TY - JOUR
T1 - Solving CNLS problems using Levenberg-Marquardt algorithm
T2 - A new fitting strategy combining limits and a symbolic Jacobian matrix
AU - Žic, Mark
AU - Subotić, Vanja
AU - Pereverzyev, Sergei
AU - Fajfar, Iztok
N1 - Funding Information:
The authors gratefully acknowledge the stimulation program “Joint Excellence in Science and Humanities” ( JESH-2017 ) of the Austrian Academy of Sciences as well as the research program “Algorithms and Optimization Methods in Telecommunications” ( P2-0246 ) of the Ministry of Education, Science and Sport of Republic of Slovenia for providing supporting funds.
Publisher Copyright:
© 2020 Elsevier B.V.
PY - 2020/6/1
Y1 - 2020/6/1
N2 - 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.
AB - 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.
KW - CNLS
KW - EIS
KW - Levenberg-Marquardt algorithm
KW - Limits
KW - Symbolic Jacobian matrix
UR - http://www.scopus.com/inward/record.url?scp=85083737699&partnerID=8YFLogxK
U2 - 10.1016/j.jelechem.2020.114171
DO - 10.1016/j.jelechem.2020.114171
M3 - Article
AN - SCOPUS:85083737699
SN - 1572-6657
VL - 866
JO - Journal of Electroanalytical Chemistry
JF - Journal of Electroanalytical Chemistry
M1 - 114171
ER -