Ill-posed Linear Inverse Problems arise in various research domains, such as control engineering and image processing. Having a fast algorithm is a great benefit when working with high-dimensional signals, such as images. However, fast convergence and iterations with low computational complexity are challenging.
In this master thesis report, we propose an exact smooth reformulation of an ill-posed Linear Inverse Problem. Subsequently, we present a novel algorithm, the Fast Linear Inverse Problem Solver (FLIPS), associated with the new problem formulation. We show that in most metrics, the algorithm outperforms state-of-the-art methods like Chambolle-Pock (CP) and the Constrained Split Augmented Lagrangian Shrinkage Algorithm (C-SALSA) in terms of speed. Finally, associated with this algorithm, we present an open-source MATLAB package that includes the proposed algorithm and state-of-the-art methods.