Influence of the Discretization Methods on the Distribution of Relaxation Times Deconvolution: Implementing Radial Basis Functions with DRTtools

Abstract

The distribution of relaxation times (DRT) is an approach that can extract time characteristics of an electrochemical system from electrochemical impedance spectroscopy (EIS) measurements. Computing the DRT is difficult because it is an intrinsically ill-posed problem often requiring regularization. In order to improve the estimation of the DRT and to better control its error, a suitable discretization basis for the regularized regression needs to be chosen. However, this aspect has been invariably overlooked in the specialized literature. Pseudo-spectral methods using radial basis functions (RBFs) are, in principle, a better choice in comparison to other discretization basis, such as piecewise linear (PWL) functions, because they may achieve fast convergence. Furthermore, they can yield improved estimation by extending the estimated DRT to the entire frequency spectrum, if the underlying DRT decays to zero sufficiently fast outside the measured frequency range. Additionally, their implementation is relatively easier than other types of pseudo-spectral methods since they do not require ad hoc collocation point distributions. The as-developed novel RBF-based DRT framework was tested against controlled synthetic EIS spectra and real experimental data. Our results indicate that the RBF discretization performance is comparable with that of the PWL discretization at normal data collection range, and with improvement when the EIS acquisition is incomplete. In addition, we also show that applying RBF discretization for deconvolving the DRT problem can lead to faster numerical convergence rate as compared with that of PWL discretization only at error free situation. As a companion to this work we have developed a MATLAB GUI toolbox, which can be used to solve DRT regularization problems.