Graph Laplacian–Based Strategies and Convex Optimization via Primal-Dual Methods

This thesis focuses on the analysis of different variational approaches for solving inverse problems. In the first part, we examine the graph Laplacian operator within an l2-l1 framework, where q is less than 1. A key challenge in using this linear operator is its dependence on an initial reconstruction, which can be obtained through a general reconstruction method. However, we demonstrate that, under very weak assumptions on the chosen reconstruction method, the resulting strategy is both convergent and stable, achieving high quality final reconstructions. Additionally, we analyze the fractional graph Laplacian operator, showing that the use of fractional powers can surpass the standard approach by providing more detailed final images. The second part of this thesis considers a more general framework, where the optimization problem consists of the sum of a differentiable term and a non-smooth but convex term. The variable metric approach we propose results in a convergent method that fixes a priori the number of nested iterations required to compute inexact approximations of the proximal gradient step. We also introduce an iterated Tikhonov-based strategy, which accelerates convergence while maintaining high-quality reconstructions. In the context of image deblurring, the variable metric approach can be reinterpreted as a right preconditioning strategy. Therefore, the final section is devoted to the analysis of a left preconditioning approach.