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Optimal control and inverse problems involving point and line functionals and inequality constraints
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Brett, Charles E. A. (2014) Optimal control and inverse problems involving point and line functionals and inequality constraints. PhD thesis, University of Warwick.
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WRAP_THESIS_Brett_2014.pdf - Submitted Version - Requires a PDF viewer. Download (12Mb) | Preview |
Official URL: http://webcat.warwick.ac.uk/record=b2752032~S1
Abstract
In this thesis we consider some problems related to the optimal control of partial differential equations (PDEs) and variational inequalities (VIs) with various constraints. Such problems are important because in real world applications we are typically more interested in optimising and controlling processes than just simulating them. We focus on developing efficient solution methods for these problems.
The first part of this thesis considers optimal control of PDEs and VIs but with the usual L2 fidelity term replaced by ones which encourages the state to take certain values at points or along surfaces of codimension 1. Such problems are related to optimal control with pointwise state constraints, which are relevant in applications. Our new fidelity terms cause complications in the formulation of the optimal control problems, as well as the analysis and the numerical analysis.
The second part of this thesis considers the inverse problem of recovering a binary function from blurred and noisy data. Such image processing problems arise in many applications, for example decoding barcodes. Our approach uses the Mumford-Shah model, but with a phase field approximation to perimeter regularisation. We develop iterative methods for solving the problem and prove convergence results. Numerical results are presented which illustrate the effectiveness of our approach and the relative merits of different phase field approximations. We finish by applying our algorithms to a problem in materials science.
| Item Type: | Thesis (PhD) | ||||
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| Subjects: | Q Science > QA Mathematics | ||||
| Library of Congress Subject Headings (LCSH): | Differential equations, Partial, Variational inequalities (Mathematics) | ||||
| Official Date: | July 2014 | ||||
| Dates: |
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| Institution: | University of Warwick | ||||
| Theses Department: | Mathematics Institute | ||||
| Thesis Type: | PhD | ||||
| Publication Status: | Unpublished | ||||
| Supervisor(s)/Advisor: | Elliott, Charles M. ; Dedner, Andreas | ||||
| Sponsors: | Engineering and Physical Sciences Research Council (EP/H023364/1) | ||||
| Extent: | ix, 177 leaves : illustrations, charts | ||||
| Language: | eng |
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