Fourth-Order Adjoint Sensitivity and Uncertainty Analysis of an OECD/NEA Reactor Physics Benchmark: II. Computed Response Uncertainties
Abstract
:1. Introduction
2. Expressions Used in the Fourth-Order Uncertainty Analysis of the PERP Leakage Response
- The expected value of a model parameter , denoted as , is defined as follows:
- 2.
- The covariance, , of two parameters, and , is defined as follows:
- 3.
- The third-order moment (skewness), , and the associated third-order correlation, , among three parameters are defined as follows:
- 4.
- The fourth-order moment (kurtosis), , and the associated fourth-order correlation, , among four parameters, are defined as follows:
- 5.
- The fifth-order moment, , and the associated fifth-order correlation, , among five parameters are defined as follows:
- 6.
- The sixth-order parameter moment, , and the associated sixth-order correlation, , among six parameters are defined as follows:
3. Numerical Results for Fourth-Order Uncertainty Analysis of the PERP Leakage Response
3.1. The Effects of the Fourth-Order Sensitivities on the Response Expectation
3.2. The Effects of the Fourth-Order Sensitivities on the Response’s Variance
- (i)
- the leakage response of the PERP benchmark;
- (ii)
- the expected value of the leakage response;
- (iii)
- the standard deviation, , for the leakage response arising solely from the 1st-order sensitivities;
- (iv)
- the standard deviation, , for the leakage response arising solely from the 2nd-order sensitivities;
- (v)
- the standard deviation, , for the leakage response stemming solely from the 3rd-order sensitivities; and
- (vi)
- the standard deviation, , for the leakage response stemming solely from the 4th-order sensitivities.
3.3. The Effects of the Fourth-Order Sensitivities on the 3rd-Order Response Moment and Skewness
4. Conclusions
- (1)
- The impact of the 4th-order sensitivities on the expected value of the leakage response varies with the value of the standard deviation of the uncorrelated microscopic total cross sections. Generally, the larger the standard deviations of the microscopic total cross sections, the higher the impact of the 4th-order sensitivities will be on the expected value. For a small relative standard deviation of 1% for the parameters under consideration, the impact of the 4th-order sensitivities on the expected response value is smaller than the impact of the lower-order sensitivities. However, for a moderate relative standard deviation of 5%, the contributions from the 4th-order sensitivities are around 56% of the expected value. When the relative standard deviation is increased to 10%, the contributions from the 4th-order sensitivities to the expected value increase to nearly 90%. Notably, for the “RSD = 10%” case, neglecting the 4th-order sensitivities would cause a large error (ca. 3400%) if the computed value were considered to be the actual expected value of the leakage response.
- (2)
- The effects of the 4th-order sensitivities on the variance of the leakage response also depends on the value of the standard deviation considered for the microscopic total cross sections. Specifically, if the microscopic total cross sections have small relative standard deviations (e.g., RSD = 1%), the 4th-order sensitivities would only contribute about 4% to the response variance . For moderate relative standard deviations of 5%, the contributions from the 4th-order sensitivities increase to 52%. For large relative standard deviations of 10%, the contributions from the 4th-order sensitivities to the response variance amounts to 59%, which is significantly larger than the contributions from the corresponding 1st-, 2nd- and 3rd-order sensitivities.
- (3)
- The impact of the 4th-order sensitivities on the standard deviation of the leakage response is as follows: (i) for small relative standard deviations (e.g., 1%) of the microscopic total cross sections, the uncertainty of the leakage response arising solely from the 1st-order sensitivities are significantly larger than the uncertainties arising solely from the 2nd-, 3rd- and 4th-order sensitivities, respectively, but the following oscillating pattern has been observed: ; (ii) when considering moderate and large relative standard deviations (e.g., 5% and 10%, respectively) for the microscopic total cross sections, the standard deviations of the leakage response appear to diverge as the order of sensitivity increases, i.e., .
- (4)
- 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. The impact of the 4th-order sensitivities on the skewness of the leakage response also changes with the value of the standard deviation of the microscopic total cross sections: larger parameter standard deviations tend to decrease the value of the skewness, causing the leakage response distribution to become more symmetrical about the mean value .
- (5)
- It was found that, among the 180 microscopic total cross sections, the parameter namely, the 30th group (i.e., the lowest energy group) of the total cross sections of isotope #6 (H) contained in the PERP benchmark, is the single most important parameter affecting the PERP benchmark’s leakage response, as it has the largest impact on the various response moments. For example, considering that all of the microscopic total cross sections are uncorrelated and have a 5% relative standard deviation, the 5% relative standard deviation of contributes around 99.8% to the expected value , 99.97% to the variance and 99.99% to the skewness .
- (6)
- Because the correlations among the group-averaged microscopic total cross sections are not available for the PERP benchmark under consideration, it was not possible to quantify in this work the impact of the mixed 4th-order sensitivities. As discussed in [2], correlations among the microscopic total cross sections would provide additional contributions to the various response moments (e.g., expected value, variance and skewness).
- (7)
- While the general mathematical expressions presented in this work can be applied to any nuclear reactor system, the numerical results reported in this work are evidently specific to the PERP benchmark.
