convex differential inclusions. When the right hand side in the differential inclusion is a bounded, Lipschitz set-valued function it is shown that the convergence in Hausdorff-distance of attainable sets for a Forward Euler discretization is linear in the time step. This implies that dynamic programming using Forward Euler discretizations of optimal control problems converge with a linear rate when all the functions involved are bounded and Lipschitz continuous.

Place, publisher, year, edition, pages
Stockholm: KTH, 2006. v, 17 p.
Series
Trita-MAT. MA, ISSN 1401-2278 ; 06:03
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-4066 (URN)91-7178-412-8 (ISBN)
Public defence
2006-08-30, Sal F3, Lindstedtsvägen 12, Stockholm, 10:00
Opponent
Supervisors
Note
QC 20100917Available from: 2006-07-21 Created: 2006-07-21 Last updated: 2011-12-14Bibliographically approved
3. Pontryagin approximations for optimal design
Open this publication in new window or tab >>Pontryagin approximations for optimal design
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
Topology Optimization, Inverse Problems, Hamilton-Jacobi, Regularization, Error Estimates, Impedance Tomography
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-4089 (URN)91-7178-417-9 (ISBN)
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

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