Jacobian of solutions to the conductivity equation in limited view. (1st February 2023)

• Record Type: Journal Article
• Authors: Salo, Mikko; Schlüter, Hjørdis
• Is Part Of: Inverse problems. Volume 39:Number 2(2023)
• Publication Date: 2023-02-01
• Subjects: acousto-electric tomography -- current density imaging -- hybrid inverse problems -- coupled physics imaging -- non-vanishing Jacobian -- conductivity equation; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/aca904
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: The aim of hybrid inverse problems such as Acousto-Electric Tomography or Current Density Imaging is the reconstruction of the electrical conductivity in a domain that can only be accessed from its exterior. In the inversion procedure, the solutions to the conductivity equation play a central role. In particular, it is important that the Jacobian of the solutions is non-vanishing. In the present paper we address a two-dimensional limited view setting, where only a part of the boundary of the domain can be controlled by a non-zero Dirichlet condition, while on the remaining boundary there is a zero Dirichlet condition. For this setting, we propose sufficient conditions on the boundary functions so that the Jacobian of the corresponding solutions is non-vanishing. In that regard we allow for discontinuous boundary functions, which requires the use of solutions in weighted Sobolev spaces. We implement the procedure of reconstructing a conductivity from power density data numerically and investigate how this limited view setting affects the Jacobian and the quality of the reconstructions.