Exact Shape-Reconstruction by One-Step Linearization in Electrical Impedance Tomography

For electrical impedance tomography (EIT), the linearized reconstruction method using the Fréchet derivative of the Neumann-to-Dirichlet map with respect to the conductivity has been widely used in the last three decades. However, few rigorous mathematical results are known regarding the errors caused by the linear approximation. In this work we prove that linearizing the inverse problem of EIT does not lead to shape errors for piecewise-analytic conductivities. If a solution of the linearized equations exists, then it has the same outer support as the true conductivity change, no matter how large the latter is. Under an additional definiteness condition we also show how to approximately solve the linearized equation so that the outer support converges toward the right one. Our convergence result is global and also applies for approximations by noisy finitedimensional data. Furthermore, we obtain bounds on how well the linear reconstructions and the true conductivity difference agree on the boundary of the outer support.