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September 2021, 17(5): 2367-2387. doi: 10.3934/jimo.2020073
A Primal-dual algorithm for unfolding neutron energy spectrum from multiple activation foils
Yu-Hong Dai 1,2, , Zhouhong Wang 3,, and Fengmin Xu 4,
LESC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
School of Science, Beijing Jiaotong University, Beijing, 100044, China
School of Economics and Finance, Xian Jiaotong University, Xi'an, 710061, China
* Corresponding author: Zhouhong Wang
Received February 2018 Revised August 2019 Published September 2021 Early access April 2020
Fund Project: This work was partly supported by the Chinese NSF grants (nos. 11631013, 11991021, 11971372 and 11991020) and partly supported by the CSC scholarship and Beijing Academy of Artificial Intelligence (BAAI). The authors are grateful to the editor and the referees for their valuable comments and suggestions
Figure(2) / Table(1)
In this paper we propose a robust and efficient primal-dual interior-point method for a nonlinear ill-conditioned problem with associated errors which are arising in the unfolding procedure for neutron energy spectrum from multiple activation foils. Based on the maximum entropy principle and Boltzmann's entropy formula, the discrete form of the unfolding problem is equivalent to computing the analytic center of the polyhedral set $ P = \{x \in R^n \mid Ax = b, x \ge 0\} $, where the matrix $ A \in R^{m\times n} $ is ill-conditioned, and both $ A $ and $ b $ are inaccurate. By some derivations, we find a new regularization method to reformulate the problem into a well-conditioned problem which can also reduce the impact of errors in $ A $ and $ b $. Then based on the primal-dual interior-point methods for linear programming, we propose a hybrid algorithm for this ill-conditioned problem with errors. Numerical results on a set of ill-conditioned problems for academic purposes and two practical data sets for unfolding the neutron energy spectrum are presented to demonstrate the effectiveness and robustness of the proposed method.
Keywords: neutron spectrum unfolding, Boltzmann's entropy, ill-condition, numerical stability, interior point algorithm, analytic center.
Mathematics Subject Classification: Primary: 65K05, 90C51; Secondary: 65F22.
Citation: Yu-Hong Dai, Zhouhong Wang, Fengmin Xu. A Primal-dual algorithm for unfolding neutron energy spectrum from multiple activation foils. Journal of Industrial & Management Optimization, 2021, 17 (5) : 2367-2387. doi: 10.3934/jimo.2020073
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Figure 1. The standard spectrum and the spectrum solved by PDUP for "data1" in logarithmic coordinates
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Figure 2. The spectrums solved by SAND-II and PDUP for "data2" in logarithmic coordinates
Table 1. Numerical Results for $ A_m = [H_m, H_m], b = H_m e $
m Algo. No. $ f $ $ g $ Opt. Time(s)
10 linprog-1 6 $ +\infty $ 4.110e-07 Y 0.235
linprog-2 5 $ +\infty $ 1.421e-09 Y 0.243
fmincon 39 13.863 1.286e-13 Y 1.392
PDUP 11 13.863 8.674e-09 Y 0.071
20 linprog-1 1001 — — N 0.285
100 linprog-1 337 — — N 1.062
fmincon 13 138.629 1.483e-13 Y 1.117
PDUP 18 138.629 1.096e-10 Y 0.390
300 linprog-1 1001 — — N 41.905
linprog-2 19 — — N 1.401
fmincon 156 415.888 3.855e-13 N 166.530
500 linprog-1 676 — — N 149.613
linprog-2 8 — — N 3.548
fmincon 77 693.147 6.652e-13 P 385.377
PDUP 24 693.147 6.288e-11 Y 11.828
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