1st Edition

# Advances in High-Order Predictive Modelling Methodologies and Illustrative Problem

By Dan Gabriel Cacuci Copyright 2025
424 Pages 33 Color & 7 B/W Illustrations
by Chapman & Hall

Continuing the author’s previous work on modeling, this book presents the most recent advances in high-order predictive modeling. The author begins with the mathematical framework of the 2nd-BERRU-PM methodology, an acronym that designates the “second-order best-estimate with reduced uncertainties (2nd-BERRU) predictive modeling (PM).” The 2nd-BERRU-PM methodology is fundamentally anchored in physics-based principles stemming from thermodynamics (maximum entropy principle) and information theory, being formulated in the most inclusive possible phase-space, namely the combined phase-space of computed and measured parameters and responses.

The 2nd-BERRU-PM methodology provides second-order output (means and variances) but can incorporate, as input, arbitrarily high-order sensitivities of responses with respect to model parameters, as well as arbitrarily high-order moments of the initial distribution of uncertain model parameters, in order to predict best-estimate mean values for the model responses (i.e., results of interest) and calibrated model parameters, along with reduced predicted variances and covariances for these predicted responses and parameters.

CHAPTER 1: 2nd-BERRU-PM: Second-Order Maximum Entropy Predictive Modeling Methodology for Reducing Uncertainties in Predicted Model Responses and Parameters

1.1. Introduction

1.2. Generic Mathematical Modeling of a Physical System

1.3. Construction of the Minimally Discrepant Maximum Entropy Distribution

1.4. Construction of the Second-Order Minimally Discrepant Maximum Entropy Distribution of Experimentally Measured Responses and Parameters

1.6.3. Practical Case 2nd-BERRU-PMP: Inclusion of Response Measurements

1.7. Inter-Comparison: 2nd-BERRU-PMP vs. 2nd-BERRU-PMD

1.7.1. Inter-Comparison: Best-Estimate Predicted Mean Values for Responses

1.7.2. Inter-Comparison: Best-Estimate Predicted Mean Values for Parameters

1.7.3. Inter-Comparison: Best-Estimate Predicted Response Covariances

1.7.4. Inter-Comparison: Best-Estimate Predicted Parameter Covariances

1.7.5. Inter-Comparison: Best-Estimate Predicted Correlations Between Parameters and Responses

1.8. Review of Principles Underlying the Data Adjustment and Data Assimilation Procedures

1.8.1. Principles Underlying the Data Adjustment Procedure

1.8.2. Principles Underlying the Data Assimilation Procedure

1.9. Discussion and Conclusions

CHAPTER 2: Application of the 2nd-BERRU-PM Methodology to the PERP Reactor Physics Benchmark

2.1. Introduction

2.2. Mathematical Modeling of the OECD/NEA Polyethylene-Reflected Plutonium Metal Sphere (PERP) Reactor Physics Benchmark

2.3.1. “High precision” parameters; uniform relative standard deviations

2.3.2. “Typical precision” parameters; uniform relative standard deviations

2.3.3. “Low precision” parameters; uniform relative standard deviations

2.4: Illustrative Application of the 2nd-BERRU-PM Methodology to the PERP Benchmark: Mathematical Expressions for the Best Estimate Predicted Mean and Variance for the PERP Leakage Response

2.4.1. Best-Estimate Predicted Mean Value, ,  for the PERP Leakage Response

2.4.2. Best-Estimate Predicted Standard Deviation for PERP Leakage Response

2.5: Typical-Precision Consistent Measured Response (neutrons/sec; )

2.5.1. High-precision (3% relative standard deviations) parameters

2.5.2: Typical precision (5% relative standard deviations) parameters

2.5.3. Low precision (10% relative standard deviations) parameters

2.6: Low-Precision Consistent Measured Response (neutrons/sec; ); High Precision Parameters (relative SD=3%)

2.7: Typical-Precision Inconsistent Measured Response ( neutrons/sec;

)

