Solution of the LIMM equation by the polynomial regularization method and the L-curve algorithm for selection of the regularization parameter

The determination of the polarization distribution by means of the laser intensity modulation method (LIMM) requires a solution of a Fredholm integral equation of the 1st kind. This is an ill-conditioned problem that can lead to multiple and very different solutions. One of the more frequently used solution methods is based upon Tikhonov regularization. Previously, a new regularization method was developed for solving the LIMM equation with an 8th degree polynomial using smoothing to achieve a stable and optimal solution. This was named the polynomial regularization method (PRM). In the current study, several simulated polarization distributions for polyvinylidene fluoride (PVDF) were selected. LIMM current versus frequency data were simulated using these distributions and Gaussian noise was added so as to emulate experimental data. The LIMM equation was solved using PRM. An algorithm based upon the L-curve method (LCM) was developed for the prediction of the optimal regularization parameter. Calculated distribution functions using PRM and LCM were in good agreement with the simulated polarization distributions.