Parametric derivatives in inverse conductivity problems with total variation regularization

Abstract

The focus of this paper is an improved differentiability result for the forward map in inverse problems involving elliptic partial differential equations, and examination of its significance in the context of the electrical impedance tomography (EIT) problem with total variation (TV) regularization. We base our analysis on the Fréchet derivative of the mapping which takes a given conductivity function (spatially varying) in an electrostatic model to a corresponding elliptic PDE solution, and we develop the implications of a certain compactness property of the parameter space. By following this approach, we show Fréchet differentiability with a weaker norm (the L¹ norm) for the parameter space than is usually used (the L ^∞ norm), thus improving the Fréchet differentiability result. The EIT problem with TV regularization is well studied in the literature, and several authors have addressed the Fréchet differentiability question. However, to the best of our knowledge and as we argue, our result is the strongest analytical result in this context. Many derivative-based methods such as Gauss–Newton and Levenburg-Marquardt lie at the heart of many proposed methods for EIT, and the results described herein for these derivative calculations provide a firm theoretical footing for them.