The accompanying works [1] [2] have computed exactly and efficiently the (180)³ = 5,832,000 third-order sensitivities of the leakage response, of the OECD/NEA subcritical polyethylene-reflected plutonium (acronym: PERP) metal fundamental physics benchmark [3], with respect to the benchmark’s 180 group-averaged microscopic total cross sections. This work presents the third-order uncertainty analysis of the PERP benchmark’s leakage response by using the results obtained in [2]. Specifically, the effects of the 3^rd-order sensitivities on the PERP benchmark’s leakage response distribution moments (e.g., expected values, variance and skewness, etc.) in the space of parameters (microscopic total cross sections) will be quantified and compared with the contributions stemming from the corresponding 1^st- and 2^nd-order sensitivities, underscoring the importance of the 3^rd-order sensitivities.

This work is organized as follows: Section 2 presents the expressions used for the 3^rd-order uncertainty analysis of the PERP leakage response, which is derived from the general expressions for third-order uncertainty analysis conceived by Cacuci [4]. Section 3 presents the numerical results for the contributions of the 3^rd-order sensitivities to the leakage response distribution’s first three moments (i.e., expected value, variance and skewness) in the space of parameters (microscopic total cross section). The 3^rd-order contributions are compared with the corresponding effects stemming from the 1^st- and 2^nd-order sensitivities. Section 4 concludes and highlights the important findings obtained in this work, which is the first third-order uncertainty analysis ever performed in reactor physics.

Knowledge of the first-, second-, and third-order sensitivities is required to compute the following moments of the response distribution:

1) Up to 3^rd-order response sensitivities, the 1^st-order moment (expected value) of a response R k has the following expression [4]:

E ( r k ) = r k ( α 0 ) + 1 2 ∑ i = 1 N α ∑ j = 1 N α ∂ 2 r k ( α 0 ) ∂ α i ∂ α j ρ i j σ i σ j     + 1 6 ∑ i = 1 N α ∑ j = 1 N α ∑ k = 1 N α ∂ 3 r k ( α 0 ) ∂ α i ∂ α j ∂ α k t i j k σ i σ j σ k + O ( σ 4 ) , (1)

where ρ i j denotes the correlation coefficient between parameters α i and α j , while σ i denotes the standard deviation of the model parameter α i .

2) Up to 3^rd-order response sensitivities, the 2^nd-order moment (covariance) of two responses ( R k , R l ) has the following expression [4]:

cov ( R k , R l ) = ∑ i = 1 N α ∑ j = 1 N α ( ∂ R k ∂ α i ∂ R l ∂ α j ) ρ i j σ i σ j   + 1 2 ∑ i = 1 N α ∑ j = 1 N α ∑ μ = 1 N α ( ∂ 2 R k ∂ α i ∂ α j ∂ R l ∂ α μ + ∂ R k ∂ α i ∂ 2 R l ∂ α j ∂ α μ ) t i j μ σ i σ j σ μ   + 1 4 ∑ i = 1 N α ∑ j = 1 N α ∑ μ = 1 N α ∑ ν = 1 N α ( ∂ 2 R k ∂ α i ∂ α j ) ( ∂ 2 R l ∂ α μ ∂ α ν ) ( q i j μ ν − ρ i j ρ μ ν ) σ i σ j σ μ σ ν   + 1 6 ∑ i = 1 N α ∑ j = 1 N α ∑ μ = 1 N α ∑ ν = 1 N α ( ∂ R k ∂ α i ∂ 3 R l ∂ α j ∂ α μ ∂ α ν + ∂ R l ∂ α i ∂ 3 R k ∂ α j ∂ α μ ∂ α ν )   q i j μ ν σ i σ j σ μ σ ν , (2)

where t i j μ and q i j μ ν denote the triple-correlations and, respectively, the quadruple correlations among the respective parameters.

