The accompanying works [1] [2] have presented the second-order predictive modelling (acronym: PM) methodology conceived by Cacuci for obtaining “best-estimate results with reduced uncertainties” (acronym: BERRU). This methodology, designated by the acronym “2^nd-BERRU-PM,” relies fundamentally on the maximum entropy (MaxEnt) principle of thermodynamics [3] and Shannon’s information theory [4] . These conceptual underpinnings of the 2^nd-BERRU-PM methodology are in contradistinction to the data assimilation methodology [5] , which relies on minimizing a subjective user-defined functional which is meant to represent, in the energy-norm, the differences between measured and computed results of interest (called “responses”). As has been presented in [1] [2] , there are two complementary methodological frameworks for constructing the 2^nd-BERRU-PM methodology, leading to equivalent, but not identical results. One framework is constructed by incorporating the representation of the computational model deterministically; the resulting methodology has been designated by the acronym “2^nd-BERRU-PMD” [1] , where the letter “D” indicates “deterministic”. The alternative framework is constructed by incorporating the representation of the computational model probabilistically; the resulting methodology has been designated by the acronym “2^nd-BERRU-PMP” [2] , where the last letter of the acronym (“P”) indicates “probabilistic”.

In this work, the 2^nd-BERRU-PMD [1] methodology (which deterministically incorporates the representation of the computational model) will be used to illustrate quantitatively the effects of second- and higher-order sensitivities for obtaining best-estimate results with reduced uncertainties for the polyethylene-reflected plutonium (acronym: PERP) OECD/NEA reactor physics benchmark [6] . This benchmark is modeled using the neutron transport Boltzmann equation, the solution of which is representative of “large-scale computations” and involves a large number (21,976) of uncertain parameters. The Boltzmann forward and adjoint neutron transport equations are solved using the software packages PARTISN [7] and SOURCES4C [8] in conjunction with the MENDF71X [9] cross section data. The characteristics of the PERP benchmark are briefly presented in Section 2, along with the computational method for solving the Boltzmann transport equation. This equation models the neutron distribution within this benchmark, including the neutron leakage from the outer surface of the benchmark, which will be the “response” of interest in this illustrative application. The specific particular forms taken on by the 2^nd-BERRU-PMD [1] mathematical formulas for the best-estimate predicted results (responses) along with the corresponding reduction in the accompanying predicted standard deviation will also be presented in Section 2. The numerical results produced by the 2^nd-BERRU-PMD [1] will be presented in Section 3, highlighting the essential impact and contributions of the 2^nd-, 3^rd- and 4^th-order sensitivities on the results. In particular, it is shown in Section 3 that mistaken indications of inconsistency between measurements and computations—when only 1^st-order sensitivities are considered—are actually corrected to the contrary by the inclusion of higher-order sensitivities (i.e., using only 1^st-order sensitivities indicates that the computations are inconsistent with the measurements, but the contrary becomes apparent when the contributions from the higher-order sensitivities are included). The discussion in Section 4 of the significance of the 2^nd-BERRU-PM methodology, as illustrated by using the PERP reactor physics benchmark, concludes this work.

This Section illustrates the steps involved in applying the 2^nd-BERRU-PMD methodology [1] to the spherical polyethylene-reflected plutonium (acronym: PERP) OECD/NEA reactor physics benchmark [6] . This benchmark is modeled using the neutron transport equation, the solution of which is representative of “large-scale computations” and involves a large number (21,976) of uncertain parameters. The PERP benchmark comprises a metallic inner sphere (“core”) containing the following 4 isotopes: Isotope 1 (239Pu), Isotope 2 (240Pu), Isotope 3 (69Ga) and Isotope 4 (71Ga). This core (which is designated as “material 1”) is surrounded by a spherical shell of polyethylene (designated as “material 2”), containing two isotopes, designated as Isotope 5 (C) and Isotope 6 (1H), respectively. The characteristics of the PERP benchmark are presented in Table 1, below.

