Discrete layer-stripping algorithms and feasibilityconditions for the 2D inverse conductivity problem

Abstract

We develop a discrete layer-stripping algorithm for the 2Dinverse conductivity problem. Unlike previous algorithms, thisalgorithm transforms the problem into a time-varying 1DSchrödinger equation inversescattering problem, discretizesthis problem and then solves the discrete problem exactly.This approach has three advantages: (i) the poor conditioninginherent in the problem is concentrated in the solution of alinear integral transform at the beginning of the problem, towhich standard regularization techniques may be applied and(ii) feasibility conditions on the transformed data are obtained,satisfaction of which ensures that (iii) the solution of thediscrete nonlinear inverse scattering problem is exact andstable. Other contributions include solution of discreteSchrödinger equation inverse potential problems withtime-varying potentials by both layer-stripping algorithms andsolution of nested systems of equations which amount to atime-varying discrete version of the Gel'fand-Levitan equation.An analytic and numerical example is supplied to demonstrate theoperation of the algorithm.