Lipschitz stability of an inverse boundary value problem for a   Schr\"{o}dinger type equation
  In this paper we study the inverse boundary value problem of determining the potential in the Schr\"{o}dinger equation from the knowledge of the Dirichlet-to-Neumann map, which is commonly accepted as an ill-posed problem in the sense that, under general settings, the optimal stability estimate is of logarithmic type. In this work, a Lipschitz type stability is established assuming a priori that the potential is piecewise constant with a bounded known number of unknown values. 
Elena Beretta and Maarten V. de Hoop and Lingyun Qiu
math.AP