The neutron flux distribution within the PERP benchmark, as well as the leakage of neutrons out of the benchmark’s outer surface, has been modeled using the standard multigroup form of the Boltzmann neutron transport equation with an internal spontaneous fission source, and subject to the boundary condition of no incoming flux, which can be written in the following matrix-form:

B ( α ) φ ( r , Ω ) = Q ( c ) ,         φ ( r , Ω ) = 0 ,         r = r d ,       Ω ⋅ n < 0. (1)

The quantities appearing in Equation (1) are defined as follows: r d denotes the radius of the PERP sphere; the matrix B ( α ) and the vectors Q ( α ) and φ ( r , Ω ) are defined as follows:

B ( α ) ≜ ( B 11 ( α ) · B 1 g ( α ) · B 1 G ( α ) · · · · · B g 1 ( α ) · B g g ( α ) · B g G ( α ) · · · · · B G 1 ( α ) · B G g ( α ) · B G G ( α ) ) ;       φ ( r , Ω ) ≜ ( φ 1 · φ g · φ G ) ;       Q ( α ) ≜ ( Q 1 · Q g · Q G ) ; (2)

having components defined below:

B g h ( α ) ≜ δ g , h B g g 0 ( α ) − B g h 1 ( α ) ;       g , h = 1 , ⋯ , G ; B g g 0 ( α ) φ g ( r , Ω ) ≜ [ Ω ⋅ ∇ + Σ t g ( α ; r ) ] φ g ( r , Ω ) ; B g h 1 ( α ) φ h ( r , Ω ′ ) ≜ ∫ 4 π d Ω ′ [ Σ s h → g ( α ; r , Ω ′ → Ω ) + χ g ( α ; r ) ( ν Σ f ) h ( α ; r ) ] ; (3)

Q g ( c ) ≜ ∑ k = 1 N f λ k N k F k S F ν k S F ( 2 π a k 3 b k e − a k b k 4 ) ∫ E g + 1 E g d E     e − E / a k sinh b k E . (4)

The notation δ g , h , which appears in Equation (3), denotes the Kronecker delta-functional, which is defined as usual, i.e., δ g , h = 1 , if g = h and δ g , h = 0 , if g ≠ h . The quantities appearing in Equations (1)-(4) are defined as follows:

1) The quantity φ g ( r , Ω ) is the customary “group-flux” for group g, g = 1 , ⋯ , G .

2) The source Q g ( c ) depends on the vector of model parameters c , defined as follows:

c ≜ [ c 1 , ⋯ , c J Q ] † ≜ [ λ 1 , λ 2 ; F 1 S F , F 2 S F ; a 1 , a 2 ; b 1 , b 2 ; ν 1 S F , ν 2 S F ] † ,       J Q = 10. (5)

3) As indicated in Table 1, the PERP benchmark comprises 2 materials: “material 1” comprises 4 isotopes, numbered 1 through 4, while “material 2” comprises 2 isotopes, numbered 5 and 6. These materials contain only isotopes that are distinct from each other, so the atomic number density N i of an isotope i, i = 1 , ⋯ , I = 6 , is computed as follows:

N i = ρ 1 w i , 1 N A A i ;     for     i = 1 , 2 , 3 , 4 ; ​ ​     N i = ρ 2 w i , 2 N A A i ;     for     i = 5 , 6 , (6)

where ρ m denotes the mass density of material m, m = 1 , 2 ; w i , m denotes the weight fraction of isotope i in material m; A i denotes the atomic weight of

isotope i; N A denotes Avogadro’s number. The atomic number densities N i , i = 1 , ⋯ , I = 6 will be considered to be components of the vector denoted as N and defined as follows: N ≜ [ N 1 , N 2 , N 3 , N 4 , N 5 , N 6 ] † .

4) The scattering transfer cross section from energy group g ′ , g ′ = 1 , ⋯ , G , into energy group g, g = 1 , ⋯ , G , is denoted as Σ s g ′ → g ( α ; r , Ω ′ → Ω ) and is computed in terms of the l-th order Legendre coefficient σ s , l , i g ′ → g using the following 3^rd-order expansion in Legendre functions :

Σ s g ′ → g ( α ; r , Ω ′ → Ω ) = ∑ i = 1 I = 6 N i ∑ l = 0 I S C T = 3 ( 2 l + 1 ) σ s , l , i g ′ → g P l ( Ω ′ ⋅ Ω ) , ​ ​ ​     g ,   g ′ = 1 , ⋯ , G , (7)

where I S C T = 3 denotes the order of the expansion in Legendre polynomials. The microscopic scattering cross sections σ s , l , i g ′ → g for isotope i, and from energy group g ′ into energy group g, are tabulated parameters. The zeroth-order (i.e., l = 0 ) scattering cross sections must be considered separately from the higher order (i.e., l ≥ 1 ) scattering cross sections, since the former contribute to the total cross sections (as noted below), while the latter do not. The microscopic scattering cross sections σ s , l , i g ′ → g will be considered to be components of a vector σ s defined below:

