Inverse problems are ubiquitous in signal processing applications. They provide an often simplified but convenient way to link measurements to unknown signals of interest. In this thesis, we focus on regularizing ill-posed linear inverse problems by means of sparsity-promoting optimization techniques. Both classical discrete-domain and more general continuous-domain problems are considered. With optimization-based sparsity as a guiding principle, this thesis covers a broad range of subjects that include the design of algorithms, the study of sparsity-promoting optimization problems, and the application of sparse recovery techniques to the challenging application of radio interferometric imaging, using both simulated and observational data. Accordingly, the thesis is structured in three parts.

The first part focuses on discrete finite-dimensional models--on the grid. In this context, the LASSO problem is indisputably the by-default optimization method for sparse recon struction. We review the relevant classical results and numerical solvers before introducing our Polyatomic Frank-Wolfe algorithm, a new optimization method for convex optimization problems. Our procedure extends on the vanilla version of the algorithm by enabling many atoms to be placed at each iteration, providing performance boosts in solving time and scalability.

The second part of the thesis considers sparse inverse problems formulated on the continuum--off the grid. These problems assume that the solutions contain sparse information with infinite precision, as opposed to finite-dimensional reconstruction on grids. In this context, we propose another Polyatomic Frank-Wolfe algorithm to solve the B-LASSO, the continuous-domain counterpart of the LASSO. We empirically demonstrate the interest of this method in moderate- to large-scale problems over the baseline Sliding Frank-Wolfe algorithm, an existing numerical procedure that exhibits remarkable convergence properties but may be limited by its computational cost. Additionally, we study composite continuous-domain problems, for which the solution is modeled as the sum of two components, one being sparse while the second is smooth. We analytically demonstrate an explicit decoupling between the two components, leading to a significant acceleration of the numerical solving methods.

The third and final part is dedicated to the application of the Polyatomic Frank-Wolfe algorithm developed in the first part to the challenging problem of sky image reconstruction in radio interferometry. This task can be modeled as a discrete sparse inverse problem whose forward operator corresponds to a non-uniform Fourier sampling. We develop PolyCLEAN as the specialization of our polyatomic algorithm for this purpose. The atomic paradigm of PolyCLEAN is used to solve a LASSO problem, combining the benefits of atomic methods already common in the field and principled optimization : scalability and accuracy. Overall, this thesis contributes to the exploration of atomic numerical methods for sparse inverse problems in various applied and theoretical settings, aiming to improve the efficiency of sparse recovery by aligning the means and the goal in a principled approach.