Before revision:  During the offline stage, given (i) a family of parameterized domains \{ \mu: \mu \in P \} \subset R^D where \mu \in P \subset R^P denotes a vector of parameters, (ii) a parameterized mapping \Phi _{\mu} between a reference domain \Omega and the parameter-dependent domain \mu, and (iii) a finite element triangulation of \Omega, we resort to an empirical quadrature procedure to select a subset of the elements of the grid.
After revision:  During the offline stage, given (i) a family of parameterized domains \{ \mu: \mu \in P \} \subset R^D where \mu \in P \subset R^P denotes a vector of parameters, (ii) a parameterized mapping \Phi _{\mu} between a reference domain \Omega and the parameter-dependent domain \mu, and (iii) a finite element triangulation of \Omega, we resort to an empirical quadrature procedure to select a subset of the elements of the grid.