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

Yu-Hong Dai; Zhouhong Wang; Fengmin Xu

doi:10.3934/jimo.2020073

Article Contents

2021, Volume 17, Issue 5: 2367-2387. Doi: 10.3934/jimo.2020073

This issue Previous Article Principal component analysis with drop rank covariance matrix Next Article Sustainable closed-loop supply chain network optimization for construction machinery recovering

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
^* Corresponding author: Zhouhong Wang

Received: January 31, 2018

Revised: July 31, 2019

Early access: April 2020

Published: September 2021

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

Abstract Full Text(HTML) Figure(2) / Table(1) Related Papers Cited by

Abstract

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:

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

Citation:

Full Text(HTML)

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

Download: Full-size image PowerPoint slide

Figure 2. The spectrums solved by SAND-II and PDUP for "data2" in logarithmic coordinates

Download: Full-size image PowerPoint slide

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

| Show Table

DownLoad: CSV

Related Papers

Cited by

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.
[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.
[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.
[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.
[5]	S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511804441.
[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.
[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.
[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.
[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.
[10]	M. P. Friedlander and P. Tseng, Exact regularization of convex programs, SIAM Journal on Optimization, 18 (2008), 1326-1350. doi: 10.1137/060675320.
[11]	G. H. Golub and C. F. Van Loan, Matrix Computations, 3$^{rd}$ edition, The Johns Hopkins University Press, Baltimore, 1996.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[21]	S. Mizuno, Polynomiality of infeasible-interior-point algorithms for linear programming, Mathematical Programming, 67 (1994), 109-119. doi: 10.1007/BF01582216.
[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.
[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.
[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.
[25]	R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, 1970.
[26]	C. Roos, T. Terlaky and J.-P. Vial, Interior Point Methods for Linear Optimization, 2$^{nd}$ edition, Springer, Berlin, 2005.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[35]	S. J. Wright, Primal-Dual Interior-Point Methods, SIAM, Philadelphia, PA, 1996. doi: 10.1137/1.9781611971453.
[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.
[37]	Y. Ye, Interior Point Algorithms: Theory and Analysis, John Wiley & Sons, New Jersey, NJ, 1997. doi: 10.1002/9781118032701.
[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.
[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.
[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.
[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.