The nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology, Volume II: Overcoming the Curse of Dimensionality: Large-Scale Application

✍ Scribed by Dan Gabriel Cacuci, Ruixian Fang

• Publisher: Springer
• Year: 2023
• Tongue: English
• Leaves: 474
• Category: Library

✦ Synopsis

This text describes a comprehensive adjoint sensitivity analysis methodology (nth-CASAM), developed by the author, which enablesthe efficient and exact computation of arbitrarily high-order functional derivatives of model responses to model parameters in large-scale systems. The nth-CASAM framework is set in linearly increasing Hilbert spaces, each of state-function-dimensionality, as opposed to exponentially increasing parameter-dimensional spaces, thereby overcoming the so-called “curse of dimensionality” in sensitivity and uncertainty analysis. The nth-CASAM is applicable to any model; the larger the number of model parameters, the more efficient the nth-CASAM becomes for computing arbitrarily high-order response sensitivities. The book will be helpful to those working in the fields of sensitivity analysis, uncertainty quantification, model validation, optimization, data assimilation, model calibration, sensor fusion, reduced-order modelling, inverse problems and predictive modelling.

This Volume Two, the second of three, presents the large-scale application of the nth-CASAM to perform a representative fourth-order sensitivity analysis of the Polyethylene-Reflected Plutonium benchmark described in the Nuclear Energy Agency (NEA) International Criticality Safety Benchmark Evaluation Project (ICSBEP) Handbook. This benchmark is modeled mathematically by the Boltzmann particle transport equation, involving 21,976 imprecisely-known parameters, the numerical solution of which requires representative large-scale computations. The sensitivity analysis presented in this volume is the most comprehensive ever performed in the field of reactor physics and the results presented in this book prove, perhaps counter-intuitively, that many of the 4th-order sensitivities are much larger than the corresponding 3rd-order ones, which are, in turn, much larger than the 2nd-order ones, all of which are much larger than the 1st-order sensitivities. Currently, the nth-CASAM is the only known methodology which enables such large-scale computations of exactly obtained expressions of arbitrarily-high-order response sensitivities.

