The Study And Application Of Proximal Newton-type Methods For Ill-posed Problems

Inverse problems are involved in geophysics, image reconstruction, biomedicine, signal processing, control theory, and many other fields. There are some challenges when handing the inverse problem because of the ill-posedness of the inverse problem. More and more attentions are focused on the algorithms solving the ill-posed problems, which greatly promote the development of the theory and practice of solving the ill-posed problems.In this paper, we mainly study the proximal Newton-type methods for ill-posed problems. Small disturbances in the observed data may lead to huge errors in the numerical results, hence it is very difficult to solved the inverse problems. Usually, the ill-posed problems are generally represented as an optimization processing. Many optimization problems can be represented as the summation of some convex functions. Using this additive structure of the optimization problems, a complex problem can be decomposed into several subproblems, which can make each subproblem contains the structure of a specific added convex functional. Thus, it can simplify the calculation and improve the efficiency.The proximal Newton-type methods are used to solve the optimization problems in a composite form, namely a smooth functional and a nonsmooth functional with a simple proximal mapping. The proximal Newton-type methods inherit the excellent convergence behavior of Newton-type methods. Meanwhile, the proximal mapping operator is very simple and easy to understand in terms of mathematics concept.This paper applies the proximal Newton-type methods to the image reconstruction problems. In order to verify the validity of the method, two examples on the imaging reconstruction are selected. The numerical results indicate that the method could effectively solve the ill-posed problem, and these results also enrich the theoretical references in the literature of solving the inverse problem.