Regularization For Solving Ill-posed Problems

Since the 1960s, there are many inverse problems in the fields of natural science and engineering technology such as geophysics, life science, materials science, pattern recognition, signal (image) processing, industrial controls, and economic decision-making. Many scholars and researchers focus on the research of inverse problems. The main difficulty about the solution of inverse problems lies in ill-posedness, which mainly is the instability of approximate solution, that is, the solution of a equation (if existing) doesn't continuously rely on the data in right hand side. There is a bigger error between the approximate solution and the correct solution when the data in right hand side exists error. A ordinary method of solving ill-posed problems is regularization method. Therefore, how to establish an effective regularization method is very important part.Based on some cases, basic definitions of inverse problems and ill-posed problems and the general theory of regularization are proposed. The regularization method is one of the methods for solving ill-posed problems, including the well-known Tikhonov regularization method and the Landweber iteration. By the singular value decomposition theory of compact operator, it is shown that the singular values of the operator trend to zero. Therefore, theoretical basis on establishing regularization method is given in the paper that the influence that the property of the singular value being close to zero has on the stability of the solution is weakened or filtrated by introducing regularizing filter. From the idea, a new regularization method is given by establishing a new filtering function, and the regularized solution of the error estimation and regularization parameter's selection are discussed. The error estimates between the exact solution and approximate solution are given while there is perturbation both the operator and the right hand.