Extraction of Distribution Function of Relaxation Times by using DRT-RBLM Tools: A New Approach to Combine Levenberg-Marquardt Algorithm and Radial Basis Functions for Discretization Basis

Electrochemical Impedance Spectroscopy (EIS) is a powerful tool for the analysis of different power sources and various materials. One of the methods used for studying EIS data is the distribution function of relaxation times (DRT). EIS data can be converted into a Fredholm integral of the first kind; and DRT extraction is known to be an inverse ill-posed problem. Herein, a new strategy to extract DRT by applying the Levenberg-Marquardt algorithm (LMA) is proposed. The Jacobian matrix appearing in LMA is partially numerically approximated by applying the radial basis function as a basis for the discretization. DRT data are smoothed by the application of the finite difference matrix and the negative values are avoided by the limits application. The tests conducted with ZARCs/FRACs synthetic data show that the extracted DRT profiles correspond well to their analytical counterparts. The application of LMA in solving Fredholm integral equation of the first kind (i.e., DRT extraction) resulted in the automatic tuning of the regularization parameter. The aforementioned findings show that by modifying LMA it is possible to both solve the Fredholm integral equation of the first kind in a completely data-driven way and to obtain the applicable DRT data for general EIS study.