The methods of constrained approximation in Hilbert spaces of analytic functions are applied to the solution of the inverse problems of detecting cracks or sources in a two-dimensional material by means of boundary measurements. Issues of well-posedness are discussed, and results on continuity and robustness with respect to the given data are established. Constructive and efficient methods for resolution of the above approximation problems are presented. The techniques are illustrated by numerical examples incorporating a further rational approximation step.