# American Institute of Mathematical Sciences

September  2021, 17(5): 2367-2387. doi: 10.3934/jimo.2020073

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

 1 LESC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China 2 School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China 3 School of Science, Beijing Jiaotong University, Beijing, 100044, China 4 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

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.

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
##### References:
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Friedlander and P. Tseng, Exact regularization of convex programs, SIAM Journal on Optimization, 18 (2008), 1326-1350.  doi: 10.1137/060675320.  Google Scholar [11] G. H. Golub and C. F. Van Loan, Matrix Computations, 3$^{rd}$ edition, The Johns Hopkins University Press, Baltimore, 1996.  Google Scholar [12] C. C. Gonzaga and R. A. Tapia, On the convergence of the Mizuno-Todd-Ye algorithm to the analytic center of the solution set, SIAM Journal on Optimization, 7 (1997), 47-65.  doi: 10.1137/S1052623493243557.  Google Scholar [13] M. D. González-Lima, R. A. Tapia and F. A. Potra, On effectively computing the analytic center of the solution set by primal-dual interior-point methods, SIAM Journal on Optimization, 8 (1998), 1-25.  doi: 10.1137/S1052623495291793.  Google Scholar [14] S. Itoh and T. Tsunoda, Neutron spectra unfolding with maximum entropy and maximum likelihood, Journal of Nuclear Science and Technology, 26 (1989), 833-843.  doi: 10.1080/18811248.1989.9734394.  Google Scholar [15] M. Kojima, N. Megiddo and S. Mizuno, A primal-dual infeasible-interior-point algorithm for linear programming, Mathematical Programming, 61 (1993), 263-280.  doi: 10.1007/BF01582151.  Google Scholar [16] M. Kojima, S. Mizuno and A. Yoshise, A primal-dual interior point algorithm for linear programming, in Progress in Mathematical Programming: Interior-Point and Related Methods (ed. N. Megiddo), Springer, New York, (1989), 29–47. doi: 10.1007/978-1-4613-9617-8_2.  Google Scholar [17] I. J. Lustig, R. E. Marsten and D. F. Shanno, On implementing Mehrotra's predictor-corrector interior-point method for linear programming, SIAM Journal on Optimization, 2 (1992), 435-449.  doi: 10.1137/0802022.  Google Scholar [18] M. 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Ye, On adaptive-step primal-dual interior-point algorithms for linear programming, Mathematics of Operations Research, 18 (1993), 964-981.  doi: 10.1287/moor.18.4.964.  Google Scholar [23] B. Mukherjee, A high-resolution neutron spectra unfolding method using the genetic algorithm technique, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 476 (2002), 247-251.  doi: 10.1016/S0168-9002(01)01440-1.  Google Scholar [24] M. Reginatto, P. Goldhagen and S. Neumann, Spectrum unfolding, sensitivity analysis and propagation of uncertainties with the maximum entropy deconvolution code maxed, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 476 (2002), 242-246.  doi: 10.1016/S0168-9002(01)01439-5.  Google Scholar [25] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, 1970.   Google Scholar [26] C. Roos, T. Terlaky and J.-P. Vial, Interior Point Methods for Linear Optimization, 2$^{nd}$ edition, Springer, Berlin, 2005.  