PhaseFieldDiffCoeffRecovery2.pdf (995.59 kB)
Double obstacle phase field approach to an inverse problem for a discontinuous diffusion coefficient
journal contribution
posted on 2023-06-09, 01:17 authored by Klaus Deckelnick, Charles M Elliott, Vanessa StylesVanessa StylesWe 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 s. 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 ?. This phase field approach is justified by proving G-convergence to the functional with perimeter regularisation as ? ? 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.
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Publication status
- Published
File Version
- Accepted version
Journal
Inverse ProblemsISSN
0266-5611Publisher
Institute of PhysicsExternal DOI
Issue
4Volume
32Article number
a045008Department affiliated with
- Mathematics Publications
Full text available
- Yes
Peer reviewed?
- Yes
Legacy Posted Date
2016-05-17First Open Access (FOA) Date
2017-03-16First Compliant Deposit (FCD) Date
2016-05-17Usage metrics
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