✦ Table of Contents

Preface
Contents
Chapter 1: First-Order Adjoint Sensitivity and Uncertainty Analysis of the OECD/NEA PERP Reactor Physics Benchmark
1.1 Introduction
1.2 Mathematical Modeling of the OECD/NEA PERP Reactor Physics Benchmark
1.3 First-Order Sensitivity Analysis of the PERP Benchmark´s Total Leakage Response
1.3.1 Microscopic Total Cross Sections: L(α)/σt
1.3.2 Microscopic Scattering Cross Sections: L(α)/σs
1.3.2.1 First-Order Sensitivities , j=1,,Jσs,l=0
1.3.2.2 First-Order Sensitivities
1.3.3 Microscopic Fission Cross Sections: L(α)/σf
1.3.4 Average Number of Neutrons per Fission: L(α)/ν
1.3.5 Fission Spectrum Parameters: L(α)/p
1.3.6 Spontaneous Source Parameters: L(α)/q
1.3.7 Isotopic Number Densities: L(α)/N
1.3.8 First-Order Sensitivities: Numerical Results
1.4 Timing Comparisons and Chapter Summary: Adjoint Sensitivity Analysis Versus Finite Differences for Computing First-Order S...
Chapter 2: Second-Order Analysis: Effects of Total Cross Sections
2.1 Introduction
2.2 Derivation of the Second-Order Sensitivities 2L(α)/tα
2.3 Second-Order Sensitivities of the PERP Leakage Response with Respect to the Total Cross Sections
2.3.1 Second-Order Unmixed Relative Sensitivities of the Leakage Response with Respect to the Microscopic Total Cross Sections...
2.3.2 Second-Order Mixed Relative Sensitivities of the Leakage Response with Respect to the Microscopic Total Cross Sections o...
2.3.3 Second-Order Mixed Relative Sensitivities of the Leakage Response with Respect to the Microscopic Total Cross Sections o...
2.3.4 Second-Order Mixed Relative Sensitivities of the Leakage Response with Respect to the Microscopic Total Cross Sections o...
2.3.5 Second-Order Mixed Relative Sensitivities of the Leakage Response with Respect to the Microscopic Total Cross Sections o...
2.3.6 Second-Order Mixed Relative Sensitivities of the Leakage Response with Respect to the Microscopic Total Cross Sections o...
2.3.7 Second-Order Mixed Relative Sensitivities of the Leakage Response with Respect to the Microscopic Total Cross Sections o...
2.3.8 Second-Order Mixed Relative Sensitivities of the Leakage Response with Respect to the Microscopic Total Cross Sections o...
2.4 Second-Order Uncertainty Analysis of the PERP Leakage Response
2.4.1 Uncorrelated Total Microscopic Cross Sections
2.4.2 Fully Correlated Total Microscopic Cross Sections
2.4.3 Numerical Results
2.4.3.1 Very Small (1%) Relative Standard Deviations
2.4.3.2 Typical (5%) Relative Standard Deviations
2.4.3.3 Large (10%) Relative Standard Deviations
2.5 Chapter Summary
Chapter 3: Second-Order Analysis: Effects of Scattering Cross Sections
3.1 Introduction
3.2 Derivation of the Second-Order Sensitivities 2L(α)/sα
3.3 Computation of Second-Order Sensitivities of the PERP Leakage Response with Respect to the Scattering Cross Sections
3.3.1 Second-Order Sensitivities 2L(α)/σsσs
3.3.1.1 Second-Order Sensitivities ; β = 1, , Jσs, l = 0
3.3.1.2 Second-Order Sensitivities , j = 1, , Jσs, l = 0, β = 1, , Jσs, l1
3.3.1.3 Second-Order Sensitivities , j = 1, , Jσs, l 1, β = 1, , Jσs, l = 0
3.3.1.4 Second-Order Sensitivities , j = 1, , Jσs, l 1, β = 1, , Jσs, l 1
3.3.2 Numerical Results for 2L/σsσs
3.4 Mixed Second-Order Sensitivities of the PERP Leakage Response with Respect to the Scattering and Total Cross Sections
3.4.1 Second-Order Sensitivities 2L(α)/σsσt
3.4.1.1 Second-Order Sensitivities , j = 1, , Jσs, l = 0; β = 1, , Jσt
3.4.1.2 Second-Order Sensitivities , j = 1, , Jσs, l 1; β = 1, , Jσt
3.4.2 Alternative Path: Computing the Second-Order Sensitivities 2L(α)/σtσs
3.4.2.1 Second-Order Sensitivities , j = 1, , Jσt; β = 1, , Jσs, l = 0
3.4.2.2 Second-Order Sensitivities , j = 1, , Jσt; β = 1, , Jσs, l 1
3.4.3 Numerical Results for 2L(α)/σtσs
3.4.3.1 Results for the Relative Sensitivities
3.4.3.2 Results for the Relative Sensitivities
3.4.3.3 Results for the Relative Sensitivities
3.4.3.4 Results for the Relative Sensitivities
3.5 Uncertainties in the PERP Leakage Response Induced by Uncertainties in Scattering Cross Sections
3.6 Chapter Summary
Chapter 4: Second-Order Analysis: Effects of Fission Cross Sections
4.1 Introduction
4.2 Derivation of the Second-Order Sensitivities 2L(α)/fα
4.3 Computation of Second-Order Sensitivities of the PERP Leakage Response with Respect to Fission Cross Sections
4.3.1 Second-Order Sensitivities 2L(α)/σfσf
4.3.2 Numerical Results for 2L(α)/σfσf
