This paper addresses the development of parameter estimation techniques for a class of models used to characterize hysteresis and constitutive nonlinearities inherent to ferroelectric, ferromagnetic and ferroelastic materials employed in a wide range of actuators and sensors. These models are formulated as integral equations with known kernels and unknown densities to be identified through least-squares techniques. Due to the compactness of the integral operators, the resulting discretized models inherit ill-posedness which must be accommodated through regularization. The accuracy of regularized finite-dimensional models is illustrated through comparison with experimental data.