3) The 3^rd-order moment μ 3 ( R k , R l , R m ) of three responses has the following expression [4]:

μ 3 ( R k , R l , R m ) = ∑ i = 1 N α ∑ j = 1 N α ∑ μ = 1 N α ∂ R k ∂ α i ∂ R l ∂ α j ∂ R m ∂ α μ t i j μ σ i σ j σ μ   + 1 2 ∑ i = 1 N α ∑ j = 1 N α ∑ μ = 1 N α ∑ ν = 1 N α ∂ R k ∂ α i ∂ R l ∂ α j ∂ 2 R m ∂ α μ ∂ α ν ( q i j μ ν − ρ i j ρ μ ν ) σ i σ j σ μ σ ν   + 1 2 ∑ i = 1 N α ∑ j = 1 N α ∑ μ = 1 N α ∑ ν = 1 N α ∂ R k ∂ α i ∂ 2 R l ∂ α j ∂ α μ ∂ R m ∂ α ν ( q i j μ ν − ρ i ν ρ j μ ) σ i σ j σ μ σ ν   + 1 2 ∑ i = 1 N α ∑ j = 1 N α ∑ μ = 1 N α ∑ ν = 1 N α ∂ 2 R k ∂ α i ∂ α j ∂ R l ∂ α μ ∂ R m ∂ α ν ( q i j μ ν − ρ i j ρ μ ν ) σ i σ j σ μ σ ν . (3)

The 3^rd-order sensitivities would produce terms on the right-side of Equation (3) which are higher than 4^th-order in the parameters’ standard deviations and are therefore not included.

The skewness, denoted as γ 1 ( R ) of a single response R, which indicates the degree of the distribution’s asymmetry with respect to its mean, is defined as follows [4]:

γ 1 ( R ) = μ 3 ( R ) [ var ( R ) ] 3 / 2 . (4)

As has been discussed in [5], correlations among the group-averaged microscopic total cross sections are not available for the PERP benchmark under consideration. When such correlations are unavailable, the maximum entropy principle (see, e.g., Ref. [6] ) indicates that neglecting them minimizes the inadvertent introduction of spurious information into the computations of the various response moments. Also, third- and fourth-order correlations for the group-averaged microscopic total cross sections are not available in the open literature; therefore, the effects on the uncertainties they would induce in the leakage response cannot be exactly quantified. In the absence of any information whatsoever regarding the triple correlations among the group-averaged microscopic total cross sections, the most reasonable assumption (in the sense of introducing the least amount of spurious information into the system, according to the maximum entropy principle [6] ) is to set t i j μ ≡ 0 , which would be rigorously correct if the group-averaged microscopic total cross section were considered to be multivariate normally distributed quantities, even if correlated. For normally distributed parameters, the following relation also holds:

q i j μ ν = ρ i j ρ μ ν + ρ i μ ρ j ν + ρ i ν ρ j μ . (5)

When the group-averaged microscopic total cross sections are uncorrelated, the expected value of the leakage response has the following expression:

[ E ( L ) ] t ( U ) = L ( α 0 ) + [ E ( L ) ] t ( 2 , U ) , (6)

where the superscript “U” indicates contributions from uncorrelated parameters, the subscript t indicates “total” cross section, and where the quantity [ E ( L ) ] t ( 2 , U ) is defined as follows:

[ E ( L ) ] t ( 2 , U ) = 1 2 ∑ g = 1 G ∑ i = 1 I ∂ 2 L ( α ) ∂ 2 σ t , i g ( s t , i g ) 2 + 1 6 ∑ g = 1 G ∑ i = 1 I ∂ 3 L ( α ) ∂ 3 σ t , i g ( s t , i g ) 3 t i i i ,       G = 30 ,     I = 6. (7)

In Equation (7), the quantity s t , i g denotes the standard deviation associated with the parameter σ t , i g .

If the group-averaged microscopic total cross sections are uncorrelated and normally-distributed [which will be indicated by using the superscript “(U,N)” in the following equations], the expected value of the leakage response does not depend on the 3^rd-order sensitivities and has the following expression:

[ E ( L ) ] t ( U , N ) = L ( α 0 ) + [ E ( L ) ] t ( 2 , U , N ) , (8)

where

[ E ( L ) ] t ( 2 , U , N ) ≜ 1 2 ∑ g = 1 G ∑ i = 1 I ∂ 2 L ( α ) ∂ σ t , i g ∂ σ t , i g ( s t , i g ) 2 ,       G = 30 ,       I = 6. (9)

When the group-averaged microscopic total cross sections are uncorrelated and normally-distributed, the variance of the leakage response of the PERP benchmark takes on the following particular form:

