Solving CNLS problems using Levenberg-Marquardt algorithm: A new fitting strategy combining limits and a symbolic Jacobian matrix

Abstract

The Levenberg-Marquardt algorithm (LMA) is generally used to solve diverse complex nonlinear least square (CNLS) problems and is one of the most used algorithms to extract equivalent electrochemical circuit (EEC) parameters from electrochemical impedance spectroscopy (EIS) data. It is a well-known fact that the convergence properties of the algorithm can be boosted by applying limits on EEC parameter values. However, when EEC parameter values are low (i.e., of the order of magnitude of 10−4 or smaller), the applied limits increase the first derivatives approximation errors which occur when using a numerical Jacobian matrix. In this work, we discuss the importance of the Jacobian matrix in LMA and propose a design of a new EIS fitting engine. The new engine is based on a novel fitting scheme using limits and a symbolic Jacobian matrix instead of the numerical one, i.e. a strategy that has not yet been reported in any EIS study. We show that using a symbolic Jacobian matrix the algorithm convergence is superior to the one with a numerical Jacobian matrix. We also investigate how to improve poor convergence properties when we still have to use a numerical Jacobian matrix when analytic derivatives are not available.