Third-Order Adjoint Sensitivity Analysis of an OECD/NEA Reactor Physics Benchmark: I. Mathematical Framework

This work extends to third-order previously published work on developing the adjoint sensitivity and uncertainty analysis of the numerical model of a polyethylene-reflected plutonium (acronym: PERP) OECD/NEA reactor physics benchmark. The PERP benchmark comprises 21,976 imprecisely known (uncertain) model parameters. Previous works have used the adjoint sensitivity analysis methodology to compute exactly and efficiently all of the 21,976 first-order and (21,976) 2 second-order sensitivities of the PERP benchmark’s leakage response to all of the benchmark’s uncertain parameters, showing that the largest and most consequential 1 st - and 2 nd -order response sensitivities are with respect to the total microscopic cross sections. These results have motivated extending the previous adjoint-based derivations to third-order, leading to the derivation, in this work, of the exact mathematical expressions of the (180) 3 third-order sensitivities of the PERP leakage response with respect to these total microscopic cross sections. The formulas derived in this work are valid not only for the PERP benchmark but can also be used for computing the 3 rd -order sensitivities of the leakage response of any nuclear system involving fissionable material and internal or external neutron sources. Subsequent works will use the adjoint-based mathematical expressions obtained in this work to compute exactly and efficiently the numerical values of these (180) 3 third-order sensitivities (which turned out to be very large and consequential) and use them for a third-order uncertainty analysis of the PERP benchmark’s leakage response.


Introduction
Until recently, only the first-order sensitivities (i.e., functional derivatives) of a computational model's responses (i.e., quantities of interest) to the respective model's imprecisely known (i.e., uncertain) parameters have been taken into account when assessing the uncertainties induced in the respective responses by the parameter uncertainties. The second-and higher-order sensitivities could not be computed, except for very simple models comprising a handful of parameters, so these sensitivities were ignored. The Second-Order Adjoint Sensitivity Analysis Methodology (2 nd -ASAM) recently conceived by Cacuci [1] is the only practical method that enables the exact computation of the large number of 2 nd -order sensitivities arising in large-scale problems comprising many parameters. The application of the 2 nd -ASAM to a multiplying nuclear system with source [2] [3] [4] has opened the way for the large-scale application presented in [5]- [10] to a polyethylene-reflected plutonium (acronym: PERP) OECD/NEA reactor physics benchmark [11]. The numerical model of the PERP benchmark includes 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.
All of the 21,976 first-order sensitivities and (21,976) 2 second-order sensitivities of the PERP leakage response with respect to the benchmark's parameters were computed, ranked, and analyzed in [5]- [10]. The results obtained in [5]- [10] showed that the contributions stemming from the second-order sensitivities of the leakage response with respect to the group-averaged microscopic total cross sections are the largest, by a significant margin, by comparison to the contributions from the other uncertain model parameters, including the number of densities, fission parameters, microscopic scattering cross sections, source parameters, etc. For the extreme case of fully correlated microscopic total cross sections, for example, neglecting the 2 nd -order sensitivities of the leakage response with respect to the total cross sections would cause an error as large as 2000% in the expected value of the leakage response and up to 6000% in the variance of the leakage response [5]. Given that the effects of these 2 nd -order sensitivities are much larger than the effects of the 1 st -order sensitivities [5] [10], it is logical to posit the question of quantifying the magnitudes and contributions that would stem from the 3 rd -order sensitivities of the PERP benchmark's total leakage response with respect to the microscopic total cross sections. To enable to compute such 3 rd -order sensitivities, Cacuci [12] has recently conceived the "third-order adjoint sensitivity analysis methodology for reaction rate responses in a multiplying nuclear system with source." Cacuci's results [12] are applied in this work to the PERP benchmark in order to derive the exact analytical expressions of the 3 rd -order sensitivities of the PERP benchmark's leakage response with respect to this benchmark's microscopic total cross sections. Two subsequent works, designated as Part II [13] and Part III [14], respectively, will report numerical results as follows: 1) Part II [13] will present the numerical values the 3 rd -order sensitivities derived in the present work, showing that the largest of these is over 400 times larger than the largest 2nd-order sensitivity and is over 20,000 times larger than the largest 1st-order sensitivity; and 2) Part III [14] will quantify the effects of the 3 rd -order sensitivities on the expected values, the standard deviation and the skewness of the PERP's leakage response, and will compare these effects to those produced by the 1 st -order and, respectively, 2 nd -order sensitivities.
This work is organized as follows: Section 2 describes the methodology for computing the leakage response of the PERP benchmark. Section 3 presents the derivation of the exact analytical expressions of the third-order mixed sensitivities of the PERP leakage response to total cross sections. Section 4 concludes this work.