2.7.1. High-precision (2% relative standard deviations) parameters

2.7.2. Typical-precision (5% relative standard deviations) parameters

2.7.3. Low-precision (10% relative standard deviations) parameters

2.8: High-Precision Apparently Inconsistent Measured Response ( neutrons/sec; ) and High Precision Parameters (SD=3%)

2.8.1. Including Only Contributions from the 1st -Order Sensitivities of the Leakage Response to the Total Cross Sections

2.8.2. Including Contributions from the 1st + 2nd -Order Sensitivities of the Leakage Response to the Total Cross Sections

2.8.3. Including Contributions from the 1st + 2nd + 3rd-Order Sensitivities of the Leakage Response to the Total Cross Sections

2.8.4. Including Contributions from the 1st + 2nd + 3rd + 4th -Order Sensitivities of the Leakage Response to the Total Cross Sections

2.9: High-Precision Possibly Inconsistent Measured Response ( neutrons/sec; ) and Low Precision Parameters (SD=10%)

2.9.1. Including Only Contributions from the 1st -Order Sensitivities of the Leakage Response to the Total Cross Sections

2.9.2. Including Contributions from the 1st + 2nd -Order Sensitivities of the Leakage Response to the Total Cross Sections

2.10: Low-Precision Apparently Inconsistent Measured Response ( neutrons/sec; ); Typical Precision Parameters (SD=5%)

2.10.1. Including Only Contributions from the 1st -Order Sensitivities of the Leakage Response to All Important Parameters

2.10.2. Including Contributions from the 1st + 2nd -Order Sensitivities of the Leakage Response to All Important Parameters

2.10.3. Including Contributions from the 1st + 2nd + 3rd-Order Sensitivities of the Leakage Response to the Total Cross Sections

2.10.4. Including Contributions from the 1st + 2nd + 3rd + 4th -Order Sensitivities of the Leakage Response to the Total Cross Sections

2.11: Measured Response Value Coincides with Nominally Computed Response Value

2.12. Concluding Remarks

CHAPTER 3: A Novel Generic Fourth-Order Moment-Constrained Maximum Entropy Distribution

3.1. Introduction

3.2. Construction of the Fourth-Order Moment-Constrained Maximum Entropy (MaxEnt) Representation of Uncertain Multivariate Quantities

3.3. Concluding Remarks

Appendix 3.A. Auxiliary Computations for Constructing the Moment-Constrained Fourth-Order MaxEnt Distribution

Appendix 3.B. Approximations Inherent to the Fourth-Order Maximum Entropy Distribution

CHAPTER 4: 4th-BERRU-PM: Fourth-Order Maxent Predictive Modeling Methodology for Combining Measurements with Computations to Obtain Best-Estimate Results with Reduced Predicted Uncertainties

4.1. Introduction

4.2. Construction of the Moments-Constrained Fourth-Order MaxEnt Distribution of the Computational Model Parameters and Responses

4.3. Mathematical Framework of the 4th-BERRU-PM Methodology for Obtaining Best Estimate Results with reduced Uncertainties

4.3.1. Best-Estimate Fourth-Order Expression of the Vector of Mean Values of the Predicted Responses

4.3.2. Best-Estimate Fourth-Order Expression of the Vector of Mean Values of the Predicted Calibrated Model Parameters

4.3.3. Best-Estimate Fourth-Order Expression of the Covariance Matrix of the Predicted Responses

4.3.4. Best-Estimate Fourth-Order Expression of the Covariance Matrix of the Calibrated Model Parameters

4.3.5. Best-Estimate Fourth-Order Expression of the Correlation Matrix of the Predicted Responses and Calibrated Model Parameters

4.3.6. Best-Estimate Fourth-Order Expression of the Triple Correlations among Predicted Responses and Calibrated Model Parameters

4.3.7. Best-Estimate Fourth-Order Expression of the Quadruple Correlations among Predicted Responses and Calibrated Model Parameters

4.3.8. Indicator of Consistency among Parameters and Responses

4.4. Conclusions

Appendix 4.A. Auxiliary Computations for Constructing the Fourth-Order Maximum Entropy Distribution for Model Parameters and Responses

