A regularized solution for the inverse conductivity problem using mollifiers. (English) Zbl 1189.65316
The authors consider the inverse problem of determining the isotropic conductivity \( \sigma \) in \( \Omega \) from applications of an electric current \( j = \sigma \frac{\partial \Phi}{\partial n} \) on the boundary \( \partial \Omega \). Here, \( \Omega \subset \mathbb{R}^n, \, n = 2 \) or \( n = 3 \), denotes a bounded, simply connected domain, and \( \Phi \) denotes the electric potential. This problem is modelled by the equation \( \nabla \cdot (\sigma(x) \nabla \Phi(x)) = 0, x \in \Omega \), and is reformulated in terms of a pair of coupled integral equations, one of them being the first kind integral equation \( (AX)(x) = \int_{\Omega} \mathcal{H}(x,y) X(y) dy = \zeta(x) \). Here, \( \mathcal{H} \) is a bounded solution of the Schrödinger equation with respect to the second variable, i.e., \( \Delta_y \mathcal{H}(x,y) + \lambda(y) \mathcal{H}(x,y) = 0, x,y \in \Omega \), and \( \zeta \) can be represented by a boundary integral. The first kind integral equation \( AX = \zeta \) is approximately solved by mollifier methods, where the kernel \( \mathcal{H} \) is chosen appropriately to ensure existence of the mollifier. Further analysis is provided for \( \lambda \) fixed and the unit disk. Finally, results of some numerical experiments are presented.
Reviewer: Robert Plato (Siegen)
MSC:
65R32 | Numerical methods for inverse problems for integral equations |
45Q05 | Inverse problems for integral equations |
45B05 | Fredholm integral equations |
92C55 | Biomedical imaging and signal processing |
78A70 | Biological applications of optics and electromagnetic theory |
Keywords:
inverse conductivity problem; electrical impedance tomography; first kind integral equation; mollifier method; Schrödinger equation; integral equation methods; nonlinear inverse problems; ill-posed problemsReferences:
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