Optimization of Impedance Models with Differential Evolution

Commercially available software for optimizing models to empirical impedance data are lacking in many regards. The models are commonly limited at worst to a set of predefined library of analogous, non-physically modeled impedance circuits and at best, limited to what can be programmed by the user within predetermined constraints. Thus, the optimization algorithms are limited in the number of model parameters that can be handled, meaning that the software is incapable of fitting more complicated models, such as the faradaic impedance predicted by the point defect model for the passive state[1-3]. Historically, the point defect model (PDM) faradaic impedance has been optimized on empirical data using gradient-based, nonlinear least squares methods, such as the Levenberg-Mardquardt method. Because the solution space of the point defect model is a 10-30 dimensional hypercube containing a multi-modal optimal surface of solution vectors, it is highly unlikely to obtain a fit solution corresponding to a global minimum in error using gradient-based methods, unless a near-perfect starting guess is employed. It is for this reason that the PDM is so difficult for researchers to use in their own work, often resulting in attempts at using the PDM to be either discarded, or of accepting a generally poor fit quality. Indeed, the screening criteria used for determining the quality of a fit with such methods is often done entirely by “eye” and is therefore highly subjective. It is also highly likely, even with a large number of random samples of starting guesses, to obtain a fit whereby there are alternative solutions with superior agreement. As previously noted, excellent starting guesses must be provided to the algorithm in order to obtain a satisfactory fit, meaning the solution must be known prior to attempting a curve-fit, which is particularly difficult and in many cases intractable. This is often done by using, as the initial guesses for the parameter values, the results of prior optimizations when changing only one independent parameter in a series of experiments (e.g., the passive film formation potential). Improving the fit over subsequent trials of a gradient method was generally accomplished by trial-anderror starting guesses input by the user, or by utilizing a linear grid-based set of starting conditions, evaluating the goodness of fit and choosing the best result. Furthermore, the all-inclusive point defect model has up to 30 parameters, meaning that due to software limitations, many assumptions were required in order to formulate the problem to fit in order to reduce the number of parameters to the limit, generally, 10 parameters, resulting in an incomplete search space, which, depending on the assumptions, could affect the scientific conclusion of the fit. The only advantage of gradient methods is that they are extremely fast. Now that the computational power of desktop computers have outpaced this hurdle, it is necessary to take a new approach for optimizing physico-chemical models, such as the PDM, on experimental impedance data. To this end, custom software has been developed adapting differential evolution[4] to the problem of impedance model optimization. It is currently being used by researchers at Penn State to fit impedance data to a degree of quality never before seen[5-6], with the ability to directly compare the dominance and efficiency of models, for example, a comparison of various diffusion impedance elements as seen in Figure 1