σ s ≜ [ s 1 , ⋯ , s J S X ] † ≜ [ σ s , l = 0 , i = 1 g ′ = 1 → g = 1 , σ s , l = 0 , i = 1 g ′ = 2 → g = 1 , ⋯ , σ s , l = 0 , i = 1 g ′ = G → g = 1 , σ s , l = 0 , i = 1 g ′ = 1 → g = 2 , σ s , l = 0 , i = 1 g ′ = 2 → g = 2 , ⋯ , σ s , l , i g ′ → g , ⋯ , σ s , I S C T , i = I G → G ] † , l = 0 , ⋯ , I S C T ;     i = 1 , ⋯ , I ;     g , g ′ = 1 , ⋯ , G ;     J S X = ( G × G ) × I × ( I S C T + 1 ) . (8)

5) The software package PARTISN [7] computes the quantity ( ν Σ f ) g ( α ; r ) for each isotope i and energy group g, as follows:

( ν Σ f ) g ( α ; r ) = ∑ i = 1 N f = 2 N i σ f , i g ν i g   ,       g = 1 , ⋯ , G = 30 , (9)

where σ f , i g denotes the microscopic fission cross section for isotope i and energy group g, ν i g denotes the average number of neutrons per fission for isotope i and energy group g, and N f denotes the total number of fissionable isotopes.

σ f ≜ [ σ f , i = 1 1 , σ f , i = 1 2 , ⋯ , σ f , i = 1 G , ⋯ , σ f , i g , ⋯ , σ f , i = N F 1 , ⋯ , σ f , i = N F G ] † ≜ [ f 1 , ⋯ , f J F X ] † ,     i = 1 , ⋯ , N F ;     g = 1 , ⋯ , G ;     J F X = G × N F ; (10)

ν ≜ [ ν i = 1 1 , ν i = 1 2 , ⋯ , ν i = 1 G , ⋯ , ν i g , ⋯ , ν i = N f 1 , ⋯ , ν i = N F G ] † ≜ [ f J F X + 1 , ⋯ , f J F X + J N U ] † ,     i = 1 , ⋯ , N F ;     g = 1 , ⋯ , G ;     J N U = G × N F . (11)

6) The total cross section for energy group g, g = 1 , ⋯ , G , is denoted as Σ t g ( α ) and is computed using the following expression:

Σ t g ( α ) = ∑ i = 1 I N i σ t , i g   ;         σ t , i g = [ σ f , i g + σ c , i g + ∑ g ′ = 1 G σ s , l = 0 , i g → g ′ ] , (12)

where the quantities σ t , i g , σ f , i g and σ c , i g denote, respectively, the total microscopic cross section, the tabulated group microscopic fission, and the neutron capture cross sections for isotope i and group g. Other nuclear reactions in the PERP benchmark are negligible. To reduce as much as possible the proliferation of indices when determining the higher-order (up to and including the 4^th-order) sensitivities of the PERP leakage response, it is useful to consider that the cross sections σ t , i g are the components of a vector t , having J T X ≜ G × I components defined as follows:

t ≜ [ t 1 , ⋯ , t J T X ] † ≜ [ σ t , i = 1 1 , σ t , i = 1 2 , ⋯ , σ t , i = 1 G , ⋯ , σ t , i g , ⋯ , σ t , i = I 1 , ⋯ , σ t , i = I G ] † ,       for     i = 1 , ⋯ , I = 6 ;     g = 1 , ⋯ , G = 30 ;     J T X ≜ I × G . (13)

7) The quantity χ g ( α ; r ) quantifies the fission spectrum in energy group g. The fission spectrum is considered to depend on the vector of parameters p , defined as follows:

p ≜ [ p 1 , ⋯ , p J χ ] † ≜ [ χ i = 1 g = 1 , χ i = 1 g = 2 , ⋯ , χ i = 1 G , ⋯ , χ i g , ⋯ , χ N F G ] † ,         for     i = 1 , ⋯ , N F ;       g = 1 , ⋯ , G ;       J χ = G × N F . (14)

In summary, the model parameters characterizing the PERP benchmark can all be considered to be the components of a “vector of model parameters” denoted as α ≜ [ α 1 , ⋯ , α T P ] † , where the subscript “TP” stands for “Total number of model and response Parameters”, and is defined below:

α ≜ [ α 1 , ⋯ , α T P ] † ≜ [ c ; N ; σ s ; σ t ; σ f ; ν ; p ] † ,       where     T P ≜ J Q + I + J S X + J T X + J F X + J N U + J χ . (15)

The multigroup Boltzmann (forward and adjoint) neutron transport equation was solved numerically using the software packages PARTISN [7] and SOURCES-4C [8] in conjunction with the MENDF71X [9] 618-group cross section data collapsed to G = 30 energy groups, as well as a P3 Legendre expansion of the scattering cross section and a fine-mesh spacing of 0.005 cm (comprising 759 meshes for the plutonium sphere of radius of 3.794 cm, and 762 meshes for the polyethylene shell of thickness of 3.81 cm). The first- and second-order response sensitivities were computed using an angular quadrature of S256. The 3^rd- and 4^th-order sensitivities of the leakage response with respect to the total cross sections were computed using an angular quadrature of S32. The scattering and fission terms in Equation (1) contain implicitly a factor 1 / 4 π . The group boundaries of the G = 30 energy groups are provided in Table 2.