Mathematical Methodology for Computing the Leakage Response of the Polyethylene-Reflected Plutonium (PERP) Metal Sphere Benchmark
This section presents the derivation of the exact analytical expressions of the 3 rd -order sensitivities of the polyethylene-reflected plutonium (acronym: PERP) metal sphere OECD/NEA benchmark [11] total leakage response with respect to this benchmark's total cross sections, thus continuing the work presented in [5]- [10]. The numerical modeling of the PERP benchmark has been already described in [5]- [10] but, for convenient reference, the materials of the PERP spherical core and spherical-shell reflector, respectively, are specified in Table   A1 in the Appendix. As in [5]- [10], the multigroup discrete ordinates particle transport code PARTISN [15] together with neutron sources computed using the code SOURCES4C [16] have been employed to perform the numerical computations of the various quantities to be derived in this section. The multigroup neutron fluxes computed by PARTISN are the solutions of the following multigroup neutron transport equation with a spontaneous fission source: The Boltzmann-operator ( ) g B α in Equation (3) contains implicitly a factor 1 4π in its scattering and fission terms, to conform to the convention used by PARTISN [15]. As in [5]- [10], the PARTISN [15] computations used the MENDF71X [17] 618-group cross sections collapsed to 30 energy-groups, as shown in Table A2 in the Appendix, in conjunction with a P 3 -Legendre expansion of the scattering cross section, an angular quadrature of S 256 , 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 symbols used in Equations (1) through (4) have their usual meanings and are summarized, for convenient referencing, in the Appendix. The mathematical expression of the PERP benchmark's leakage response, denoted as ( ) L α , is provided below: The vector α , which appears in the expressions of the Boltzmann-operator ( ) g B α and the leakage response ( ) L α , in Equation (3) and Equation (5), respectively, represents the "vector of imprecisely known model parameters", and is defined as follows: The components of the vector α are described in the Appendix. Since only the effects of the uncertainties in the total macroscopic cross sections will be considered in this work, only the components of the vector t for the parameters of macroscopic total cross sections will be explicitly used; they are reproduced below from the Appendix: , , , , ; , , ; , , where † † 1 2 1 1 , 1 In Equations (7) through (9), the dagger denotes "transposition," , g t i σ denotes the microscopic total cross section for isotope i and energy group g, , will be considered in this work. Thus, the numbers of sensitivities of the PERP leakage response with respect to the total microscopic cross sections are as follows: 1) 180 first-order sensitivities; 2) 32,400 second-order sensitivities, of which 16,290 are distinct; and 3) 5,832,000 third-order sensitivities, of which 988,260 are distinct. The 180 first-order and 16,290 second-order sensitivities were obtained and analyzed in [5]. The exact mathematical expressions of the 988,260 distinct third-order sensitivities will be obtained in Section 3 of this work, and their numerical values and numerical effects on the variance and skewness of the PERP leakage response will be presented in the accompanying Part II [13] and Part III [14].

Exact Analytical Expressions of the Third-Order Mixed Sensitivities of the PERP Leakage Response to Total Cross Sections
This Section will present the derivation of the 3 rd -order mixed sensitivities ( ) 3 , , , 1, , of the PERP leakage response with respect to the group-averaged microscopic total cross sections. These 3 rd -order sensitivities will be derived by using two alternative procedures, which will be presented in sections 3.1 and 3.2, respectively. In section 3.1, the sensitivities ( ) 3 , , , 1, , will be obtained as a particular case of the general expressions derived in [12]. In section 3.2, the expressions of ( ) 3 , , , 1, , by applying the concepts presented in [12] to the expression of the 2 nd -order sensitivities which were derived in Ref. [5]. It will be shown that these two alternative procedures will yield identical expressions for the corresponding 3 rd -order sensitivities, as would be expected.

