In this paper, the characterization of electromagnetic inverse problems is addressed. As it is well known, an inverse problem can be ill-posed, i.e., its solution is not necessarily unique and might be quite sensitive to the measured data. To characterize such an inverse problem a combination of a surrogate model --based on an optimal database-- and an inverse mapping of some sort, both using kriging prediction as tools, is proposed. The database (a kind of discrete representation of the electromagnetic model) is generated using an adaptive sampling strategy aiming to find a set of optimal input parameter-output data pairs. Once the latter has been computed, both qualitative and quantitative conclusions about the related inverse problem can be drawn. The illustrative examples are drawn from eddy-current nondestructive testing. month =aug