Thus, the numerical model of the PERP benchmark comprises 21,976 uncertain parameters, as follows: 180 group-averaged total microscopic cross sections, 21,600 group-averaged scattering microscopic cross sections; 120 fission process parameters; 60 fission spectrum parameters; 10 parameters describing the experiment’s nuclear sources; and 6 isotopic number densities.

The quantity of interest in this work, which will be called the “response,” is the leakage of neutrons through the outer surface of the spherical benchmark; this leakage response is denoted as L ( α ) and is defined below:

L ( α ) ≜ ∫ S b d S ∑ g = 1 G ∫ Ω ⋅ n > 0 d Ω   Ω ⋅ n   φ g ( r , Ω ) . (16)

For convenient reference, the histogram plot of the leakage for each energy group for the PERP benchmark is presented in Figure 1, below. The value of the total leakage computed using Equation (16) for the PERP benchmark is 1.7648 × 106 neutrons/sec.

As generally presented in [10] , the formulas/expressions used for computing the mean value and the variance of the leakage response are based on the Taylor series of the leakage response around the expected (nominal) parameter values α 0 , which has the following form up to fourth-order:

L ( α ) = L ( α 0 ) + ∑ j 1 = 1 T P { ∂ L ( α ) ∂ α j 1 } α 0 δ α j 1     + 1 2 ∑ j 1 = 1 T P ∑ j 2 = 1 T P { ∂ 2 L ( α ) ∂ α j 1 ∂ α j 2 } α 0 δ α j 1 δ α j 2     + 1 3 ! ∑ j 1 = 1 T P ∑ j 2 = 1 T P ∑ j 3 = 1 T P { ∂ 3 L ( α ) ∂ α j 1 ∂ α j 2 ∂ α j 3 } α 0 δ α j 1 δ α j 2 δ α j 3     + 1 4 ! ∑ j 1 = 1 T P ∑ j 2 = 1 T P ∑ j 3 = 1 T P ∑ j 4 = 1 T P { ∂ 4 L ( α ) ∂ α j 1 ∂ α j 2 ∂ α j 3 ∂ α j 4 } α 0 δ α j 1 δ α j 2 δ α j 3 δ α j 4 + ε k . (17)

The radius/domain of convergence of the series in Equation (17) determines the largest values of the parameter variations δ α j which are admissible before the respective series becomes divergent.

Considering that the model parameters underlying the PERP benchmark are uncorrelated and normally distributed, and considering only the unmixed higher-order sensitivities, the computed mean value and, respectively, computed

variance of the leakage response have the following expressions [11] when the sensitivities up to fourth-order are included:

E ( L c ) = L c ( α 0 ) + 1 2 ∑ j 1 = 1 T P ∂ 2 L ( α 0 ) ( ∂ α j 1 ) 2 σ j 1 2 + 1 8 ∑ j 1 = 1 T P ∂ 4 L ( α 0 ) ( ∂ α j 1 ) 4 σ j 1 4 , (18)

where L c ( α 0 ) = 1.7648 × 10 6 neutrons/sec denotes the value of the computed leakage response at the nominal parameter values and where T P = 21976 denotes the “total number of model parameters.”

The computed variance of the leakage response, var ( L c ) , has the following expression [11] when considering only the unmixed response sensitivities to uncorrelated and normally distributed parameters:

var ( L c ) = ∑ j 1 = 1 T P [ ∂ L ( α 0 ) ∂ α j 1 ] 2 σ j 1 2 + 1 2 ∑ j 1 = 1 T P [ ∂ 2 L ( α 0 ) ( ∂ α j 1 ) 2 ] 2 σ j 1 4     + ∑ j 1 = 1 T P [ ∂ 3 L ( α 0 ) ( ∂ α j 1 ) 3 × ∂ L ( α 0 ) ∂ α j 1 ] σ j 1 4 + 15 36 ∑ j 1 = 1 T P [ ∂ 3 L ( α 0 ) ( ∂ α j 1 ) 3 ] 2 σ j 1 6     + 1 2 ∑ j 1 = 1 T P [ ∂ 4 L ( α 0 ) ( ∂ α j 1 ) 4 × ∂ 2 L ( α 0 ) ( ∂ α j 1 ) 2 ] σ j 1 6 . (19)

The expressions provided in Equations (18) and (19) are valid only if the Taylor series shown in Equation (17) converges. The largest admissible parameter variations δ α j which are still within the radius of convergence of Equation (17) provide the largest parameter covariances/standard deviations which can be considered to ensure that the expressions provided in Equations (18) and (19) are valid.

