Research On Regularization Method Of Numerical Differention

As a classical inverse problem,numerical differential problems are widely used in many fields,such as scientific calculation and engineering technology.However,due to its ill posed nature,it is difficult to solve the problem,so it is very important to find a stable solution algorithm.Based on the idea of regularization,this paper studies several solutions of numerical differential problems.The specific research work is as follows:(1)The causes of the ill-posed of the numerical differential problem are specifically analyzed,and the relevant basic knowledge required in this article is given.(2)Using finite difference method to construct regularization strategy and give error estimation.The least squares regularization method for discrete data is obtained.Three numerical integration formulas and Galerkin method are used to discretize the integral equation into a linear system,and the Tikhonov regularization method is used to solve the first-order and second-order numerical differentiation problems.(3)The Huber function and the Log-cosh function are used as regularization terms in the total variation regularization method,which are discretized by the forward difference method.The algorithm steps are given to solve the numerical differential problem.The numerical results show that the algorithm is efficient and stable.(4)By solving the singular system of differential operators,the truncated singular value regularization method and the mixed regularization method are used to solve the numerical differential.Numerical examples are given and the results are compared and analyzed.(5)The Gauss kernel is used as a smooth kernel function,a smoothing method is adopted,and the Gauss-Legendre type numerical integration is used to obtain a regularized solution,and an error estimation expression is given.The smoothing method is improved,and the least square regularization method is used to calculate the regularized solution at the end of the interval.The result shows that the precision of the improved method is improved obviously.