Fourth-Order Comprehensive Adjoint Sensitivity Analysis (4th-CASAM) of Response-Coupled Linear Forward/Adjoint Systems: I. Theoretical Framework

2022

JNE

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This work quantifies the impact of the most important 4th-order sensitivities of the leakage response of a polyethylene-reflected plutonium (PERP) reactor physics benchmark with respect to the benchmark’s 180 group-averaged microscopic total cross sections, on the expected value, variance and skewness of the benchmark’s leakage response. This work shows that, as the standard deviations of the cross sections increase, the contributions of the 4th-order sensitivities to the response’s expected value and variance become significantly larger than the corresponding contributions stemming from the 1st-, 2nd- and 3rd-order sensitivities. Considering a uniform 5% relative standard deviation for all microscopic total cross sections, the contributions from the 4th-order sensitivities to the expected value and variance of the PERP leakage response amount to 56% and 52%, respectively. Considering 10% uniform relative standard deviations for the microscopic total cross sections, the contributions from the 4th-order sensitivities to the expected value increase to nearly 90%. Consequently, if the computed value L(a) were considered to represent the actual expected value of the leakage response and the 4th-order sensitivities were neglected, the computed value would represent the actual expected value with an error of 3400%. Furthermore, uniform relative standard deviations of 5% and larger (10%) for the microscopic total cross sections cause the higher-order sensitivities to contribute increasingly higher amounts to the response standard deviation: the contributions stemming from the 4th-order sensitivities are larger than the contributions stemming from the 3rd-order sensitivities, which in turn are larger than those stemming from the 2nd-order sensitivities, which are themselves larger than the contributions stemming from the 1st-order sensitivities. This finding evidently underscores the need for computing sensitivities of order higher than first-order. The results obtained in this work also indicate that the 4th-order sensitivities produce a positive response skewness, causing the leakage response distribution to be skewed towards the positive direction from its expected value. Increasing the parameter standard deviations tends to decrease the value of the response skewness, causing the leakage response distribution to become more symmetrical about the mean value. The results presented in this work highlight the finding that the microscopic total cross section for hydrogen (H) in the lowest (“thermal”) energy group is the single most important parameter among the 180 microscopic total cross sections of the PERP benchmark, as it contributes most to the various response moments.

Section: Discussionmentioning

confidence: 99%

Fourth-Order Adjoint Sensitivity and Uncertainty Analysis of an OECD/NEA Reactor Physics Benchmark: II. Computed Response Uncertainties

Fang

2022

JNE

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“…Extending the methodology presented in [5] to enable the exact and efficient computation of third-order (and higher-order) sensitivities for large-scale nonlinear systems is currently in progress. For large-scale (response-coupled) linear forward/adjoint systems, all of the response sensitivities up to and including the fourth-order sensitivities can be computed exactly and most efficiently by applying the methodology (fourth CASAM), recently developed by Cacuci [30].…”

Section: Discussionmentioning

confidence: 99%

On the Need to Determine Accurately the Impact of Higher-Order Sensitivities on Model Sensitivity Analysis, Uncertainty Quantification and Best-Estimate Predictions

2021

Energies

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This work aims at underscoring the need for the accurate quantification of the sensitivities (i.e., functional derivatives) of the results (a.k.a. “responses”) produced by large-scale computational models with respect to the models’ parameters, which are seldom known perfectly in practice. The large impact that can arise from sensitivities of order higher than first has been highlighted by the results of a third-order sensitivity and uncertainty analysis of an OECD/NEA reactor physics benchmark, which will be briefly reviewed in this work to underscore that neglecting the higher-order sensitivities causes substantial errors in predicting the expectation and variance of model responses. The importance of accurately computing the higher-order sensitivities is further highlighted in this work by presenting a text-book analytical example from the field of neutron transport, which impresses the need for the accurate quantification of higher-order response sensitivities by demonstrating that their neglect would lead to substantial errors in predicting the moments (expectation, variance, skewness, kurtosis) of the model response’s distribution in the phase space of model parameters. The incorporation of response sensitivities in methodologies for uncertainty quantification, data adjustment and predictive modeling currently available for nuclear engineering systems is also reviewed. The fundamental conclusion highlighted by this work is that confidence intervals and tolerance limits on results predicted by models that only employ first-order sensitivities are likely to provide a false sense of confidence, unless such models also demonstrate quantitatively that the second- and higher-order sensitivities provide negligibly small contributions to the respective tolerance limits and confidence intervals. The high-order response sensitivities to parameters underlying large-scale models can be computed most accurately and most efficiently by employing the high-order comprehensive adjoint sensitivity analysis methodology, which overcomes the curse of dimensionality that hampers other methods when applied to large-scale models involving many parameters.

“…Section 3.1 illustrates the need for computing high-order sensitivities by reviewing the results obtained in [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30] for an OECD/NEA reactor physics benchmark [25], comprising a subcritical polyethylene-reflected plutonium metal sphere, which is called the "PERP" (polyethylene-reflected plutonium) benchmark. The PERP benchmark is modeled by using the deterministic neutron transport Boltzmann equation, which is solved numerically (after discretization in the energy, spatial, and angular independent variables) using the software package PARTISN [58].…”

Section: High-order Sensitivity and Uncertainty Analysis Of Linear Mo...mentioning

confidence: 99%

“…The findings reported in [19][20][21][22][23][24] motivated the investigation [26,27] of the largest thirdorder sensitivities, many of which were found to be even larger than the second-order ones. This finding, in turn, has motivated the development [28] of the mathematical framework for determining and computing the fourth-order sensitivities for the OECD/NEA benchmark, many of which were found [29,30] to be larger than the third-order ones. The need for computing the third-and fourth-order sensitivities has been underscored in [31,32].…”

Section: Introductionmentioning

confidence: 99%

“…If the negative impact of the curse of dimensionality is not alleviated when computing sensitivities, this negative impact is transmitted to the computation of the aforementioned correlations. Section 3.1 illustrates the need for computing high-order sensitivities by reviewing the results obtained in [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30] for an OECD/NEA reactor physics benchmark [25], which is representative of linear models/systems. These results underscore the efficiency of the n th -CASAM-L in overcoming the curse of dimensionality while producing the respective results.…”

Section: Introductionmentioning

confidence: 99%

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Overview of Arbitrarily High-Order Adjoint Sensitivity and Uncertainty Quantification Methodology for Large-Scale Systems

2022

Energies

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This work reviews from a unified viewpoint the concepts underlying the “nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Response-Coupled Forward/Adjoint Linear Systems” (nth-CASAM-L) and the “nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems” (nth-CASAM-N) methodologies. The practical application of the nth-CASAM-L methodology is illustrated for an OECD/NEA reactor physics benchmark, while the practical application of the nth-CASAM-N methodology is illustrated for a nonlinear model of reactor dynamics that exhibits periodic and chaotic oscillations. As illustrated both by the general theory and by the examples reviewed in this work, both the nth-CASAM-L and nth-CASAM-N methodologies overcome the curse of dimensionality in sensitivity analysis. The availability of efficiently and exactly computed sensitivities of arbitrarily high order can lead to major advances in all areas that need such high-order sensitivities, including data assimilation, model calibration, uncertainty reduction, and predictive modeling.