基于SVD求解病态线性方程组的正则因子分步选取方法

Stepwise selection method of regularization factors for solving ill-posed linear equations based on SVD

为提高奇异值分解法求解病态线性方程组的有效性,研究了基于奇异值分解求解病态线性方程组的正则化因子分步选取方法。首先基于奇异值分解求解病态线性方程组构建滤波正则化方程,根据正则化因子序列求出正则解范数和正则解残差范数的L-曲线,基于L-曲线的局部特征寻找候选角点,再从候选角点中确定最佳角点,进而得到最佳正则化因子。通过对希尔伯特方程组的求解,验证了本算法的有效性。

In order to improve the effectiveness of solving ill-posed problems using the singular value decomposition method,the Stepwise selection method of the regularization factor based on the singular value decomposition to solve the ill-posed linear equations is studied in this paper.Firstly,the ill-posed linear equations is solved to construct a filtered regularization equation based on the singular value decomposition.Then the L-curve of the regular solution norm and the regular solution residual norm according to the regularization factor sequence is obtained.Meanwhile,the candidate corner points are searched according to the local inflection point feature of the L-curve.Furthermore,we determine the global optimal corner point of the L-curve from the candidate corner points,and obtain the optimal regularization factor.Finally,it is substituted into the filtered regularization equation to obtain the global optimal solution of the ill-posed problem.The effectiveness of this algorithm is verified by solving the Hilbert equations.