I Publication I J. Tiilikainen, J.-M. Tilli, V. Bosund, M. Mattila, T. Hakkarainen, V.-M. Airaksinen and H. Lipsanen, Nonlinear fitness-space-structure adaptation and principal component analysis in genetic algorithms: an application to x-ray reflectivity anal-Abstract Two novel genetic algorithms implementing principal component analysis and an adaptive nonlinear fitness-space-structure technique are presented and compared with conventional algorithms in x-ray reflectivity analysis.Principal component analysis based on Hessian or interparameter covariance matrices is used to rotate a coordinate frame. The nonlinear adaptation applies nonlinear estimates to reshape the probability distribution of the trial parameters. The simulated x-ray reflectivity of a realistic model of a periodic nanolaminate structure was used as a test case for the fitting algorithms. The novel methods had significantly faster convergence and less stagnation than conventional non-adaptive genetic algorithms. The covariance approach needs no additional curve calculations compared with conventional methods, and it had better convergence properties than the computationally expensive Hessian approach. These new algorithms can also be applied to other fitting problems where tight interparameter dependence is present.
A novel genetic algorithm (GA) utilizing independent component analysis (ICA) was developed for x-ray reflectivity (XRR) curve fitting. EFICA was used to reduce mutual information, or interparameter dependences, during the combinatorial phase. The performance of the new algorithm was studied by fitting trial XRR curves to target curves which were computed using realistic multilayer models. The median convergence properties of conventional GA, GA using principal component analysis and the novel GA were compared. GA using ICA was found to outperform the other methods with problems having 41 parameters or more to be fitted without additional XRR curve calculations. The computational complexity of the conventional methods was linear but the novel method had a quadratic computational complexity due to the applied ICA method which sets a practical limit for the dimensionality of the problem to be solved. However, the novel algorithm had the best capability to extend the fitting analysis based on Parratt's formalism to multiperiodic layer structures.
IIIPublication III J. Tiilikainen, V. Bosund, M. Mattila, T. Hakkarainen, J. Sormunen and H. Lipsanen, Fitness function and nonunique solutions in x-ray reflectivity curve fitting: Crosserror between surface roughness and mass density, AbstractNonunique solutions of the x-ray reflectivity (XRR) curve fitting problem were studied by modelling layer structures with neural networks and designing a fitness function to handle the nonidealities of measurements. Modelled atomic-layer-deposited aluminium oxide film structures were used in the simulations to calculate XRR curves based on Parratt's formalism. This approach reduced the dimensionality of the parameter space and allowed the use of fitness landscapes in the study of nonunique solutions. Fitness landscapes, where the height in a map represents the fitness value as a function of the process parameters, revealed tracks where the local fitness optima lie. The tracks were projected on the physical parameter space thus allowing the construction of the crosserror equation between weakly determined parameters, i.e. between the mass density and the surface roughness of a layer. The equation gives the minimum error for the other parameters which is a consequence of the nonuniqueness of the solution if noise is present. Furthermore, the existence of a possible unique solution in a certain parameter range was found to be dependent on the layer thickness and the signal-to-noise ratio.
The influence of Poisson noise on the accuracy of x-ray reflectivity analysis is studied with an aluminium oxide (AlO) layer on silicon. A null hypothesis which argues that other than the exact solution gives the best fitness is examined with a statistical p-value test using a significance level of α = 0.01. Simulations are performed for a fit instead of a measurement since the exact error caused by noise cannot be determined from the measurement. The p-value is studied by comparing trial curves to 1000 'measurements', each of them including synthetic Poisson noise. Confidence limits for the parameters of Parratt's formalism and the Nevot-Croce approximation are determined in (mass density, surface roughness), (thickness, surface roughness) and (thickness, mass density) planes. The most significant result is that the thickness determination accuracy of AlO is approximately ±0.09 nm but the accuracy is better for materials having higher mass density. It is also shown that the accuracy of mass density determination can be significantly improved using a suitably designed fitness measure. Although the power of the presented method is demonstrated only in one case, it can be used in any parameter region for a plethora of single layer systems to find the lower limit of the error made in x-ray reflectivity analysis.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.