病态问题解算的直接正则化方法比较

Comparison of Direct Regularization Methods of Ill-conditioned Problems Solution

为了解算病态问题,需正确选择适合的正则化方法,为此分析了截断奇异值法和Tik-honov正则化方法的异同点。在此基础上,阐述了L曲线法和GCV法确定最优正则化参数的基本原理。通过数值算例分析表明:截断奇异值法和Tikhonov法可以有效消除观测方程的病态性;利用L曲线法和GCV法不仅可以对Tikhonov方法中的连续正则化参数进行合理确定,而且还可以准确确定截断奇异值法中的离散正则化参数。最后,比较研究了四种组合方法的正则化解的精度和稳健性。

In order to resolve ill-conditioned problems,suitable regularization methods are needed to choose correctly.The characteristics of truncated singular value decomposition（TSVD） method and Tikhonov regularization method are discussed respectively.On the basis,L-curve method and generalized cross validation（GCV） method are both employed to attain the optimal regularization parameters.Numerical results show that TSVD method and Tikhonov method can eliminate ill-condition of observation equation effectively.Through applying to L-curve method and GCV method,continuous regularization parameter for Tikhonov method can be confirmed reasonably.Furthermore,discrete regularization parameter for TSVD method can be determined accurately.Finally,the accuracy and robustness of regularization solution for four combined methods are investigated.