Well-posed Questions for Ill-posed Inverse Problems: a Note in Memory of Pierre Sabatier
Analysis of PDEs
2025-04-25 v1
Abstract
Professor Pierre Sabatier contributed much to the study of inverse problems in theory and practice. Two of these contributions were a focus on theory that actually supports practice, and the identification of well-posed aspects of inverse problems that may quite ill-posed. This paper illustrates these two themes in the context of Electrical Impedance Tomography (EIT), which is both very ill-posed and very practical. We show that for a highly constrained version of this inverse problem, in which a small elliptical inclusion in a homogeneous background is to be identified, optimization of the experimental design (that is, electrode locations) vastly improves the stability of the solution.
Keywords
Cite
@article{arxiv.2504.17592,
title = {Well-posed Questions for Ill-posed Inverse Problems: a Note in Memory of Pierre Sabatier},
author = {Gaoming Chen and Fadil Santosa and William W. Symes},
journal= {arXiv preprint arXiv:2504.17592},
year = {2025}
}