- (8)
- The general methodology underlying the specific computations of the 4th-order sensitivities used in this work has been presented in [5]. This 4th-order methodology has been recently generalized [6,7] to enable the most efficient computation of exactly obtained mathematical expressions of arbitrarily-high-order (nth-order) sensitivities of a generic system response with respect to all of the parameters (including uncertain domain boundaries) underlying the respective forward/adjoint systems. The mathematical framework underlying this arbitrarily-high order methodology, called the “nth-CASAM-L” methodology, is developed in linearly increasing higher-dimensional Hilbert spaces, as opposed to the exponentially increasing “parameter-dimensional” spaces in which response sensitivities are computed by other methods, thus providing the basis for overcoming the “curse of dimensionality” in sensitivity analysis and all other fields (uncertainty quantification, predictive modeling, etc.) which need such sensitivities. Thus, for the response of a model which comprises a total number of -parameters—and hence admits first-order sensitivities—the 1st-CASAM-L requires 1 additional large-scale adjoint computation (as opposed to at least large-scale computations, as required by other methods) for computing exactly all of the 1st-order response sensitivities. All of the second-order response sensitivities are computed exactly by the 2nd-CASAM-L in at most computations, as opposed to at least computations required by finite-difference and/or other methods, and so on. For every lower-order sensitivity of interest, the nth-CASAM-L computes the “next-higher-order” sensitivities in one adjoint computation performed in a linearly increasing higher-dimensional Hilbert space, thus providing a leap forward in the quest to overcome the “curse of dimensionality” in sensitivity analysis, uncertainty quantification and predictive modeling.
- (9)
- The need for computing higher-order (i.e., higher than first-order) sensitivities (functional derivatives) of model responses with respect to the model parameters has been underscored in [8,9]. Using an analytically solvable model of neutron scattering in a hydrogenous medium for which all of the response’s relative sensitivities had the same absolute value of unity, it was shown in [9] that the wider the distribution of model parameters, the higher the order of sensitivities needed to achieve a desired level of accuracy in representing the response and in computing the response’s expectation, variance, skewness and kurtosis. If only first-order sensitivities are considered, the third-order moment of the response is always zero. Hence, a “first-order sensitivity and uncertainty quantification” will always produce an erroneous third moment (and, hence, skewness) of the predicted response distribution, unless the unknown response distribution happens to be symmetrical. At least second-order sensitivities must be used in order to estimate the third-order moment (and, hence, the skewness) of the response distribution. With pronounced skewness, standard statistical inference procedures such as constructing a confidence interval for the mean (expectation) of a computed/predicted model response will be not only incorrect, in the sense that the true coverage level will differ from the nominal (e.g., 95%) level, but the error probabilities will be unequal on each side of the predicted mean. Thus, the truncation of Taylor expansion of the response (as a function of parameters) depends both on the magnitudes of the response sensitivities to parameters and the parameter uncertainties involved: if the uncertainties are small, then a 4th-order expansion suffices, in most cases, for obtaining relatively accurate results. In any case, the truncation error of a convergent Taylor-series can be quantified a priori. If the parameter uncertainties are large, the Taylor series may diverge, so one would need to consider asymptotic expansions. Of course, if the parameter uncertainties are large, all statistical methods are doomed to produce unreliable results for large-scale, realistic problems, involving many uncertain parameters.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Appendix A. General Expressions for the Expectation, Variance, and Third-Order Response Moment, up to Fourth-Order Response Sensitivities
Appendix A.1. General Expression for the Expectation
Appendix A.2. General Expressions for the Response Variance
Appendix A.3. General Expressions for the Third-Order Response Moment
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Expected Value | RSD = 1% | RSD = 5% | RSD = 10% |
---|---|---|---|
1.765 × 106 | 1.765 × 106 | 1.765 × 106 | |
4.598 × 104 | 1.149 × 106 | 4.598 × 106 | |
0.0 | 0.0 | 0.0 | |
6.026 × 103 | 3.766 × 106 | 6.026 × 107 | |
1.817 × 106 | 6.681 × 106 | 6.662 × 107 |
Variances | RSD = 1% | RSD = 5% | RSD = 10% |
---|---|---|---|
3.419 × 1010 | 8.549 × 1011 | 3.419 × 1012 | |
2.879 × 109 | 1.799 × 1012 | 2.879 × 1013 | |
9.841 × 109 | 2.338 × 1013 | 1.236 × 1015 | |
1.825 × 109 | 2.852 × 1013 | 1.825 × 1015 | |
4.874 × 1010 | 5.456 × 1013 | 3.093 × 1015 |
3rd-Order Moment and Skewness | RSD = 1% | RSD = 5% | RSD = 10% |
---|---|---|---|
0 | 0 | 0 | |
6.663 × 1015 | 1.070 × 1019 | 4.982 × 1020 | |
3.948 × 1015 | 6.169 × 1019 | 3.948 × 1021 | |
1.973 × 1015 | 3.083 × 1019 | 1.973 × 1021 | |
1.258 × 1016 | 1.032 × 1020 | 6.419 × 1021 | |
1.169 | 0.256 | 0.037 |
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Fang, R.; Cacuci, D.G. Fourth-Order Adjoint Sensitivity and Uncertainty Analysis of an OECD/NEA Reactor Physics Benchmark: II. Computed Response Uncertainties. J. Nucl. Eng. 2022, 3, 1-16. https://doi.org/10.3390/jne3010001
Fang R, Cacuci DG. Fourth-Order Adjoint Sensitivity and Uncertainty Analysis of an OECD/NEA Reactor Physics Benchmark: II. Computed Response Uncertainties. Journal of Nuclear Engineering. 2022; 3(1):1-16. https://doi.org/10.3390/jne3010001
Chicago/Turabian StyleFang, Ruixian, and Dan Gabriel Cacuci. 2022. "Fourth-Order Adjoint Sensitivity and Uncertainty Analysis of an OECD/NEA Reactor Physics Benchmark: II. Computed Response Uncertainties" Journal of Nuclear Engineering 3, no. 1: 1-16. https://doi.org/10.3390/jne3010001