In engineering, many system design problems come down to finding design parameters for the system to operate under desired conditions. When time-consuming numerical simulations are used to assess the conditions of operation of such systems, it is essential to treat this problem—broadly known as "inversion"—using search methods that are very sparing in the number of simulations. With this constraint in mind, we propose here a sequential strategy for a particular robust inversion problem that we call "reliability-based inversion" (RBI), in which the system has both deterministic and uncertain inputs. In this formulation, the objective is to retrieve the set of deterministic inputs such that the probability of the outputs exceeding a given level is greater than a threshold.