Particularizing the General Expressions Obtained in Ref. [12] for the PERP Leakage Response Sensitivities to Total Cross Sections
The general expression of the 3 rd -order sensitivities of a reaction-rate type response to the nuclear data characterizing a physical system modeled by the multigroup neutron transport equation has been derived in the Appendix of [12]. This general expression can be specialized for the PERP benchmark's leakage response and total microscopic cross sections by introducing the following correspondences: 1, , , ; , , , for 1, , ; 1, , ; 1, , , where the inner product Ω Ω Ω , is defined as follows: In Equation (14), the 3 rd -level adjoint functions are the solutions of the following multigroup form of the "third-level adjoint sensitivity system" (3 rd -LASS) presented in the Appendix of [12]: , ; , ,  3 ,  2 ,  3 ,  3 ,  1  1 2  41  4  1 2 , ; , , ; , , and where the various quantities serving as "sources" on the right-sides of the 3 rd -LASS are defined as follows: ; , , ; , ,   .
The particular forms taken on by the general expressions given Equations (25) through (33) when considering solely the group-averaged microscopic total cross sections are obtained specializing these expressions to the particular cases Thus, the expression in Equation (25) becomes: , , . 0, for , .
i k In view of Equation (36), the expression in Equation (28) becomes: The expression in Equation (31)    ; , d The expression in Equation (33) becomes: Next, each of the quantities on the right-side of Equation (14) will be specialized to obtain the specific expression for computing the 3 rd -order sensitivities of the leakage response in the PERP benchmark to the 180 total cross sections , , 1, , 6; 1, , 30 defined in Equation (8). Thus, in view of Equation (10), it follows that the first term on the right-side of Equation (14) vanishes, since ( ) ( ) In view of Equation (35), the second term on the right-side of Equation (14) vanishes, i.e., In view of Equation (35), the third term on the right-side of Equation (14) In view of Equation (36), the fourth term on the right-side of Equation (14) vanishes, i.e., American Journal of Computational Mathematics In view of Equation (34), the fifth term on the right-side of Equation (14) takes on the following particular form: In view of Equation (36), the sixth term on the right-side of Equation (14) takes on the following particular form: , ; , , , ; , , , , The seventh term on the right-side of Equation (14) takes on the following particular form: The result obtained in Equation (64) ; , ; , Ω Ω , and from the fact that the second term on the right side of Equation (64) vanishes, as shown below: ; , d , The eighth term on the right-side of Equation (14) takes on the following particular form: Collecting the results obtained in Equations (57)-(63), and (66), and replacing them in Equation (14) yields the following expression: In Ref. [5], the specific 2 nd -order sensitivities of the PERP leakage response to the group-averaged microscopic total cross sections were shown to have the following expression: where the first-level adjoint functions The total G-differential of Equation (70) provides the 3 rd -order sensitivities of the leakage response involving the 2 nd -order derivatives to the total cross sections. Since this work is limited to computing the sensitivities of the PERP leakage response solely with respect to the group-averaged microscopic total cross sections, it follows that the sensitivities ( ) 3 , , , 1, , are obtained by taking the following G-differential of Equation (70), limited to variations just in the group-averaged total microscopic cross sections, namely: The source ( ) ( ) 1 , , ; g Q ϕ δ α α on the right side in Equation (72) is defined as follows: The 3 rd -Level Forward Sensitivity System (3 rd -LFSS) comprises Equations (72) through (79). The 3 rd -Level Adjoint Sensitivity System (3 rd -LASS) that corresponds to the 3 rd -LFSS is derived by following the general procedure described in Ref. [12], which involves the following sequence of operations: A. Use the definition of the inner product provided in Equation (15) to perform the following operations: 1) Form the inner product of Equation (72) with a yet undefined vector-valued function 2) Form the inner product of Equation (74) with a yet undefined vector-valued function 3) Form the inner product of Equation (76) with a yet undefined vector-valued function