Google Scholar [27] V. Suman and P. Sarkar, Neutron spectrum unfolding using genetic algorithm in a Monte Carlo simulation, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 737 (2014), 76-86.  doi: 10.1016/j.nima.2013.11.012.  Google Scholar [28] S. Tripathy, C. Sunil, M. Nandy, P. Sarkar, D. Sharma and B. Mukherjee, Activation foils unfolding for neutron spectrometry: Comparison of different deconvolution methods, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 583 (2007), 421-425.  doi: 10.1016/j.nima.2007.09.028.  Google Scholar [29] A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Mathematical Programming, 106 (2006), 25-57.  doi: 10.1007/s10107-004-0559-y.  Google Scholar [30] Y. Wang and Y. Yuan, Convergence and regularity of trust region methods for nonlinear ill-posed inverse problems, Inverse Problems, 21 (2005), 821-838.  doi: 10.1088/0266-5611/21/3/003.  Google Scholar [31] Y. Wang, Y. Yuan and H. Zhang, A trust region-CG algorithm for deblurring problem in atmospheric image reconstruction, Science in China Series A: Mathematics, 45 (2002), 731-740.   Google Scholar [32] K. Weise and M. Matzke, A priori distributions from the principle of maximum entropy for the monte carlo unfolding of particle energy spectra, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 280 (1989), 103-112.  doi: 10.1016/0168-9002(89)91277-1.  Google Scholar [33] M. Wright, Ill-conditioning and computational error in interior methods for nonlinear programming, SIAM Journal on Optimization, 9 (1998), 84-111.  doi: 10.1137/S1052623497322279.  Google Scholar [34] S. Wright, Effects of finite-precision arithmetic on interior-point methods for nonlinear programming, SIAM Journal on Optimization, 12 (2001), 36-78.  doi: 10.1137/S1052623498347438.  Google Scholar [35] S. J. Wright, Primal-Dual Interior-Point Methods, SIAM, Philadelphia, PA, 1996. doi: 10.1137/1.9781611971453.  Google Scholar [36] Y. Ye, O. Güler, R. A. Tapia and Y. Zhang, A quadratically convergent $o(\sqrt{n}l)$-iteration algorithm for linear programming, Mathematical Programming, 59 (1993), 151-162.  doi: 10.1007/BF01581242.  Google Scholar [37] Y. Ye, Interior Point Algorithms: Theory and Analysis, John Wiley & Sons, New Jersey, NJ, 1997. doi: 10.1002/9781118032701.  Google Scholar [38] Y. Zhang and R. A. Tapia, On the Convergence of Interior-Point Methods to the Center of Solution Set in Linear Programming, Technical Report TR91-30, Dept. Mathematical Sciences, Rice University, Houston, TX, 1991. Available from: https://www.researchgate.net/publication/235075603_On_the_Convergence_of_Interior-Point_Methods_to_the_Center_of_the_Solution_Set_in_Linear_Programming. doi: 10.1007/BF01581087.  Google Scholar [39] Y. Zhang, On the convergence of a class of infeasible interior-point methods for the horizontal linear complementarity problem, SIAM Journal on Optimization, 4 (1994), 208-227.  doi: 10.1137/0804012.  Google Scholar [40] Y. Zhang, Solving large-scale linear programs by interior-point methods under the matlab environment, Optimization Methods and Software, 10 (1998), 1–31. doi: 10.1080/10556789808805699.  Google Scholar [41] Y. Zhangsun, Unfolding Method Based on Entropy Theory for the Determination of Neutron Spectrum (in Chinese), Master's thesis, Northwest Institute of Nuclear Technology, Xi'an, Shanxi, P. R. China, 2015. Google Scholar

