Inverse problems for PDE via infinite dimensional compressed sensing

Compressed sensing stands for a series of techniques whose aim is to recover a sparse signal from a small number of measurements. Since the 2006 seminal papers of Candes-Romberg-Tao and Donoho, which concerned the recovery of a sparse vector from few discrete Fourier coefficients, the subject has been extensively studied and generalized. In this talk we will present new results concerning generalization of compressed sensing in the framework of Hilbert spaces: in particular, the measurement operator does not need to be a orthonormal transformation and the unknown is assumed to be sparse in a frame. Applications to inverse problems for PDE, such as electrical impedance tomography, will be discussed as well. This is a joint work with Giovanni S. Alberti.