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.

中文翻译：

---

用于从多个激活箔展开中子能谱的原始对偶算法

在本文中，我们针对非线性病态问题提出了一种鲁棒且有效的原始对偶内点方法，该问题具有相关错误，这些错误是在多个激活箔的中子能谱展开过程中产生的。基于最大熵原理和玻尔兹曼熵公式，展开问题的离散形式等价于计算多面体集的解析中心 $ P = \{x \in R^n \mid Ax = b, x \ge 0 \} $，其中矩阵 $ A \in R^{m\times n} $ 是病态的，并且 $ A $ 和 $ b $ 都不准确。通过一些推导，我们找到了一种新的正则化方法，将问题重新表述为条件良好的问题，这也可以减少 $ A $ 和 $ b $ 中错误的影响。然后基于线性规划的原始对偶内点方法，我们为这个带有错误的病态问题提出了一种混合算法。提供了一组用于学术目的的病态问题的数值结果和两个用于展开中子能谱的实用数据集，以证明所提出方法的有效性和鲁棒性。