Applications of sparsity to the regularization of ill-posed inverse problems in physics This research project focuses on a very important and timely problem in applied mathematics and computational physics, namely on the role played by sparsity in inverse problems. Modern technology provides us with an impressive capacity for data acquisition and data storage-- more than our algorithms can handle, despite the constant speed-up in computation described by Moore's law. On the other hand, the very abundance of data increasingly leads to situations where we know that the objects under study have, intrinsically, a much more parsimonious description than the number of degrees of freedom permitted by the data or the model. In this research programme we investigate the use of sparse techniques and representations to reduce the computational complexity of the solution of ill-posed inverse problems. It is expected that the need for such new sparse techniques will increase very strongly in the near future; there is therefore a growing amount of activity in this emerging domain, as witnessed by special sessions, addressing the theme of exploiting sparsity, at e.g. ICASSP 2005, 2006 and 2008, as well as EUSIPCO 2008, and international workshops, such as SPARS05, exclusively devoted to this topic and its applications to signal analysis. See also the preprint repository [1]. In this project, we will direct our attention to two concrete physical problems, in seismic tomography and in medical imaging. Because of their enormous size, we shall also develop new applied mathematical methods.