In this paper we describe the polynomial and regularized discrete solutions of the thermal wave probing problem found in the investigation of polarization and space charge in dielectrics. We demonstrate that approximate low order polynomial solutions can be obtained in the presence of noise and without regularization. We also demonstrate a more general discrete solution with explicit regularization. In this case, the regularization parameter can be selected with the aid of the L-curve. We observe that the L-curve needs to be used with care when applied to experimental data.