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A Primal-dual algorithm for unfolding neutron energy spectrum from multiple activation foils

  • * Corresponding author: Zhouhong Wang

    * Corresponding author: Zhouhong Wang 

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

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  • 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.

    Mathematics Subject Classification: Primary: 65K05, 90C51; Secondary: 65F22.

    Citation:

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  • Figure 1.  The standard spectrum and the spectrum solved by PDUP for "data1" in logarithmic coordinates

    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
    linprog-2 5 $ +\infty $ 1.101e-08 Y 0.266
    fmincon 15 27.726 1.528e-13 Y 0.763
    PDUP 13 27.726 3.589e-11 Y 0.092
    50 linprog-1 1001 N 0.820
    linprog-2 6 $ +\infty $ 6.305e-08 Y 0.241
    fmincon 14 69.315 2.336e-13 Y 1.021
    PDUP 16 69.315 1.780e-10 Y 0.127
    100 linprog-1 337 N 1.062
    linprog-2 5 $ +\infty $ 1.299e-07 Y 0.246
    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
    PDUP 22 415.888 2.379e-11 Y 3.600
    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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