5.1. Introduction

5.2. 4th-BERRU-PM Predicted Best-Estimate Posterior Mean and Variance for the PERP Leakage Response

5.2.1. Low Precision (Standard Deviation=10%) Measurement of the Leakage Response

5.2.1.a. “High precision” parameters (relative standard deviations )

5.2.1.b. “Typical precision” parameters (relative standard deviations )

5.2.1.c. “Low precision” parameters (standard deviations )

5.2.2. Typical Precision (Standard Deviation=5%) Measurement of the Leakage Response

5.2.2.a. “High” precision parameters (relative standard deviations )

5.2.2.b. “Typical” precision parameters (relative standard deviations )

5.3. 4th-BERRU-PM Best-Estimate Posterior Mean Values of Calibrated Model Parameters

5.3.1. “High” precision parameters (relative standard deviations )

5.3.2. “Typical” precision parameters (relative standard deviations )

5.4. 4th-BERRU-PM Best-Estimate Posterior Correlations Between Predicted Responses and Calibrated Model Parameters

5.4.1. “High” precision parameters (relative standard deviations )

5.4.2. “Typical precision” parameters (relative standard deviations )

5.5. 4th-BERRU-PM Best-Estimate Posterior Covariance Matrix of Calibrated Model Parameters

5.5.1. “High” precision parameters (relative standard deviations )

5.5.2. “Typical” precision parameters (relative standard deviations )

5.5.3. “Low” precision parameters (relative standard deviations ); low precision measurement ()

5.6. 4th-BERRU-PM Best-Estimate Posterior Skewness of Predicted Responses and Calibrated Model Parameters

5.7. 4th-BERRU-PM Best-Estimate Posterior Kurtosis of Predicted Responses and Calibrated Model Parameters

5.8. Concluding Remarks

CHAPTER 6: Fourth-Order Comprehensive Adjoint Sensitivity Analysis Methodology: Mathematical Framework

6.1. Introduction

6.2. First-Order Comprehensive Adjoint Sensitivity Analysis Methodology

6.3. Second-Order Comprehensive Adjoint Sensitivity Analysis Methodology

6.4. Third-Order Comprehensive Adjoint Sensitivity Analysis Methodology

6.5. Fourth-Order Comprehensive Adjoint Sensitivity Analysis Methodology

6.6. Concluding Remarks

CHAPTER 7: Polyethylene Reflected Plutonium (PERP) Reactor Physics Benchmark: Sensitivities of the Neutron Leakage Response to Total Cross Sections

7.1. Introduction

7.2. Computation of First-Order Sensitivities

7.3. Computation of Second-Order Sensitivities

7.4. Computation of Third-Order Sensitivities

7.5. Computation of Fourth-Order Sensitivities

7.6 Computational Considerations

7.7. Concluding Remarks

### Biography

Dan Gabriel Cacuci is a Distinguished Professor Emeritus in the Department of Mechanical Engineering at the University of South Carolina and the Karlsruhe Institute of Technology, Germany. He received his Ph.D. in Applied Physics, Mechanical and Nuclear Engineering from Columbia University. He is also the recipient of many awards including four honorary doctorates, the Ernest Orlando Lawrence Memorial award from the U.S. Dept. of Energy and the Arthur Holly Compton, Eugene P. Wigner and the Glenn Seaborg Awards from the American Nuclear Society. He was named an “Inaugural Highly Ranked Scholar” by Scholar GPS, being ranked #2 in the world in the field of Uncertainty Analysis, #5 in the world in the field of Sensitivity Analysis, and ranked in the top 0.05% of all scholars worldwide.

This is Dr. Cacuci’s fifth book for CRC Press. The others include, The Second-Order Adjoint Sensitivity Analysis Methodology (2018); Computational Methods for Data Evaluation and Assimilation with Ionel Michael NavonMihaela Ionescu-Bujor (2013); Sensitivity and Uncertainty Analysis, Volume I Applications to Large-Scale Systems (2003) and Volume II (2005) also with Mihaela Ionescu-BujorIonel Michael Navon.