Variational methods for data decomposition

Separating characteristic features of signals and images plays a fundamental role in a wide range of applications and it serves as a highly valuable pre-processing step for data analysis. This dissertation explores the inverse problem of data decomposition that aims to estimate morphologically distinct components of observed signals and images. In this thesis, we develop novel and robust variational models formulated as constrained optimization problems presenting non-convex and non-smooth cost functions which though entail careful initialization strategies and efficient optimization algorithms to determine good solutions. The numerical optimization methods proposed can easily be applied to a wider range of ill-posed inverse problems. In particular, we present a predictor corrector strategy that efficiently computes locally optimal solutions to these problems inspired by the graduated non convexity approach.