[ var ( L ) ] t ( U , N ) = [ var ( L ) ] t ( 1 , U , N ) + [ var ( L ) ] t ( 2 , U , N ) + [ var ( L ) ] t ( 3 , U , N ) , (10)

where the first-order contribution term, [ var ( L ) ] t ( 1 , U , N ) , to the variance [ var ( L ) ] t ( U , N ) is defined as follows,

[ var ( L ) ] t ( 1 , U , N ) ≜ ∑ g = 1 G ∑ i = 1 I [ ∂ L ( α ) ∂ σ t , i g ] 2 ( s t , i g ) 2 ,         G = 30 ,       I = 6 ; (11)

while the second-order contribution term, [ var ( L ) ] t ( 2 , U , N ) , to the variance [ var ( L ) ] t ( U , N ) is defined as follows:

[ var ( L ) ] t ( 2 , U , N ) ≜ 1 2 ∑ g = 1 G ∑ i = 1 I [ ∂ 2 L ( α ) ∂ σ t , i g ∂ σ t , i g ( s t , i g ) 2 ] 2 ,         G = 30 ,       I = 6 ; (12)

and the third-order contribution term, [ var ( L ) ] t ( 3 , U , N ) , to the variance [ var ( L ) ] t ( U , N ) is defined as follows:

[ var ( L ) ] t ( 3 , U , N ) ≜ ∑ g = 1 G ∑ i = 1 I ∑ g ′ = 1 G ∑ j = 1 I [ ∂ L ( α ) ∂ σ t , i g ∂ 3 L ( α ) ∂ σ t , i g ∂ σ t , j g ′ ∂ σ t , j g ′ ] ( s t , i g ) 2 ( s t , j g ′ ) 2 ,   G = 30 ,     I = 6. (13)

For uncorrelated and normally distributed microscopic total cross sections, the third-order moment, [ μ 3 ( L ) ] t ( U , N ) , of the leakage response for the PERP benchmark takes on the following particular form:

[ μ 3 ( L ) ] t ( U , N ) = 3 ∑ g = 1 G ∑ i = 1 I [ ∂ L ( α ) ∂ σ t , i g ] 2 ∂ 2 L ( α ) ∂ σ t , i g ∂ σ t , i g ( s t , i g ) 4 ,       G = 30 ,       I = 6. (14)

Since the third-order correlations for the group-averaged microscopic total cross sections are not available, the contributions of the third-order sensitivities are neglected in Equation (14).

The skewness, [ γ 1 ( L ) ] t ( U , N ) , of a single response L ( α ) , which indicates the degree of the distribution’s asymmetry with respect to its mean, is given by the following expression for uncorrelated and normally distributed parameters:

[ γ 1 ( L ) ] t ( U , N ) = [ μ 3 ( L ) ] t ( U , N ) / { [ var ( L ) ] t ( U , N ) } 3 / 2 . (15)

As Equation (14) indicates, if the 2^nd-order sensitivities were unavailable, the third moment [ μ 3 ( L ) ] t ( U , N ) and hence the skewness [ γ 1 ( L ) ] t ( U , N ) of the leakage response would vanish and the response distribution would by default be assumed to be Gaussian.

The effects of the first- and second-order sensitivities of the leakage response with respect to the microscopic total cross sections on the response’s expected value, variance and skewness have been quantified and documented in the previous works [5] [7]. In this work, the effects of the third-order sensitivities on the response’s variance and skewness are quantified by considering typical values for the standard deviations of the group-averaged microscopic total cross sections, and using these values together with the respective first-, second- and third-order sensitivities in Equations (6)-(15). The results thus obtained are presented in Table 1, considering uniform parameter standard deviations of 1% (very small), 5% (moderate), and 10% (large), respectively.

As shown in Equations (9) and (14), the third-order sensitivities do not affect to the expected value [ E ( L ) ] t ( U , N ) and, respectively, the third-order moment [ μ 3 ( L ) ] t ( U , N ) when the parameters are uncorrelated and normally distributed. However, the third-order sensitivities do affect the values of the variances [ var ( L ) ] t ( 3 , U , N ) and [ var ( L ) ] t ( U , N ) , as well as the skewness [ γ 1 ( L ) ] t ( U , N ) , as shown in Table 1. Note that the numerical values shown in Table 1 for the first- and second-order contributions to the response moments, namely, L ( α 0 ) , [ E ( L ) ] t ( 2 , U , N ) , [ E ( L ) ] t ( U , N ) , [ var ( L ) ] t ( 1 , U , N ) , [ var ( L ) ] t ( 2 , U , N ) and [ μ 3 ( L ) ] t ( U , N ) , are the same as those presented in Table 25 from the previous work [5].

