An alternative approach to solve complex nonlinear least-squares problems

Abstract

Abstract In this work, a hybrid electrochemical impedance spectroscopy (EIS) strategy was developed with the aim to solve complex nonlinear least-square (CNLS) problems. For the first time the CLNS problems were solved by an alternative approach, i.e. by using Python to merge the following tactics: circuit description code (CDC), central-difference formula, Nielsen's modification of the Levenberg–Marquardt algorithm (LMA) and visual inspection of the χ2-function minima. It was presented that the Nielsen's damping (λ) factor updating strategy can be used to design a topical fitting engine in which the change in the λ factor is continuous. The commonly used forward-difference formula was replaced with a more accurate central-difference formula to approximate the first derivates. It was elucidated that by combining the Nielsen's λ updating tactics and central-difference formula in the hybrid EIS strategy resulted in the design of a more robust fitting engine. The hybrid EIS strategy was enhanced by the CDC routine which allows researchers to effortlessly generate a great variety of electrical equivalent circuits (EECs). The credibility of the EEC parameters was increased by visual inspection of the χ2-function minima. Additionally, a novel convergence in the symmetry criterion was proposed to avoid errors during the visual inspection. The applicability of the hybrid EIS strategy was evaluated after it was integrated into a new software solution and compared to the more widely used MEISP and EQUIVCRT software packages. It was presented that the three fundamentally different software solutions yielded approximately equal EEC data. However, only EIS strategies with the similar LMA tactics produced the identical χ2-values in all case studies.