The best-estimate predicted response (leakage) value, denoted as L b e , which is obtained after combining the computed response value with a measured response value, is given by the following particular form of the expression derived in [1] :

L b e = L e + var ( L e ) E ( L c ) − L e var ( L e ) + var ( L c ) = L e var ( L c ) + E ( L c ) var ( L e ) var ( L e ) + var ( L c ) , (20)

where: L e denotes the experimentally measured mean value of the leakage response; var ( L e ) denotes the experimentally measured variance of the leakage response; E ( L c ) denotes the computed mean value of the leakage response; and var ( L c ) denotes the computed variance of the leakage response. It is important to note that the relation provided in Equation (20) implies the following inequalities:

If ​ ​     0 < L e < E ( L c ) ,         then     L e < L b e < E ( L c ) , (21)

If     0 < E ( L c ) < L e ,         then     E ( L c ) < L b e < L e . (22)

The inequalities shown in Equations (21) and (22) indicate that the best-estimate predicted value L b e of the leakage response will fall in between the initially measured value, L e , and the computed value, E ( L c ) , of the leakage response.

The predicted variance, var ( L b e ) , of the best-estimate (leakage) response L b e is given by the following particular form of the expression derived in [1] :

var ( L b e ) = var ( L e ) { 1 − var ( L e ) [ var ( L e ) + var ( L c ) ] − 1 } = var ( L e ) var ( L c ) var ( L e ) + var ( L c ) . (23)

Note that the expression in Equation (23) ensures the reduction of the predicted variance var ( L b e ) by comparison to either the variance var ( L e ) of the experimentally measured response or the variance var ( L c ) of the computed response, since Equation (23) implies the following inequalities:

var ( L b e ) < var ( L e ) ;         var ( L b e ) < var ( L c ) . (24)

The first-order partial sensitivities ∂ L ( α 0 ) / ∂ α j , for all α j , j = 1 , ⋯ , T P , have the following expressions [10] :

∂ L ( α 0 ) ∂ q j = ∑ g = 1 G ∫ 0 r d 4 π r 2 d r ∫ 4 π d Ω   ψ ( 1 ) , g ( r , Ω ) ∂ Q g ( α 0 ) ∂ q j ,       j = 1 , ⋯ , J Q = 10 ; (25)

∂ L ( α 0 ) ∂ N j = ∑ g = 1 G ∫ 0 r d 4 π r 2 d r ∫ 4 π d Ω   ψ ( 1 ) , g ( r , Ω ) ∂ Σ t g ( α 0 ) ∂ N j φ g ( r , Ω ) ​ − ∑ h = 1 G ∫ 0 r d 4 π r 2 d r ∫ 4 π d Ω ′ φ h ( r , Ω ′ ) { ∂ Σ s h → g ( α 0 ; Ω ′ → Ω ) ∂ N j + { ∂ [ χ g ( ν Σ f ) h ] ∂ N j } α 0 } ,                   for     j = 1 , ⋯ , I = 6 ; (26)

∂ L ( α 0 ) ∂ s j = ∑ g = 1 G ∫ 0 r d 4 π r 2 d r ∫ 4 π d Ω     ψ ( 1 ) , g ( r , Ω )   × ∑ h = 1 G ∫ 0 r d 4 π r 2 d r ∫ 4 π d Ω ′ ∂ Σ s h → g ( α 0 ; Ω ′ → Ω ) ∂ s j φ h ( r , Ω ′ ) ,       for     j = 1 , ⋯ , J S X = ( G × G ) × I × ( I S C T + 1 ) = 21600 ; (27)

∂ L ( α 0 ) ∂ t j = − ∑ g = 1 G ∫ 0 r d 4 π r 2 d r ∫ 4 π d Ω   ψ ( 1 ) , g ( r , Ω ) φ g ( r , Ω ) ∂ Σ t g ( α 0 ) ∂ t j ,       for     j = 1 , ⋯ , J T X = 180 ; (28)

