Regulatization methods in Banach spaces

A large array of physical and technical challenges can be modelled by inverse problems, which currently form the subject of a rapidly growing scientific field. Most of the inverse problems can be expressed as operator equations which are usually ill-posed in the sense that small data perturbations produce large oscillations in the solution. Therefore, some form of regularization is needed, where the original ill-posed problem is replaced by a family of neighboring problems which are well- posed and supply stable approximations of the true solution. Today`s complex problems arising in biology, medicine, chemistry and data mining require mathematical models involving infinite dimensional Banach spaces that are not Hilbertian, reflexive, or separable (e.g., the Lebesgue spaces, the space of bounded variation functions, the Besov spaces). Thus, a broad convergence theory, covering also these settings, has become necessary. This theory can be partly developed by extending classical regularization methods to such spaces, which requires sustained research efforts. In this project, we intend to contribute to the foundations of a general regularization theory, by proposing and analyzing various nonquadratic stabilization methods for solving ill-posed problems in Banach spaces. Among these, we address generalized Landweber procedures, and regularization based on nonquadratic data-fitting terms and on surrogate functionals. Special attention is devoted to the Expectation-Maximization (EM) algorithm for Positron Emission Tomography (PET), where convergence results in infinite dimensional settings are still missing. The proposed methods will be applicable to practical situations such as image deblurring/denoising, sparse functions recovery and PET. The analysis will comprise qualitative aspects (stability, convergence) and quantitative aspects (convergence rates, numerical experiments). The proximal point method in optimization will be a key tool in achieving these goals.

Inverse Problems (IP) is a very dynamic research eld, with direct applications to real system problems based on indirect observations/measurements. The mathematical model employed here involves an (operator) equation, where one has to nd the unknown based on the inexact observed data. Such an equation is ill-posed in most of the cases, which means that small perturbations in the data lead to high oscillations in the solution. Consequently, solutions computed in a direct way are inaccurate. IP research seeks to stabilize such equations by means of regularization, where the ill-posed problem is replaced by a family of well-posed problems. The core of this research project addresses such regularization methods; the results cover several of the initially proposed goals, and subscribe to the main strategical directions of the project proposal. The theoretical results have been obtained from the applications perspective, covering the following topics: Approaching inverse problems by optimization techniques: This has been a major interest, leading to deep results for a well-known inverse problems method and to interesting new procedures for image reconstruction problems. For instance, deblurring images contaminated with various types of noise (such as Gaussian, Laplace and Poisson) has been dealt with in a unied framework relying on a rigorous analysis and relevant numerical experiments. The convergence speed for some methods could also be established due to optimization tools. A variety of inverse problems arising in physics (e.g., the uorescence problem), in computed tomography require positive functions/vectors as solutions. We proposed two methods which guarantee approximating such solutions. Treating a mathematical problem in an innite dimensional context may help in nding a solution; however, practical implementations usually require discretization (a nite dimensional approach). Such issues have also been considered and analysed in more detail for some cases. Approximating sparse solutions (which have only few nonzero components - as in signal processing or gene networks) is promoted by context-specic regularization procedures. Part of our work investigates solution properties (and their practical meaning) that ensure certain speed of convergence for some stabilization of the corresponding problem. We also introduce a new form of stabilization of the sparse solutions reconstruction problem, which has an intuitive motivation. Heuristic ways for stable image reconstruction have also been considered. These are methods which practitioners would use in their regular work because they yield good experimental results although no theory may exist around them. The performed numerical experiments hint on how to try establishing theoretical results.