Nonlinear implicit iterative method for solving nonlinear ill-posed problems

In the paper, we extend the implicit iterative method for linear ill-posed operator equations to solve nonlinear ill-posed problems. We show that under some conditions the error sequence of solutions of the nonlinear implicit iterative method is monotonically decreasing and, with this monotonicity, prove convergence of the new method for both the exact and perturbed equations.     

一种广义插值法

本文考虑一种广义插值问题，插值条件为小区间上的积分值，以弥补现有的插值方法在L2空间不再适用的不足，除了多项式插值外，还讨论了两种一次样条插值方法。     

求解非线性不适定问题的隐式迭代法

将处理线性不适定算子方程的隐式迭代法推广到非线性不适定问题,证明了迭代解误差序列的单调性,并进一步利用迭代误差的单调性得出求解非线性不适定问题隐式迭代法对精确方程和扰动方程的收敛性.     

求解非线性不适定的混合Newton-Tikhonov迭代法

鉴于Newton型方法在实际计算中计算量可能非常大,因此提出了一种一步Newton结合若干步简化Newton的混合Newton-Tikhonov方法,并且在一定条件下证明了该方法的收敛性和稳定性.数值试验表明,在减少计算量方面该方法相对于经典的Newton方法有明显的改善.     

A mixed Newton-Tikhonov method for nonlinear ill-posed problems

Newton type methods are one kind of the efficient methods to solve nonlinear ill-posed problems, which have attracted extensive attention. However, computational cost of Newton type methods is high because practical problems are complicated. We propose a mixed Newton-Tikhonov method, i.e., one step Newton-Tikhonov method with several other steps of simplified Newton-Tikhonov method. Convergence and stability of this method are proved under some conditions. Numerical experiments show that the proposed method has obvious advantages over the classical Newton method in terms of computational costs.