4.3.2.1 Second-Order Unmixed Relative Sensitivities S(2) g = 1, , 30
4.3.2.2 Second-Order Relative Sensitivities
4.4 Mixed Second-Order Sensitivities of the PERP Leakage Response with Respect to Fission and Total Cross Sections
4.4.1 Computing the Second-Order Sensitivities 2L(α)/σfσt
4.4.2 Alternative Path: Computing the Second-Order Sensitivities 2L(α)/σtσf
4.4.3 Numerical Results for 2L(α)/σfσt
4.4.3.1 Second-Order Relative Sensitivities
4.4.3.2 Second-Order Relative Sensitivities
4.4.3.3 Second-Order Relative Sensitivities
4.5 Mixed Second-Order Sensitivities of the PERP Leakage Response with Respect to Fission and Scattering Cross Sections
4.5.1 Computing the Second-Order Sensitivities 2L(α)/σfσs
4.5.1.1 Second-Order Sensitivities
4.5.1.2 Second-Order Sensitivities
4.5.2 Alternative Path: Computing the Second-Order Sensitivities 2L(α)/σsσf
4.5.2.1 Second-Order Sensitivities
4.5.2.2 Second-Order Sensitivities
4.5.3 Numerical Results for 2L(α)/σfσs
4.5.3.1 Results for the Relative Sensitivities
4.5.3.2 Results for the Relative Sensitivities
4.5.3.3 Results for the Relative Sensitivities
4.5.3.4 Results for the Relative Sensitivities
4.6 Quantification of Uncertainties in the PERP Leakage Response Due to Uncertainties in Fission Cross Sections
4.7 Chapter Summary
Chapter 5: Second-Order Analysis: Effects of Average Number of Neutrons Per Fission
5.1 Introduction
5.2 Computation of the Second-Order Sensitivities of the PERP Leakage Response with Respect to the Average Number of Neutrons ...
5.2.1 Second-Order Sensitivities 2L(α)/νν
5.2.2 Numerical Results for 2L(α)/νν
5.2.2.1 Second-Order Unmixed Relative Sensitivities g = 1, , 30
5.2.2.2 Second-Order Relative Sensitivities
5.3 Mixed Second-Order Sensitivities of the PERP Total Leakage Response with Respect to the Average Number of Neutrons Per Fis...
5.3.1 Second-Order Sensitivities 2L(α)/νσt
5.3.2 Alternative Path: Computing the Second-Order Sensitivities 2L(α)/σtν
5.3.3 Numerical Results for 2L(α)/νσt
5.3.3.1 Second-Order Relative Sensitivities
5.3.3.2 Second-Order Relative Sensitivities
5.3.3.3 Second-Order Relative Sensitivities
5.3.3.4 Second-Order Relative Sensitivities
5.4 Mixed Second-Order Sensitivities of the PERP Leakage Response with Respect to the Average Number of Neutrons Per Fission a...
5.4.1 Computation of the Second-Order Sensitivities 2L(α)/νσs
5.4.1.1 Computation of the Second-Order Sensitivities
5.4.1.2 Second-Order Sensitivities
5.4.2 Alternative Path: Computing the Second-Order Sensitivities 2L(α)/σsν
5.4.2.1 Second-Order Sensitivities
5.4.2.2 Second-Order Sensitivities
5.4.3 Numerical Results for 2L(α)/νσs
5.4.3.1 Results for the Relative Sensitivities
5.4.3.2 Results for the Relative Sensitivities
5.4.3.3 Results for the Relative Sensitivities
5.4.3.4 Results for the Relative Sensitivities
5.5 Mixed Second-Order Sensitivities of the PERP Leakage Response with Respect to the Average Number of Neutrons Per Fission a...
5.5.1 Computing the Second-Order Sensitivities 2L(α)/νσf
5.5.2 Alternative Path: Computing the Second-Order Sensitivities 2L(α)/σfν
5.5.3 Numerical Results for 2L(α)/νσf
5.6 Uncertainties in the PERP Leakage Response Stemming from Uncertainties in the Average Number of Neutrons Per Fission
5.7 Chapter Summary
Chapter 6: Second-Order Analysis: Effects of Source Parameters
6.1 Introduction
6.2 Derivation of the Second-Order Sensitivities 2L(α)/qα
6.3 Computation of Second-Order Sensitivities of the PERP Leakage Response with Respect to the Source Parameters
6.3.1 Computing 2L(α)/qq
6.3.2 Numerical Results for 2L(α)/qq
6.4 Mixed Second-Order Sensitivities of the PERP Leakage Response with Respect to the Source Parameters and Total Cross Sectio...
6.4.1 Computing 2L(α)/qσt
6.4.2 Alternative Path: Computing 2L(α)/σtq
6.4.3 Numerical Results for 2L(α)/qσt
6.5 Mixed Second-Order Sensitivities of the PERP Leakage Response with Respect to the Source Parameters and Scattering Cross S...
6.5.1 Computing 2L(α)/qσs
6.5.1.1 Second-Order Sensitivities β = 1, , Jσs, l = 0
6.5.1.2 Second-Order Sensitivities β = 1, , Jσs, l 1
6.5.2 Alternative Path: Computing 2L(α)/σsq
6.5.2.1 Second-Order Sensitivities β = 1, , Jq
6.5.2.2 Second-Order Sensitivities β = 1, , Jq
6.5.3 Numerical Results for 2L(α)/qσs
6.6 Mixed Second-Order Sensitivities of the PERP Leakage Response with Respect to the Source Parameters and Fission Cross Sect...
