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). a045008. ISSN 0266-5611

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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 σ. 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 Є. This phase field approach is justified by proving Γ−convergence to the functional
with perimeter regularisation as Є → 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: Article
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Subjects: Q Science > QA Mathematics > QA0297 Numerical analysis
Depositing User: Vanessa Styles
Date Deposited: 17 May 2016 10:07
Last Modified: 24 Mar 2017 11:42
URI: http://srodev.sussex.ac.uk/id/eprint/61051

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