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Numerical implementation of a variational method for electrical impedance tomography. (English) Zbl 0718.65089
Let D be an inhomogeneous body with conductivity $$\sigma$$. The electrical impedance tomography problem consists in determining $$\sigma$$ from boundary measurements $$f_ i$$, $$\phi_ i$$ of the current flux and voltage. With J the current density and $$\Phi$$ the electric potential within D we have $$\nabla \cdot J_ i=0,\quad J_ i=-\sigma \nabla \Phi_ i,\quad J_ i\cdot n|_{\partial \Omega}=f_ i,\quad \Phi_ i|_{\partial \Omega}=\phi_ i.$$ The authors suggest to compute $$\sigma$$ as the minimizer of $$\sum_{i}\int_{D}| \sigma^{1/2}\Phi_ i+\sigma^{1/2}J_ i|^ 2dx$$ with respect to $$\Phi_ i|_{\partial \Omega}=\phi_ i,\quad J_ i-n=f_ i,\quad \nabla \cdot J_ i=0.$$ Several minimization methods are discussed. Numerical examples based on synthetic data are given.

##### MSC:
 65Z05 Applications to the sciences 35R30 Inverse problems for PDEs 92C55 Biomedical imaging and signal processing 35Q60 PDEs in connection with optics and electromagnetic theory 78A70 Biological applications of optics and electromagnetic theory
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