广义共轭梯度算法

GENERALIZED CONJUGATE GRADIENT ALGORITHM

在地球物理反问题的求解过程中，共轭梯度（ＣＧ）法是一种经典的、很有价值的主要算法之一。本文在经典的共轭梯度标准基础上，将进一步发展，推导出了求解阻尼最小二乘（ＬＳ）解和最小绝对值偏差（ＬＡＤ）意义下的迭代再加权最小平方（ＩＲＬＳ）解的标准算法形式，从而使得ＣＧ法的应用更具一般性。为了更好地理解ＧＣＧ法的性能，文中还给出了两个例子，并将计算结果与公认的、好的求解病态问题的奇异值分解（ＳＶＤ）算法的计算结果进行了比较，结果表明：ＧＣＧ法亦具有很强的求解病态问题的能力，精度高，且运算速度快。此外，ＧＣＧ法还具有两个显著的特点：①算法简单、编程灵活；②可以保持系数矩阵的稀疏特征。

In the process of solving inverse gaphysical problems, the conjugate gradient (CG) method is one of the classical and valuable main algorithms. Based on classical conjugate gradient standard algorithm, the present paper made further development and deduced the standard algorithm forms for figuring out damping(IRLS)solution under the meaning of least absolute deviation(LAD), which resulted in more generalization of the application of the CG method. This method is called generalized conjugate gradient (GCG) method.In order to render the functions of GCG method more understandable. the paper has given two examples and compared the calculations with the calculations obtained by the generally recognized singular value decompeition(SVD)algorithm for figuring out morbid problems. The results show that the GCG method is characterized by strong capacity for solving morbid problems, high precision and fast operation speed. In addition, the GCG method obviously has two more features:(1) simple algorithm and flexible programming; (2) the capacity for maintaining spare characteristics of the coefficient matrix.