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Double obstacle phase field approach to an inverse problem for a discontinuous diffusion coefficient
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Deckelnick, Klaus, Elliott, Charles M. and Styles, Vanessa (2016) Double obstacle phase field approach to an inverse problem for a discontinuous diffusion coefficient. Inverse Problems, 32 (4). 045008. doi:10.1088/0266-5611/32/4/045008 ISSN 0266-5611.
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Official URL: http://dx.doi.org/10.1088/0266-5611/32/4/045008
Abstract
We propose a double obstacle phase field approach to the recovery of piece-wise constant diffusion coefficients for elliptic partial differential equations. The approach to this inverse problem is that of optimal control in which we have a quadratic fidelity term to which we add a perimeter regularisation weighted by a parameter sigma. This yields a functional which is optimised over a set of diffusion coefficients subject to a state equation which is the underlying elliptic PDE. In order to derive a problem which is amenable to computation the perimeter functional is relaxed using a gradient energy functional together with an obstacle potential in which there is an interface parameter epsilon. This phase field approach is justified by proving. We propose a double obstacle phase field approach to the recovery of piece-wise constant diffusion coefficients for elliptic partial differential equations. The approach to this inverse problem is that of optimal control in which we have a quadratic fidelity term to which we add a perimeter regularisation weighted by a parameter sigma. This yields a functional which is optimised over a set of diffusion coefficients subject to a state equation which is the underlying elliptic PDE. In order to derive a problem which is amenable to computation the perimeter functional is relaxed using a gradient energy functional together with an obstacle potential in which there is an interface parameter epsilon. This phase field approach is justified by proving Gamma-convergence to the functional with perimeter regularisation as epsilon ->0. The computational approach is based on a finite element approximation. This discretisation is shown to converge in an appropriate way to the solution of the phase field problem. We derive an iterative method which is shown to yield an energy decreasing sequence converging to a discrete critical point. The efficacy of the approach is illustrated with numerical experiments.
Item Type: | Journal Article | ||||||||
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Subjects: | Q Science > QA Mathematics | ||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
Library of Congress Subject Headings (LCSH): | Inverse problems (Differential equations) -- Numerical solutions, Differential equations, Elliptic | ||||||||
Journal or Publication Title: | Inverse Problems | ||||||||
Publisher: | Institute of Physics Publishing Ltd. | ||||||||
ISSN: | 0266-5611 | ||||||||
Official Date: | 16 March 2016 | ||||||||
Dates: |
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Volume: | 32 | ||||||||
Number: | 4 | ||||||||
Article Number: | 045008 | ||||||||
DOI: | 10.1088/0266-5611/32/4/045008 | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||
Date of first compliant deposit: | 21 June 2016 | ||||||||
Date of first compliant Open Access: | 17 March 2017 |
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