A Novel In Silico Monte Carlo Approach to Optimize a PSD Estimation Problem. Generation of Data Fusion Experiment Rules
DOI:
https://doi.org/10.5540/tcam.2022.023.04.00749Keywords:
Inverse problem, particle size distribution, DOE, data fusionAbstract
This article analyzes the performance of combining information from Scanning Electron Microscopy (SEM) micrographs with Static Light Scattering (SLS) measurements for retrieving the so-called Particle Size Distribution (PSD) in terms of experimental features. The corresponding data fusion is implemented using a novel Monte Carlo-based method consisting in a SMF (Sampling-Mapping-Filtering) approach. This approach provides an important reference to assess the strategy of the experiment for this specific problem by means of solving an inverse problem. Furthermore, low levels of volume fraction and a PSD represented by log-normal distributions are considered in order to reduce processing and model errors due to ill-posedness. The prior statistics corresponding to the SEM micrographs have been achieved by means of the Jackknife procedure used as a resampling technique. The likelihood term considers iid normal measurements generated from the Local Monodisperse Approximation (LMA) and also makes use of the same model as forward linear model, in an inversion case known as inverse crime. However, it has been proved that the LMA performs well in practice for low fraction volume systems as considered here. The PSD retrieval is measured in terms of improvement in precision with respect to one of the log-normal parameters in SEM micrographs, i.e., the desirability. Estimates are expressed as a function of a typical system parameter such as polydispersity, as well as experimental variables, i.e., number of particles per micrograph (PPM) and noise level $\varepsilon$ in the SLS measurements. These estimations are then analyzed by means of the Box-Behnken (BB) design and the response surface methodology (RSM) in order to generate a surrogate model from which rules for the optimization of the experiment are made when desirability is maximized. Finally, a Rule-Based System (RBS) is proposed for future use.References
R. A. Reynolds, D. Stramski, V. M. Wright, and S. B. Woźniak, ``Measurements and characterization of particle size distributions in coastal waters,'' J. Geophys. Res., vol. 115, pp. 1-19, 2010.
L. A. Clementi, J. R. Vega, L. M. Gugliotta, and H. R. B. Orlande, ``A bayesian inversion method for estimating the particle size distribution of latexes from multiangle dynamic light scattering measurements,'' Chemometr. Intell. Lab. Syst., vol. 107, pp. 165-173, 2011.
F. A. Otero, H. R. B. Orlande, G. L. Frontini, and G. E. Eliçabe, ``Bayesian approach to the inverse problem in a light scattering application,'' J. Appl. Stat., vol. 42, pp. 994-1016, 2015.
S. A. Sisson, Y. Fan, and M. A. Beaumont, Handbook of Approximate Bayesian Computation, ch. 1: Overview of ABC, p. 4. Boca Raton, Florida: Chapman & Hall/CRC, 2018.
T. J. Santner, B. J. Williams, and W. I. Notz, The Design and Analysis of computer Experiments, ch. Preface, p. 7. New York: Springer-Verlag, 2003.
H. Martens, G. B. Dijksterhuis, and D. V. Byrne, ``Power of experimental designs, estimated by monte carlo simulation,'' J. Chemometrics, vol. 14, pp. 441-462, 2010.
A. Ligêza, Logical Foundations for Rule-Based Systems, ch. Preface, pp. 5-6. Berlin: Springer-Verlag, 2006.
W. L. Martinez and A. R. Martinez, Exploratory Data Analysis with MATLAB, ch. 1: Introduction to Exploratory Data Analysis, pp. 3-4. Boca Raton, Florida: Chapman & Hall/CRC Press, 2005.
J. W. Tukey, ``We need both exploratory and confirmatory,'' The American Statistician, vol. 34, pp. 23-25, 1980.
F. A. Otero, G. L. Frontini, and G. E. Eliçabe, `` PSD retrieval by bayesian data fusion via Metropolis-Hastings and the Jackknife procedure,'' in 17th Workshop of Information and Control (RPIC 2017), (Mar del Plata, Argentina), 2017.
D. C. Joy, Monte Carlo Modeling for Electron Microscopy and Microanalysis. New York, Oxford: Oxford University Press, 1995.
M. Gu, X. Gan, and X. Deng, Microscopic Imaging through Turbid Media. Monte Carlo Modeling and Applications. Berlin, Heidelberg: Springer-Verlag, 2015.
J. S. Pedersen, ``Determination of size distribution from small-angle scattering data for systems with effective hardsphere interactions,'' J. Appl. Cryst., vol. 27, p. 595-608, 1994.
F. A. Otero, G. L. Frontini, and G. E. Eliçabe, ``Evaluation of light scattering models to characterize concentrated polymer particles embedded in a solid polymer matrix,'' Journal of Polymer Science Part B: Polymer Physics, vol. 48, p. 958-963, 2010.
G. E. Eliçabe and F. A. Otero, ``Static light scattering of concentrated particle systems in the rayleigh-debye-gans regime: Modeling and data analysis,'' Particulate Science and Technology, vol. 28, p. 485-497, 2010.