∂ L ( α 0 ) ∂ f j = ∑ g = 1 G ∫ 0 r d 4 π r 2 d r ∫ 4 π d Ω   ψ ( 1 ) , g ( r , Ω ) χ g ( α 0 ) ∑ h = 1 G ∫ 0 r d 4 π r 2 d r ∫ 4 π d Ω ′                                   × { ∂ [ ( ν Σ f ) h ] ∂ f j } α 0 φ h ( r , Ω ′ ) ,             for     j = 1 , ⋯ , J F X = 60 ; (29)

∂ L ( α 0 ) ∂ ν j = ∑ g = 1 G ∫ 0 r d 4 π r 2 d r ∫ 4 π d Ω   ψ ( 1 ) , g ( r , Ω ) χ g ( α 0 ) ∑ h = 1 G ∫ 0 r d 4 π r 2 d r ∫ 4 π d Ω ′                                     × { ∂ [ ( ν Σ f ) h ] ∂ ν j } α 0 φ h ( r , Ω ′ )   ,                   for     j = 1 , ⋯ , J N U = 60 ; (30)

∂ L ( α 0 ) ∂ p j = ∑ g = 1 G ∫ 0 r d 4 π r 2 d r ∫ 4 π d Ω   ψ ( 1 ) , g ( r , Ω ) ∂ χ g ( α 0 ) ∂ p j                         × ∑ h = 1 G ∫ 0 r d 4 π r 2 d r ∫ 4 π d Ω ′ { ( ν Σ f ) h } α 0 φ h ( r , Ω ′ ) ,       for     j = 1 , ⋯ , J χ = 60. (31)

In Equations (25)-(31), the 1^st-level adjoint function ψ ( 1 ) is the solution of the following 1^st-Level Adjoint Sensitivity System (1^st-LASS):

A ( α ) ψ ( 1 ) ( r , Ω ) = I [ Ω ⋅ n δ ( r − r d ) ] ; ​ ​ ​       I ≜ [ 1 , ⋯ , 1 , ⋯ , 1 ] † , (32)

ψ ( 1 ) ( r , Ω ) = 0 ,     r = r d ,       Ω ⋅ n > 0 , (33)

where A ( α ) denotes the operator adjoint to B ( α ) , having components A g h ( α ) ≜ [ B h g ( α ) ] * , where the symbol [ ] * indicates “formal adjoint operator.” In component form, Equations (32) and (33) are as follows:

− Ω ⋅ ∇ ψ ( 1 ) , g ( r , Ω ) + Σ t g ( α ) ψ ( 1 ) , g ( r , Ω ) − ∑ h = 1 G ∫ 4 π d Ω ′   ψ ( 1 ) , h ( r , Ω ′ ) × [ Σ s g → h ( α ; Ω → Ω ′ ) + ( ν Σ f ) g ( α ) χ h ( α ) ] = Ω ⋅ n δ ( r − r d ) ,         g = 1 , ⋯ , G ; (34)

ψ ( 1 ) , g ( r , Ω ) = 0   ,     r = r d ,       Ω ⋅ n > 0 ,       g = 1 , ⋯ , G . (35)

The expressions of the sensitivities provided in Equations (25)-(31) are to be evaluated at the nominal values u 0 ≜ ( φ 0 ; α 0 ) . Evidently, the computations of these sensitivities are inexpensive, involving only integrations using quadrature formulas after having obtained the 1^st-level adjoint function ψ ( 1 ) ( r , Ω ) . The 1^st-LASS is independent of parameter variations, so it needs to be solved just once to obtain ψ ( 1 ) ( r , Ω ) . Thus, the computation of the TP partial sensitivities ∂ L ( α 0 ) / ∂ α j , j = 1 , ⋯ , T P , requires just a single large-scale computation in order to determine ψ ( 1 ) ( r , Ω ) , followed by TP inexpensive computations to perform each integration (quadrature) involving the 1^st-level adjoint function ψ ( 1 ) ( r , Ω ) .

Each of the first-order sensitivities provided in Equations (25)-(31) gives rise to corresponding 2^nd-order sensitivities, which are computed using a 2^nd-level adjoint function that is obtained by solving a corresponding 2^nd-Level Adjoint Sensitivity System (2^nd-LASS). For example, the 2^nd-order sensitivities which arise from the 1^st-order sensitivities of the response to the total cross sections, cf. Equation (28), have the following expression [10] :

