In this talk we discuss the convergence rate of the prolate-Galerkin linear sampling method for shape-parameter identification in the context of the multi-frequency inverse source problem for a fixed observation direction and the Born inverse scattering problems. The prolate-Galerkin linear sampling method can be seen as a spectral discretization of the linear sampling method using the prolate spheroidal wave functions and their generalizations. The techniques are rooted in that the prolate spheroidal wave functions and their generalizations are eigenfunctions of a compact integral operator and of a Sturm-Liouville differential operator at the same time. We obtain convergence rates for both noiseless and noisy data, roughly speaking, under the assumption that the contrast belongs to the standard Sobolev space $H^s$ instead of less explicit functional spaces characterized by the compact data operator. Some a priori estimates on the eigenvalues of the data operator are also investigated. The convergence rate for noiseless data is explicit and the convergence rate for noisy data is semi-explicit.