R. Prabhu, W. R. Whittington, S. S. Patnaik, Y. Mao, M. T. Begonia, L. N. Williams, J. Liao, and M. F. Horstemeyer, ``A coupled experiment-finite element modeling methodology for assessing high strain rate mechanical response of soft biomaterials,'' Journal of Visualized Experiments, vol. 99, pp. 1-17, 2015.
M. H. Quenouille, ``Notes on bias in estimation,'' Biometrika, vol. 43, pp. 353-360, 1956.
C. M. Bishop, Pattern Recognition and Machine Learning, ch. 11: Sampling methods, p. 523. New York: Springer, 2006.
V. A. Morozov, ``On the solution of functional equations by the method of regularization,'' Soviet Math Dokl, vol. 7, pp. 414-417, 1966.
J. Gillard, ``Cadzow's basic algorithm, alternating projections and singular spectrum analysis,'' Statistics and Its Interface, vol. 3, pp. 335-343, 2010.
R. C. Aster, B. Borchers, and C. H. Thurber, Parameter Estimation and Inverse Problems, ch. 4: Rank deficiency and ill-conditioning, p. 67. New York: Elsevier Academic Press, 2005.
H. Khoshdast, A. Soflaeian, and V. Shojaei, ``Coupled fuzzy logic and experimental design application for simulation of a coal classifier in an industrial environment,'' Physicochem. Probl. Miner. Process., vol. 55, p. 504-515, 2019.
F. Aginab, R. Khosravaniana, M. Karimifardc, and A. Jahanshahid, ``Application of adaptive neuro-fuzzy inference system and data mining approach to predict lost circulation using doe technique (case study: Maroon oilfield),'' Petroleum, vol. 6, pp. 423-437, 2020.
G. Box and D. Behnken, ``Some new three level designs for the study of quantitative variables,'' Technometrics, vol. 2, pp. 455-475, 1960.
R. L. Mason, R. F. Gunst, and J. L. Hess, Statistical Design and Analysis of Experiments with Applications to Engineering and Science, ch. 17: Designs and Analyses for Fitting Response Surfaces, pp. 568-635. Hoboken, New Jersey: John Wiley & Sons, Inc., 2003.
A. Gosavi, Simulation-based Optimization: Parametric Optimization Techniques and Reinforcement Learning, ch. 4: Parametric Optimization: Response Surfaces and Neural Networks, p. 57. New York: Springer Science+ Business Media, 2003.
P. Jiang, Q. Zhou, and X. Shao, Surrogate Model-Based Engineering Design and Optimization, ch. 2: Classic Types of Surrogate Models, pp. 7-9. Singapore: Springer Nature, 2020.
P. P. Angelov,Evolving Rule-Based Models: A Tool for Design of Flexible Adaptive Systems, ch. 3: Flexible models, p. 25. Berlin, Heidelberg: Springer-Verlag, 2002.
Mendel, Uncertain Rule-based Fuzzy Systems. Introduction and New Directions, ch. 3: Type-1 Fuzzy Systems, pp. 101-160. Cham, Switzerland: Springer Nature, 2017.
K. Levenberg, ``A method for the solution of certain non-linear problems in least squares,'' Quarterly of Applied Mathematics, vol. 2, pp. 164-168, 1944.
D. W. Marquardt, ``An algorithm for least-squares estimation of nonlinear parameters,'' Journal of the Society for Industrial and Applied Mathematics, vol. 11, pp. 431-44, 1963.
M. J. Anderson and P. J. Whitcomb, RSM Simplified: Optimizing Processes Using Response Surface Methods for Design of Experiments. Boca Raton, Florida: CRC Press, 2017.
W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Pascal, ch. 10: Minimization or maximization of functions, p. 326. New York: Cambridge University Press, 1989.
F. Otero, G. Eliçabe, and G. Frontini, ``A modified processing method for light scattering measurements in particulate systems by means of splines and fuzzy logic,'' in 16th Workshop on Information Processing and Control (RPIC 2015), vol. 1, (Córdoba, Argentina), 2016.
E. R. Soulé and G. E. Eliçabe, ``Determination of size distributions of concentrated polymer particles embedded in a solid polymer matrix,'' Particle & Particle Systems Characterization, vol. 125, p. 84-91, 2008.
Downloads
Published
How to Cite
Issue
Section
License
Copyright
Authors of articles published in the journal Trends in Computational and Applied Mathematics retain the copyright of their work. The journal uses Creative Commons Attribution (CC-BY) in published articles. The authors grant the TCAM journal the right to first publish the article.
Intellectual Property and Terms of Use
The content of the articles is the exclusive responsibility of the authors. The journal uses Creative Commons Attribution (CC-BY) in published articles. This license allows published articles to be reused without permission for any purpose as long as the original work is correctly cited.
The journal encourages Authors to self-archive their accepted manuscripts, publishing them on personal blogs, institutional repositories, and social media, as long as the full citation is included in the journal's website version.
