Summary: A globally convergent (the so-called convexification) algorithm was previously developed for coefficient inverse problems (CIPs) with the time/frequency-dependent data. In this publication the convexification is extended to the case of a CIP for an elliptic equation with the data generated by the source running along a straight line. The data are incomplete, since they are given only at a part of the boundary. Applications to both electrical impedance and optical tomographies are feasible, which include, in particular, imaging of land mines and underground bunkers, as well as diffuse optical imaging of targets on battlefields through smogs and flames. However, our numerical setup is intended for medical applications to small animals. Numerical experiments in the 2D case are presented.