On the Parametrization of Ill - posed Inverse Problems Arising from Elliptic Partial Differential

On the Parametrization of Ill-posed Inverse Problems Arising from Elliptic Partial Differential Equations by Fernando Guevara Vasquez Electric impedance tomography (EIT) consists in finding the conductivity inside a body from electrical measurements taken at its surface. This is a severely ill-posed problem: any numerical inversion scheme requires some form of regularization. We present inversion schemes that address the instability of the problem by proper sparse parametrization of the unknown conductivity. To guide us, we consider a consistent finite difference approach to an inverse Sturm-Liouville problem arising in EIT for layered media. The method first solves a model reduction problem for the differential equation where the reduced model parameters are essentially averages of the conductivity over the cells of a grid depending on the conductivity. Fortunately the dependence is weak. Thus one can efficiently estimate conductivity averages by using the grid for a reference conductivity. This simple inversion method converges to the true solution as the number of measurements increases. We analyze the sensitivity of the reduced model parameters to small changes in the conductivity, and introduce a Newton-type iteration to improve the reconstructions of the simple inversion method. As an added bonus, our method can benefit from a priori information if available. We generalize both methods to the 2D EIT problem by considering finite volumes discretizations of size determined by the mesurement precision, but where the node locations are to be determined adaptively. This discretization can be viewed as a resistor network, where the resistors are essentially averages of the conductivity over grid cells. We show that the model reduction problem of finding the smallest resistor network (of fixed topology) that can predict meaningful measurements of the Dirichlet-to-Neumann map is uniquely solvable for a broad class of measurements. We propose a simple inversion method that, as in the simple method for the inverse Sturm-Liouville problem, is based on an interpretation of the resistors as conductivity averages over grid cells, and an iterative method that improves such reconstructions by using sensitivity information on the changes in the resistors due to small changes in the conductivity. A priori information can also be incorporated to the latter method.