∑ j 2 = 1 T P { ∂ 2 L ( α ) ∂ α j 2 ∂ t j 1 δ α j 2 } α 0 = { ∑ g = 1 G ∫ 0 r d 4 π r 2 d r ∫ 4 π d Ω   ψ 1 ( 2 ) , g ( j 1 ; r , Ω ) Q ( 1 ) ( g ; α , φ ; δ α ) } α 0 + { ∑ g = 1 G ∫ 0 r d 4 π r 2 d r ∫ 4 π d Ω   ψ 2 ( 2 ) , g ( j 1 ; r , Ω ) Q ( 2 ) ( g ; α , ψ ( 1 ) ; δ α ) } α 0 ,       j 1 = 1 , ⋯ , J T X . (36)

The 2^nd-level adjoint function ψ ( 2 ) ( j 1 ; r , Ω ) ≜ [ ψ 1 ( 2 ) ( j 1 ; r , Ω ) , ψ 2 ( 2 ) ( j 1 ; r , Ω ) ] † , ψ i ( 2 ) ≜ ( ψ i ( 2 ) , 1 , ⋯ , u i ( 2 ) , G ) † , i = 1 , 2 , is the solution of the following 2^nd-Level Adjoint Sensitivity System (2^nd-LASS):

{ A ( α ) ψ 1 ( 2 ) ( j 1 ; r , Ω ) } α 0 = − { S ( j 1 ; α ) ψ ( 1 ) ( r , Ω ) } α 0 , ​ ​ ​       j 1 = 1 , ⋯ , J T X = G × I = 180 , (37)

{ B ( α ) ψ 2 ( 2 ) ( j 1 ; r , Ω ) } α 0 = − { S ( j 1 ; α ) φ ( r , Ω ) } α 0 ;     j 1 = 1 , ⋯ , J T X , (38)

ψ 1 ( 2 ) ( j 1 ; r , Ω ) = 0 ,     r =   r d ,       Ω ⋅ n > 0 ;     j 1 = 1 , ⋯ , J T X , (39)

ψ 2 ( 2 ) ( j 1 ; r , Ω ) = 0 ,       r = r d ,     Ω ⋅ n < 0 ;       j 1 = 1 , ⋯ , J T X , (40)

where, for each j 1 = 1 , ⋯ , J T X = G × I = 30 × 6 = 180 , S ( j 1 ; α ) is a G × G diagonal matrix having non-zero elements of the form ∂ Σ t g ( α ) / ∂ t j 1 ,   g = 1 , ⋯ , G on its diagonal, i.e.,

S ( j 1 ; α ) ≜ ( ∂ Σ t 1 ( α ) / ∂ t j 1 · 0 · · · 0 · ∂ Σ t G ( α ) / ∂ t j 1 ) , (41)

Q ( 1 ) ( α ; δ α ) ≜ ( Q ( 1 ) ( 1 ; α , ψ ( 1 ) ; δ α ) · Q ( 1 ) ( G ; α , ψ ( 1 ) ; δ α ) ) , Q ( 2 ) ( α , ψ ( 1 ) ; δ α ) ≜ ( Q ( 2 ) ( 1 ; α , ψ ( 1 ) ; δ α ) · Q ( 2 ) ( G ; α , ψ ( 1 ) ; δ α ) ) , (42)

Q ( 1 ) ( g ; α , φ ; δ α ) ≜ ∂ Q g ( α ) ∂ α δ α − φ g ( r , Ω ) ∂ Σ t g ( α ) ∂ α δ α + ∑ h = 1 G ∫ 4 π d Ω ′ φ h ( r , Ω ′ ) { ∂ Σ s h → g ( α ; Ω ′ → Ω ) ∂ α δ α + ∂ [ χ g ( α ) ( ν Σ f ) h ( α ) ] ∂ α δ α }   , (43)

Q ( 2 ) ( g ; α , ψ ( 1 ) ; δ α ) ≜ − ψ ( 1 ) , g ( r , Ω ) ∂ Σ t g ( α ; r ) ∂ α δ α + ∑ h = 1 G ∫ 4 π d Ω ′   ψ ( 1 ) , h ( r , Ω ′ ) { ∂ Σ s g → h ( α ; Ω → Ω ′ ) ∂ α δ α + ∂ [ ( ν Σ f ) g ( α ) χ h ( α ) ] ∂ α δ α } . (44)

The expressions of the 3^rd- and 4^th-order sensitivities will not be reproduced here but can be found in [10] [12] [13] .

This Section presents results that will illustrate the reduction of the predicted standard deviation of the leakage response when applying the 2^nd-BERRU-PMD methodology to incorporate a measurement, even when the measurement appears initially to be inconsistent with the computation. The effects of the measurement’s precision are also investigated, as are the respective impacts of the 1^st-, 2^nd-, 3^rd- and 4^th-order sensitivities. Results are also presented for illustrating the effects of parameters standard deviations which are within, borderline, or outside of the radius of convergence of the Taylor-series provided in Equation (17).