show all references

##### References:
 [1] D. S. Atkinson and P. M. Vaidya, A scaling technique for finding the weighted analytic center of a polytope, Mathematical Programming, 57 (1992), 163-192.  doi: 10.1007/BF01581079.  Google Scholar [2] E. A. Belogorlov and V. P. Zhigunov, Interpretation of the solution to the inverse problem for the positive function and the reconstruction of neutron spectra, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 235 (1985), 146-163.  doi: 10.1016/0168-9002(85)90256-6.  Google Scholar [3] D. P. Bertsekas, Necessary and sufficient conditions for a penalty method to be exact, Mathematical Programming, 9 (1975), 87-99.  doi: 10.1007/BF01681332.  Google Scholar [4] J. F. Bonnans and C. C. Gonzaga, Convergence of interior point algorithms for the monotone linear complementarity problem, Mathematics of Operations Research, 21 (1996), 1-25.  doi: 10.1287/moor.21.1.1.  Google Scholar [5] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004.  doi: 10.1017/CBO9780511804441.  Google Scholar [6] E. J. Candes, J. Romberg and T. Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information, IEEE Transactions on Information Theory, 52 (2006), 489-509.  doi: 10.1109/TIT.2005.862083.  Google Scholar [7] X. Chen, Z. Lu and T. K. Pong, Penalty methods for a class of non-lipschitz optimization problems, SIAM Journal on Optimization, 26 (2016), 1465-1492.  doi: 10.1137/15M1028054.  Google Scholar [8] G. Cowan, A Survey of Unfolding Methods for Particle Physics, 2002. Available from: https://www.ippp.dur.ac.uk/old/Workshops/02/statistics/proceedings/cowan.pdf. Google Scholar [9] F. Z. Dehimi, A. Seghour and S. E. H. Abaidia, Unfolding of neutron energy spectra with fisher regularisation, IEEE Transactions on Nuclear Science, 57 (2010), 768-774.  doi: 10.1109/TNS.2010.2041791.  Google Scholar [10] M. P. Friedlander and P. Tseng, Exact regularization of convex programs, SIAM Journal on Optimization, 18 (2008), 1326-1350.  doi: 10.1137/060675320.  Google Scholar [11] G. H. Golub and C. F. Van Loan, Matrix Computations, 3$^{rd}$ edition, The Johns Hopkins University Press, Baltimore, 1996.  Google Scholar [12] C. C. Gonzaga and R. A. Tapia, On the convergence of the Mizuno-Todd-Ye algorithm to the analytic center of the solution set, SIAM Journal on Optimization, 7 (1997), 47-65.  doi: 10.1137/S1052623493243557.  Google Scholar [13] M. D. González-Lima, R. A. Tapia and F. A. Potra, On effectively computing the analytic center of the solution set by primal-dual interior-point methods, SIAM Journal on Optimization, 8 (1998), 1-25.  doi: 10.1137/S1052623495291793.  Google Scholar [14] S. Itoh and T. Tsunoda, Neutron spectra unfolding with maximum entropy and maximum likelihood, Journal of Nuclear Science and Technology, 26 (1989), 833-843.  doi: 10.1080/18811248.1989.9734394.  Google Scholar [15] M. Kojima, N. Megiddo and S. Mizuno, A primal-dual infeasible-interior-point algorithm for linear programming, Mathematical Programming, 61 (1993), 263-280.  doi: 10.1007/BF01582151.  Google Scholar [16] M. Kojima, S. Mizuno and A. Yoshise, A primal-dual interior point algorithm for linear programming, in Progress in Mathematical Programming: Interior-Point and Related Methods (ed. N. Megiddo), Springer, New York, (1989), 29–47. doi: 10.1007/978-1-4613-9617-8_2.  Google Scholar [17] I. J. Lustig, R. E. Marsten and D. F. Shanno, On implementing Mehrotra's predictor-corrector interior-point method for linear programming, SIAM Journal on Optimization, 2 (1992), 435-449.  doi: 10.1137/0802022.  Google Scholar [18] M. Matzke, Propagation of uncertainties in unfolding procedures, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 476 (2002), 230–241. doi: 10.1016/S0168-9002(01)01438-3.  Google Scholar [19] S. Mehrotra, On the implementation of a primal-dual interior point method, SIAM Journal on Optimization, 2 (1992), 575-601.  doi: 10.1137/0802028.  Google Scholar [20] S. Mehrotra, Quadratic convergence in a primal-dual method, Mathematics of Operations Research, 18 (1993), 741-751.  doi: 10.1287/moor.18.3.741.  Google Scholar [21] S. Mizuno, Polynomiality of infeasible-interior-point algorithms for linear programming, Mathematical Programming, 67 (1994), 109-119.  doi: 10.1007/BF01582216.  Google Scholar [22] S. Mizuno, M. J. Todd and Y. Ye, On adaptive-step primal-dual interior-point algorithms for linear programming, Mathematics of Operations Research, 18 (1993), 964-981.  doi: 10.1287/moor.18.4.964.  