A Globally Convergent Algorithm for a PDE-Constrained Optimization Problem Arising in Electrical Impedance Tomography

Abstract

We study the convergence properties of an algorithm for the inverse problem of electrical impedance tomography, which can be reduced to a partial differential equation (PDE) constrained optimization problem. The direct problem consists of the potential equation div(ϵ∇u) = 0 in a circle, with Neumann condition describing the behavior of the electrostatic potential in a medium with conductivity given by the function ϵ(x, y). We suppose that at each time a current ψ i is applied to the boundary of the circle (Neumann's data), and that it is possible to measure the corresponding potential ϕ i (Dirichlet data). The inverse problem is to find ϵ(x, y) given a finite number of Cauchy pairs measurements (ϕ i , ψ i ), i = 1,…, N. The problem is formulated as a least squares problem, and the developed algorithm solves the continuous problem using descent iterations in its corresponding finite element approximations. Wolfe's conditions are used to ensure the global convergence of the optimization algorithm for the continuous problem. Although exact data are assumed, measurement errors in data and regularization methods shall be considered in a future work.