求解离散不适定问题的正则化GMERR方法

迭代极小残差方法是求解大型线性方程组的常用方法, 通常用残差范数控制迭代过程.但对于不适定问题, 即使残差范数下降, 误差范数未必下降. 对大型离散不适定问题,组合广义最小误差(GMERR)方法和截断奇异值分解(TSVD)正则化方法, 并利用广义交叉校验准则(GCV)确定正则化参数,提出了求解大型不适定问题的正则化GMERR方法.数值结果表明, 正则化GMERR方法优于正则化GMRES方法.

A REGULARIZING GMERR METHOD FOR SOLVING DISCRETE ILL-POSED PROBLEMS

The iterative minimum-residual methods for solving large-scale linear systems are usually controlled by the norm of the residual. However, the errors do not necessarily decrease while the residuals decrease for ill-posed problems. Combining the generalized minimal error (GMERR) method with the truncated singular value decomposition (TSVD) regularization, and using the generalized cross validation (GCV) for determining the regularization parameter, we present the regularizing GMERR method for solving discrete ill-posed problem. Numerical results show that the regularizing GMERR method is superior to the regularizing GMRES method.