In this work, we propose an innovative iterative direct sampling method to solve nonlinear elliptic inverse problems from a limited number of pairs of Cauchy data. It extends the original direct sampling method (DSM) by incorporating an iterative mechanism, enhancing its performance with a modest increase in computational effort but a clear improvement in its stability against data noise. The method is formulated in an abstract framework of operator equations and is applicable to a broad range of elliptic inverse problems. Numerical results on electrical impedance tomography, optical tomography and cardiac electrophysiology etc. demonstrate its effectiveness and robustness, especially with an improved accuracy for identifying the locations and geometric shapes of inhomogeneities in the presence of large noise, when compared with the standard DSM.
Ito et al. (Sat,) studied this question.