Regularizations for an all-at-once formulation of the electrical impedance tomography problem

An all-at-once formulation of the inverse problem of electrical impedance tomography (EIT) is proposed, and three regularizations for it are analyzed. Under a set of assumptions made in the context of Banach spaces, an abstract problem is proposed aiming at generalizing the all-at-once formulation of the EIT inverse problem. This abstract problem allows to incorporate several strategies of input data, namely voltage measurements, current measurements, magnitudes of the current density field, and interior power densities. Three regularizations, based on the classic Tikhonov, Ivanov, and Morozov approaches, are proposed for this problem. The existence, stability, and convergence of regularized solutions are proved. Well-known EIT models fit into this abstraction (for instance, the complete electrode model). It turns out that the all-at-once approach provides an alternative formulation of the EIT inverse problem. Numerical tests are performed using the complete electrode model and the previously mentioned strategies.