病态问题奇异值截断阈值算法研究

Research on Truncated Singular Value Method for Ill Conditioned Least Squares Problem

针对截断奇异值方法中截断阈值的选择问题,提出了一种选择最优截断阈值的两步法,用于求解病态最小二乘问题,获得稳定可靠的解.分析了现有方法确定最优截断阈值的依据,指出了其中可能存在的不足之处:在确定截断阈值的过程中始终将残差范数作为主要方面予以考虑,这并不十分符合最优解的特征.以此为基础,提出一种确定最优截断阈值的两步法,将各级解的范数和相应的残差范数分步予以考虑,第一步只考虑残差范数,计算出残差界限值,排除那些残差范数大于界限值的解,从而获得一个小范围的最优解备选集合.第二步只考虑备选集合中各级解本身的范数,根据解的范数的稳定性来确定最优解所对应的截断阈值,稳定性指标最小的解即为最优解.数值算例结果表明,所提截断阈值算法比现有方法更加合理可靠.

In this paper,a two-step method is proposed to select the optimal truncation threshold in the singular value truncation method.The theory basis of the existing methods to determine the optimal solution are firstly discussed and the possible shortcomings are analyzed.The reason may be the residual norm is always considered as the main aspect in the process of determining the truncation threshold,which is not very consistent with the characteristics of the optimal solution.On this basis,a two-step method is proposed to determine the optimal truncation threshold.The norm of each level solution and the corresponding residual norm are considered step by step.In the first step,only the residual norm is considered.The residual limit value is calculated firstly and then those solutions whose residual norm is greater than the limit value are excluded.As a result,a small range alternative set of optimal solutions can be obtained.In the second step,only the norm of each solution in the alternative set are considered.The cut-off threshold of the optimal solution is determined by the stability of the solution norm.The solution whose stability index is the smallest is the optimal solution.Two numerical examples are used to illustrate the proposed method.The results show that the proposed method is reasonable and feasible.