6.6.1 Computing the Second-Order Sensitivities 2L(α)/qσf
6.6.2 Alternative Path: Computing the Second-Order Sensitivities 2L(α)/σfq
6.6.3 Numerical Results for 2L(α)/qσf
6.7 Mixed Second-Order Sensitivities of the PERP Leakage Response with Respect to the Source Parameters and the Average Number...
6.7.1 Computing the Second-Order Sensitivities 2L(α)/qν
6.7.2 Alternative Path: Computing the Second-Order Sensitivities 2L(α)/νq
6.7.3 Numerical Results for 2L(α)/qν
6.8 Mixed Second-Order Sensitivities of the PERP Leakage Response with Respect to the Source Parameters and Isotopic Number De...
6.8.1 Computing the Second-Order Sensitivities 2L(α)/qN
6.8.2 Alternative Path: Computing the Second-Order Sensitivities 2L(α)/Nq
6.8.3 Numerical Results for 2L(α)/qN
6.9 Quantification of Uncertainties in the PERP Leakage Response Due to Uncertainties in Source Parameters
6.10 Chapter Summary
Chapter 7: Second-Order Analysis: Effects of Isotopic Number Densities
7.1 Introduction
7.2 Computation of Second-Order Sensitivities 2L(α)/NN
7.3 Computation of Second-Order Sensitivities of the PERP Leakage Response with Respect to the Isotopic Number Densities and T...
7.3.1 Computing the Second-Order Sensitivities 2L(α)/Nσt
7.3.2 Alternative Path: Computing the Second-Order Sensitivities 2L(α)/σtN
7.3.3 Numerical Results for 2L(α)/Nσt
7.4 Computation of Second-Order Sensitivities of the PERP Leakage Response with Respect to the Isotopic Number Densities and S...
7.4.1 Computing the Second-Order Sensitivities 2L(α)/Nσs
7.4.1.1 Second-Order Sensitivities
7.4.1.2 Second-Order Sensitivities , j = 1, , Jn; β = 1, , Jσs, l 1
7.4.2 Alternative Path: Computing the Second-Order Sensitivities 2L(α)/σsN
7.4.2.1 Second-Order Sensitivities j = 1,,Jσs,l=0; β = 1, , Jn
7.4.2.2 Second-Order Sensitivities β = 1, , Jn
7.4.3 Numerical Results for 2L(α)/Nσs
7.5 Computation of Second-Order Sensitivities of the PERP Leakage Response with Respect to the Isotopic Number Densities and F...
7.5.1 Computing the Second-Order Sensitivities 2L(α)/Nσf
7.5.2 Alternative Path: Computing the Second-Order Sensitivities 2L(α)/σfN
7.5.3 Numerical Results for 2L(α)/Nσf
7.6 Computation of Second-Order Sensitivities of the PERP Leakage Response with Respect to the Isotopic Number Densities and t...
7.6.1 Computing the Second-Order Sensitivities 2L(α)/Nν
7.6.2 Alternative Path: Computing the Second-Order Sensitivities 2L(α)/νN
7.6.3 Numerical Results for 2L(α)/Nν
7.7 Quantification of Uncertainties in the PERP Leakage Response Due to Uncertainties in Isotopic Number Densities
7.8 Chapter Summary
Chapter 8: Third- and Fourth-Order Adjoint Sensitivity and Uncertainty Analysis of the PERP Benchmark
8.1 Introduction
8.2 First-Order Through Third-Order Sensitivities of the PERP Leakage Response with Respect to the Total Cross Sections
8.2.1 First-Order Sensitivities L(α)/tj
8.2.2 Second-Order Sensitivities
8.2.3 Third-Order Sensitivities : Mathematical Framework and Computational Considerations
8.2.3.1 Mathematical Framework
8.2.3.2 Computational Considerations
8.2.4 Computational Results for the Largest Mixed Third-Order Sensitivities
8.2.4.1 Third-Order Mixed Relative Sensitivities
8.2.4.2 Third-Order Mixed Relative Sensitivities
8.2.4.3 Third-Order Mixed Relative Sensitivities
8.2.4.4 Third-Order Mixed Relative Sensitivities
8.2.4.5 Third-Order Mixed Relative Sensitivities
8.2.4.6 Third-Order Mixed Sensitivities: Summary
8.3 Fourth-Order Sensitivities of the PERP Leakage Response with Respect to the Total Cross Sections
8.3.1 Mathematical Framework
8.3.2 Computational Considerations
8.3.3 Numerical Results for the First- Through Fourth-Order Unmixed Sensitivities
8.3.3.1 4th-CASAM-L Computation of the Unmixed Sensitivities
8.3.3.2 Illustrative Finite Difference Computations of Fourth-Order Unmixed Sensitivities
8.3.4 Computation of the Largest Fourth-Order Mixed Sensitivities
8.3.4.1 4th-CASAM-L Computation of the Mixed Sensitivities , i = 1, , 6; g = 1, , 30
8.3.4.2 Finite Difference Computations of the Largest Fourth-Order Mixed Sensitivities
8.4 Fourth-Order Uncertainty Analysis
8.4.1 Effects of the Fourth-Order Sensitivities on the Response Expectation
8.4.2 Effects of the Fourth-Order Sensitivities on the Response´s Variance
8.4.3 Effects of the Fourth-Order Sensitivities on the Third-Order Response Moment and Skewness
8.5 Chapter Summary
Chapter 9: Concluding Remarks
Nomenclature
References
Index