This research project focuses on a very important and timely problem in applied mathematics, namely the role played by sparsity in inverse problems and related computational issues. Modern technology provides us with an impressively growing capacity for data acquisition, data storage, and computation. As a result, even problems that until recently were "dense" will soon be "sparse", in the sense that the number of physical degrees of freedom will be small compared to the volume of data acquired. The complexity of many algorithms to handle these data (especially where "reconstruction" or inverse problems are concerned) grows so fast, as a function of the volume of the data, that it could easily outpace Moore's law for computation capacity growth. This makes it urgent that algorithms be developed that are specially adapted to the sparsity of the problem, and exploit this sparsity to reduce the computational complexity. This is the framework in which we propose our project, in which we focus on sparsity as an effective tool in regularization of ill-posed problems, optimization and on the development of relevant algorithms; we focus in particular on three interconnected case studies. Each of these aims to be vertically integrated, in the sense that the project encompasses mathematical modeling of the problem, issues of data representation and classification, and effective, fast and optimized computation.