基于迭代正则化高斯-牛顿法的非线性Urysohn积分方程数值解

On the numerical solution of Urysohn integral equation using iteratively regularized Gauss-Newton method

非线性Urysohn积分方程在许多领域中都有广泛的应用,但由于该方程具有不适定性的特点,数据的微小扰动可能导致解的巨大变化,给数值求解带来很大困难.为了获得稳定的、准确的数值解,本文利用迭代正则化高斯-牛顿法对此方程进行求解,给出了利用Sigmoid-型函数确定迭代正则化参数的方法.对一类重力测定问题进行了数值模拟,将得到的数值解和相应的精确解作比较.结果表明,本文提出的方法在求解非线性Urysohn积分方程时是可行的也是有效的.

Urysohn integral equation has been widely used in many fields,but because of the ill nature of the equation,it is possible to solve the problem,which is very sensitive to perturbation.In order to obtain a stable and accurate numerical solution,the paper use the Iteratively Regularized Gauss-Newton method.In addition,by using the Sigmoid-function to determine the regularization parameter,the numerical simulation is carried out for the gravity measurement problem in the field of geophysics.The results show that the method is feasible and effective in solving the Urysohn integral equation.