Adaptive and stochastic algorithms for electrical impedance tomography and DC resistivity problems with piecewise constant solutions and many measurements

摘要

This article develops fast numerical methods for the practical solution of the famous electrical impedance tomography and DC resistivity problems in the presence of discontinuities and potentially many experiments or data. Based on a Gauss-Newton (GN) approach coupled with preconditioned conjugate gradient (PCG) iterations, we propose two algorithms. One determines adaptively the number of inner PCG iterations required to stably and effectively carry out each GN iteration. The other algorithm, useful especially in the presence of many experiments, employs a randomly chosen subset of experiments at each GN iteration that is controlled using a cross validation approach. Numerical examples demonstrate the efficacy of our algorithms.