This work has quantified the contributions of the (180)³ third-order mixed sensitivities, ∂ 3 L / ∂ σ t , j g ∂ σ t , k g ′ ∂ σ t , l g ″ ,   j , k , l = 1 , ⋯ , 6 ;   g , g ′ , g ″ = 1 , ⋯ , 30 of the PERP benchmark’s total leakage response with respect to the benchmark’s 180 microscopic total cross sections, to the benchmark’s leakage response moments (expected values, variance and skewness), and compared these contributions with those stemming from the corresponding first- and second-order sensitivities. The following conclusions can be drawn from the results reported in this work:

1) The results shown in Table 1 indicate that the importance of the 3^rd-order sensitivities increases with increasing uncertainties in the PERP benchmark’s microscopic total cross sections.

2) The contributions of the 3^rd-order sensitivities to the response’s variance surpass the contributions of the 1^st-order and 2^nd-order sensitivities to the response’s variance already for relatively small (ca. 5%) standard deviations for the uncertain cross sections. These effects are rapidly amplified as the uncertainties in the cross sections increase. In particular, for a uniform standard deviation of 10%, the 3^rd-order sensitivities contribute ca. 80% of the response variance [ var ( L ) ] t ( U , N ) . By comparison, the first-order sensitivities contribute ca. 2%, while the second-order sensitivities contribute ca. 18% of the response variance [ var ( L ) ] t ( U , N ) . Thus, neglecting the contributions stemming from the third-order sensitivities would cause a large non-conservative error by under-reporting the response variance by a factor of 506%.

3) The 1^st-, 2^nd- and 3^rd-order sensitivities all yield a positive response skewness, indicating that the response distribution is asymmetrically “skewed” towards values higher than the response’s expected value [ E ( L ) ] t ( U , N ) . When the relative standard deviations of the microscopic total cross sections are increased, the magnitude of the response skewness remains positive but decreases, indicating that the response distribution becomes more symmetrical around the response’s expected value.

4) The effects of the 3^rd-order sensitivities are towards reducing the response skewness, thereby reducing the asymmetry of the leakage response distribution in the space of parameters (total cross sections); this effect is the more pronounced the larger the parameter’s standard deviations. In particular, for a uniform relative standard deviation of 10% for the uncorrelated microscopic total cross sections, the skewness is [ γ 1 ( L ) ] t ( U , N ) = 0.341 when only the 1^st- and 2^nd-order sensitivities are considered; however, when the 3^rd-order sensitivities are also included, the magnitude of the skewness decreases to [ γ 1 ( L ) ] t ( U , N ) = 0.030 .

5) Since the actual correlations between the microscopic total cross sections for the PERP benchmark are unavailable, the effects of the mixed 3^rd-order sensitivities on the response moments (e.g., expected value, variance and skewness) are underestimated; the same is true of the effects stemming from the 1^st- and 2^nd-order sensitivities. Therefore, it would be very important for future experimental research to obtain values for the correlations that might exist among the various cross sections. Correlations among the total cross sections could provide significantly larger contributions to the response moments in addition to those stemming from the parameters’ standard deviations.

6) Relative standard deviations of 10% in the total cross sections can produce unphysical values for the standard deviation of the response distribution, as displayed in Figure 3, which indicates that the 3^rd-order Taylor series expansion of the response around the nominal parameter values (which underlies the mathematical expressions for computing response uncertainties that stem from parameter uncertainties) is an inadequate representation of the response distribution. An investigation of the consequences of this observation is currently underway, commencing with the examination of the convergence properties of the series of higher-order sensitivities.

The authors declare no conflicts of interest regarding the publication of this paper.

Fang, R.X. and Cacuci, D.G. (2020) Third Order Adjoint Sensitivity and Uncertainty Analysis of an OECD/NEA Reactor Physics Benchmark: III. Response Moments. American Journal of Computational Mathematics, 10, 559-570. https://doi.org/10.4236/ajcm.2020.104031