Google Scholar [23] B. Mukherjee, A high-resolution neutron spectra unfolding method using the genetic algorithm technique, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 476 (2002), 247-251.  doi: 10.1016/S0168-9002(01)01440-1.  Google Scholar [24] M. Reginatto, P. Goldhagen and S. Neumann, Spectrum unfolding, sensitivity analysis and propagation of uncertainties with the maximum entropy deconvolution code maxed, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 476 (2002), 242-246.  doi: 10.1016/S0168-9002(01)01439-5.  Google Scholar [25] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, 1970.   Google Scholar [26] C. Roos, T. Terlaky and J.-P. Vial, Interior Point Methods for Linear Optimization, 2$^{nd}$ edition, Springer, Berlin, 2005.  Google Scholar [27] V. Suman and P. Sarkar, Neutron spectrum unfolding using genetic algorithm in a Monte Carlo simulation, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 737 (2014), 76-86.  doi: 10.1016/j.nima.2013.11.012.  Google Scholar [28] S. Tripathy, C. Sunil, M. Nandy, P. Sarkar, D. Sharma and B. Mukherjee, Activation foils unfolding for neutron spectrometry: Comparison of different deconvolution methods, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 583 (2007), 421-425.  doi: 10.1016/j.nima.2007.09.028.  Google Scholar [29] A. Wächter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Mathematical Programming, 106 (2006), 25-57.  doi: 10.1007/s10107-004-0559-y.  Google Scholar [30] Y. Wang and Y. Yuan, Convergence and regularity of trust region methods for nonlinear ill-posed inverse problems, Inverse Problems, 21 (2005), 821-838.  doi: 10.1088/0266-5611/21/3/003.  Google Scholar [31] Y. Wang, Y. Yuan and H. Zhang, A trust region-CG algorithm for deblurring problem in atmospheric image reconstruction, Science in China Series A: Mathematics, 45 (2002), 731-740.   Google Scholar [32] K. Weise and M. Matzke, A priori distributions from the principle of maximum entropy for the monte carlo unfolding of particle energy spectra, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 280 (1989), 103-112.  doi: 10.1016/0168-9002(89)91277-1.  Google Scholar [33] M. Wright, Ill-conditioning and computational error in interior methods for nonlinear programming, SIAM Journal on Optimization, 9 (1998), 84-111.  doi: 10.1137/S1052623497322279.  Google Scholar [34] S. Wright, Effects of finite-precision arithmetic on interior-point methods for nonlinear programming, SIAM Journal on Optimization, 12 (2001), 36-78.  doi: 10.1137/S1052623498347438.  Google Scholar [35] S. J. Wright, Primal-Dual Interior-Point Methods, SIAM, Philadelphia, PA, 1996. doi: 10.1137/1.9781611971453.  Google Scholar [36] Y. Ye, O. Güler, R. A. Tapia and Y. Zhang, A quadratically convergent $o(\sqrt{n}l)$-iteration algorithm for linear programming, Mathematical Programming, 59 (1993), 151-162.  doi: 10.1007/BF01581242.  Google Scholar [37] Y. Ye, Interior Point Algorithms: Theory and Analysis, John Wiley & Sons, New Jersey, NJ, 1997. doi: 10.1002/9781118032701.  Google Scholar [38] Y. Zhang and R. A. Tapia, On the Convergence of Interior-Point Methods to the Center of Solution Set in Linear Programming, Technical Report TR91-30, Dept. Mathematical Sciences, Rice University, Houston, TX, 1991. Available from: https://www.researchgate.net/publication/235075603_On_the_Convergence_of_Interior-Point_Methods_to_the_Center_of_the_Solution_Set_in_Linear_Programming. doi: 10.1007/BF01581087.  Google Scholar [39] Y. Zhang, On the convergence of a class of infeasible interior-point methods for the horizontal linear complementarity problem, SIAM Journal on Optimization, 4 (1994), 208-227.  doi: 10.1137/0804012.  Google Scholar [40] Y. Zhang, Solving large-scale linear programs by interior-point methods under the matlab environment, Optimization Methods and Software, 10 (1998), 1–31. doi: 10.1080/10556789808805699.  Google Scholar [41] Y. Zhangsun, Unfolding Method Based on Entropy Theory for the Determination of Neutron Spectrum (in Chinese), Master's thesis, Northwest Institute of Nuclear Technology, Xi'an, Shanxi, P. R. China, 2015. Google Scholar
The standard spectrum and the spectrum solved by PDUP for "data1" in logarithmic coordinates
The spectrums solved by SAND-II and PDUP for "data2" in logarithmic coordinates
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
 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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