Using the maximum entropy (MaxEnt) principle of thermodynamics [3] and concepts of information theory [4] , Cacuci [1] [2] has conceived the 2^nd-BERRU-PM methodology; this acronym stands for “second-order predictive modelling methodology conceived by for obtaining “best-estimate results with reduced uncertainties.” The conceptual underpinnings of the 2^nd-BERRU-PM methodology are in contradistinction to the data assimilation methodology [5] , which relies on minimizing a subjective user-defined functional which is meant to represent, in the energy-norm, the differences between measured and computed results of interest (called “responses”). The 2^nd-BERRU-PM methodology can be constructed either by incorporating the representation of the computational model deterministically (yielding the “2^nd-BERRU-PMD” [1] methodology) or by incorporating the representation of the computational model probabilistically (yielding the alternative “2^nd-BERRU-PMP” [2] methodology). These methodologies yield equivalent but not identical expression for the predicted results (second-order best-estimate predicted values for the predicted model response and calibrated model parameters, along with reduced predicted best-estimate standard deviations).

This work has illustrated the application of the 2^nd-BERRU-PMD [1] methodology to illustrate quantitatively the effects of second- and higher-order sensitivities for obtaining best-estimate results with reduced uncertainties for the polyethylene-reflected plutonium (acronym: PERP) OECD/NEA reactor physics benchmark [6] . This benchmark is modeled using the neutron transport Boltzmann equation, the solution of which is representative of “large-scale computations” and involves a large number (21,976) of uncertain parameters. Several representative practical situations have been considered, as follows: 1) high-precision measured response and high precision parameters; 2) high-precision measured response and low precision parameters; 3) low-precision measured response and borderline (in terms of convergence of the expressions underlying the computation of the expected value and, respectively, the standard deviation of the computed response) parameter precision. In all situations, as predicted by the 2^nd-BERRU-PMD [1] methodology, the predicted best-estimate value of the response has fallen in between the computed and measured values of the leakage response. Also as predicted by the 2^nd-BERRU-PMD [1] methodology, it has been observed that the predicted standard deviation of the predicted response is smaller than either the measure or the computed standard deviation, regardless of the order of sensitivities included in the respective computation, i.e., S D ( b e ) < S D ( e ) and S D ( b e ) < S D ( c o m p ) . It has also been observed that when the parameters are known with high precision, the contributions of the higher-order sensitivities diminish with increasing order, so that the inclusion of the 2^nd-order sensitivities (in addition to the 1^st-order sensitivities), while necessary, may suffice for obtaining accurate predicted best-estimate response values and best-estimate standard deviations. On the other hand, when the parameters’ standard deviations are sufficiently large to approach (or be outside of) the radius of convergence of the multivariate Taylor-series which represents the response in the phase-space of model parameters, the contributions of the 3^rd- and even 4^th-order sensitivities are necessary to ensure consistency between the computed and measured response. In such cases, the use of only the 1^st-order sensitivities invariably indicates erroneously that the computed results are inconsistent with the respective measured response.

The results presented in this work indicate the general conclusion that at least the 1^st- and 2^nd-order sensitivities need to be computed and included in any methodology that combines computational and experimental information. While the higher-order sensitivities may contribute little in special situations, e.g., when the model parameters are known with high precision, at least the 2^nd-order sensitivities are needed to be included, while the 3^rd-order sensitivities would be needed in order to quantitatively confirm or infirm the necessity of including sensitivities of order three and/or higher-order.

Ongoing research is dedicated to computing the best-estimate calibrated parameters, and their corresponding best-estimate reduced uncertainties, which are to be obtained by applying the 2^nd-BERRU-PM methodology for the illustrative example presented in this work. Once the best-estimate calibrated parameters and their accompanying best-estimate reduced uncertainties are obtained, they will enable the subsequent computations, via a corresponding forward computation using the PARTISN-software, that will produce the best-estimate responses values and their accompanying reduced uncertainties. Ongoing research also aims at extending Cacuci’s 2^nd-BERRU-PM [1] [2] methodology to fourth-order, thus enabling the computation of third-order response correlations (skewness) and fourth-order response correlations (kurtosis).

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

Fang, R.X. and Cacuci, D.G. (2023) Second-Order MaxEnt Predictive Modelling Methodology. III: Illustrative Application to a Reactor Physics Benchmark. American Journal of Computational Mathematics, 13, 295-322. https://doi.org/10.4236/ajcm.2023.132015