Fourth-Order Adjoint Sensitivity Analysis of an OECD/NEA Reactor Physics Benchmark: I. Mathematical Expressions and CPU-Time Comparisons for Computing 1<sup>st</sup>-, 2<sup>nd</sup>- and 3<sup>rd</sup>-Order Sensitivities

Abstract

This work extends to fourth-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 7477 imprecisely known (uncertain) model parameters which have nonzero values. These parameters are as follows: 180 microscopic total cross sections; 7101 microscopic scattering sections; 60 microscopic fission cross sections; 60 parameters that characterize the average number of neutrons per fission; 60 parameters that characterize the fission spectrum; 10 parameters that characterize the fission source; and 6 parameters that characterize the isotope number densities. Previous works have used the adjoint sensitivity analysis methodology to compute exactly and efficiently all of the 7477 first-order and 27,956,503 second-order sensitivities of the PERP benchmark's leakage response to all of the benchmark's uncertain parameters. These works showed that largest response sensitivities involve the total microscopic cross sections, which motivated the recent computation of all of the (180) 3 third-order sensitivities of the PERP leakage response with respect to these total microscopic cross sections. It turned out that some of these 3 rd -order cross sections were far larger than the corresponding 2 nd -order ones, thereby having the largest impact on the uncertainties induced in the PERP benchmark's response. This finding has motivated the development of the original 4 th -order formulas presented in this work, which are valid not only for the PERP benchmark but can also be used for computing the 4 th -order sensitivities of response of any nuclear sys-How to cite this paper: Cacuci, D.G. and Fang, R.X. (2021) Fourth-Order Adjoint Sensitivity Analysis of an OECD/NEA Reactor Physics Benchmark: I. Mathematical Expressions and CPU-Time Comparisons for Computing 1 st -, 2 nd -and 3 rd -Order Sensitivities.

Fourth-Order Adjoint Sensitivity and Uncertainty Analysis of an OECD/NEA Reactor Physics Benchmark: I. Computed Sensitivities

“…, j3, of the PERP leakage response with respect to the benchmark's microscopic total cross sections are computed by applying the 4th-Order Comprehensive Sensitivity Analysis Methodology (4th-CASAM). The mathematical expressions used for the computations in this work have been derived in [13] and are reproduced below, for convenience: (1) ;ψ (2) ;ψ (3) ;ψ (4)…”

“…The dagger in Equation ( 3) denotes "transposition", σ g t,i denotes the group-averaged microscopic total cross section for isotope i and energy group g, and JTX denotes the total number of microscopic total cross sections. (iii) The vectors ψ (1) ; ψ (2) ; ψ (3) ; ψ (4) denote the 1st-level, 2nd-level, 3rd-level, and 4thlevel adjoint functions, respectively. Specifically:…”

“…The uniquely advantageous features of the 2nd-ASAM have been demonstrated by applying this methodology to a polyethylene-reflected plutonium (acronym: PERP) OECD/NEA reactor physics benchmark [2]. As has been described in [3,4], the numerical model of the PERP benchmark comprises 21,976 uncertain parameters, of which 7477 parameters have non-zero values. These non-zero parameters are as follows: 180 group-averaged microscopic total cross sections; 7101 non-zero group-averaged microscopic scattering cross sections; 120 fission process parameters; 60 fission spectrum parameters; 10 parameters describing the experiment's nuclear sources; and 6 isotopic number densities.…”

“…Since the results obtained in [4][5][6][7][8][9] indicated that the effects of the 2nd-order sensitivities of the PERP leakage response to the total microscopic group cross sections are much larger than the effects of the corresponding 1st-order sensitivities, Cacuci and Fang [10][11][12] have extended the concepts underlying the 2nd-ASAM in order to compute the (180) 3 3rd-order sensitivities of the PERP benchmark's total leakage response with respect to the microscopic total cross sections. It turned out that some of these 3rd-order sensitivities were far larger than the corresponding 2nd-order ones, thereby having the largest impact on the uncertainties induced in the PERP benchmark's leakage response.…”

“…Section 2 also presents a comparison of the 4th-order unmixed sensitivities to the corresponding 1st-, 2nd-and 3rd-order ones. Furthermore, Section 2 also reports and analyzes the numerical results for the 360 fourth-order mixed sensitivities that correspond to the largest 3rd-order unmixed sensitivity S (3) and the largest 3rd-order mixed sensitivity S (3) σ 30 t,1 , σ 30 t,6 , σ 30 t, 6 . Section 3 presents the verification of the 4th-order mixed sensitivities obtained in this work by using their symmetry properties and also by using fourth-order finite-difference formulas.…”