Fourth-Order Predictive Modelling: II. 4th-BERRU-PM Methodology for Combining Measurements with Computations to Obtain Best-Estimate Results with Reduced Uncertainties
This work presents a comprehensive fourth-order
predictive modeling (PM) methodology that
uses the MaxEnt principle to incorporate fourth-order moments (means,
covariances, skewness, kurtosis) of model parameters, computed and
measured model responses, as well as fourth (and higher) order sensitivities of
computed model responses to model parameters. This new methodology is
designated by the acronym 4th-BERRU-PM,
which stands for “fourth-order best-estimate results with reduced
uncertainties.” The results predicted by the 4th-BERRU-PM incorporates, as particular
cases, the results previously predicted by the second-order predictive modeling
methodology 2nd-BERRU-PM,
and vastly generalizes the results produced by extant data assimilation and
data adjustment procedures.
References
[1]
Cacuci, D.G. (2023) Second-Order MaxEnt Predictive Modelling Methodology. I: Deterministically Incorporated Computational Model (2nd-BERRU-PMD). American Journal of Computational Mathematic, 13, 236-266. https://doi.org/10.4236/ajcm.2023.132013
[2]
Cacuci, D.G. (2023) Second-Order MaxEnt Predictive Modelling Methodology. II: Probabilistically Incorporated Computational Model (2nd-BERRU-PMP). American Journal of Computational Mathematic, 13, 267-294. https://doi.org/10.4236/ajcm.2023.132014
[3]
Kuroi, H. and Mitani, H. (1975) Adjustment to Cross-Section Data to Fit Integral Experiments by Least Squares Method. American Journal of Computational Mathematic, 12, 663-680. https://doi.org/10.1080/18811248.1975.9733172
[4]
Dragt, J.B., Dekker, J.W.M., Gruppelaar, H. and Janssen, A.J. (1977) Methods of Adjustment and Error Evaluation of Neutron Capture Cross Sections. Nuclear Science and Engineering, 62, 117-129. https://doi.org/10.13182/NSE77-3
[5]
Lewis, J.M., Lakshmivarahan, S. and Dhall, S.K. (2006) Dynamic Data Assimilation: A Least Square Approach. Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9780511526480
[6]
Lahoz, W., Khattatov, B. and Ménard, R. (2010) Data Assimilation: Making Sense of Observations. Springer Verlag, Berlin.
[7]
Cacuci, D.G., Navon, M.I. and Ionescu-Bujor, M. (2014) Computational Methods for Data Evaluation and Assimilation. Chapman & Hall/CRC, Boca Raton.
[8]
Fang, R. and Cacuci, D.G. (2023) Second-Order MaxEnt Predictive Modelling Methodology. III: Illustrative Application to a Reactor Physics Benchmark. American Journal of Computational Mathematics, 13, 295-322. https://doi.org/10.4236/ajcm.2023.132015
[9]
Cacuci, D.G. and Fang, R. (2023) Demonstrative Application to an OECD/NEA Reactor Physics Benchmark of the 2nd-BERRU-PM Method. I: Nominal Computations Consistent with Measurements. Energies, 16, Article 5552. https://doi.org/10.3390/en16145552
[10]
Fang, R. and Cacuci, D.G. (2023) Demonstrative Application to an OECD/NEA Reactor Physics Benchmark of the 2nd-BERRU-PM Method. II: Nominal Computations Apparently Inconsistent with Measurements. Energies, 16, Article 5614. https://doi.org/10.3390/en16155614
[11]
Valentine, T.E. (2006) Polyethylene-Reflected Plutonium Metal Sphere Subcritical Noise Measurements. SUB-PU-METMIXED-001, International Handbook of Evaluated Criticality Safety Benchmark Experiments, NEA/NSC/DOC(95)03/I-IX, Organization for Economic Co-Operation and Development, Nuclear Energy Agency, Paris.
[12]
Cacuci, D.G. (2023) Fourth-Order Predictive Modelling: I. General-Purpose Closed-Form Fourth-Order Moments-Constrained MaxEnt Distribution. American Journal of Computational Mathematics.
[13]
Cacuci, D.G. (2022) The nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology (nth-CASAM): Overcoming the Curse of Dimensionality in Sensitivity and Uncertainty Analysis, Volume I: Linear Systems. Springer, Cham. https://doi.org/10.1007/978-3-030-96364-4
[14]
Cacuci, D.G. (2023) The nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology (nth-CASAM): Overcoming the Curse of Dimensionality in Sensitivity and Uncertainty Analysis, Volume III: Nonlinear Systems. Springer, Cham. https://doi.org/10.1007/978-3-031-22757-8
[15]
Jaynes, E.T. (1957) Information Theory and Statistical Mechanics. Physical Review Journals Archive, 106, 620-630. https://doi.org/10.1103/PhysRev.106.620
[16]
Shannon, C.E. (1948) A Mathematical Theory of Communication. The Bell System Technical Journal, 27, 379-423. https://doi.org/10.1002/j.1538-7305.1948.tb01338.x
[17]
Shun, Z. and McCullagh. P. (1995) Laplace Approximation of High Dimensional Integrals. Journal of the Royal Statistical Society: Series B, 57, 749-760. https://doi.org/10.1111/j.2517-6161.1995.tb02060.x
[18]
Evangelou, E., Zhu, Z. and Smith, R.L. (2011) Estimation and Prediction for Generalized Linear Mixed Models Using High-Order Laplace Approximation. Journal of Statistical Planning and Inference, 141, 3564-3577. https://doi.org/10.1016/j.jspi.2011.05.008