Advances in High-Order Predictive Modeling : Methodologies and Illustrative Problems (Hardcover)

CHAPTER 1: 2^nd-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????????????????????????????????????????????????

1.5. 2^nd-BERRU-PMD: Second Order MaxEnt Predictive Modeling Methodology with Deterministically Included Computed Responses ?????????????????????????????????????????????????????????????????????

1.6. 2^nd-BERRU-PMP: Second-Order MaxEnt Predictive Modeling Methodology with Probabilistically Included Computed Responses ?????????????????????????????????????? ????????????????

1.6.1. Second-Order MaxEnt Probabilistic Representation of the Computational Model ??????????????????????????????????????????????????????????????????????????????????????? ????????????????????????????????????????????????????

1.6.2. General Case 2^nd-BERRU-PMP: Inclusion of Additional External Measurements for Both Responses and Parameters ????????????????????????????????????????? ????????????????????????????????????????

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

1.7. Inter-Comparison: 2^nd-BERRU-PMP vs. 2^nd-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 ??????????????????????????????????????????????????????????????????????? ????????????????

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CHAPTER 2: Application of the 2^nd-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: Mean and Variance of the PERP Benchmark’s Computed Leakage Response ????????????????

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 2^nd-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 1^st -Order Sensitivities of the Leakage Response to the Total Cross Sections??????????????????????????????????????????????????????????????????????????

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

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

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

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

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

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

2.10.4. Including Contributions from the 1^st + 2^nd + 3^rd + 4^th -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????????????????????????????????????????????????????????????????????????????????? ????????????????

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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 ??????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? ?????????????????????????????????????

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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 4^th-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???????????? ????????????????????????????????????????

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CHAPTER 5 :Application of the 4^th-BERRU-PM Methodology to the PERP Reactor Physics Benchmark???????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????

5.1. Introduction???????????????????????????????????????????????????????????????????????????????????????????????? ????

5.2. 4^th-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. 4^th-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. 4^th-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. 4^th-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. 4^th-BERRU-PM Best-Estimate Posterior Skewness of Predicted Responses and Calibrated Model Parameters ???????????????????????????????????????????????????????????????????????????? ????????????????????????????

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

5.8. Concluding Remarks??????????????????????????????????????????????????????????????????????????????????? ????????????????

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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 ?????????????????????????????????????????????????????????????????????????????????? ????????????????

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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 ?????????????????????????????????????????????????????????????????????????????????????????????????????