During this thesis, we investigated the phase retrieval problem of real-valued signals in finite dimension, a challenge encountered across various scientific and engineering disciplines. Our work explores two complementary approaches: retrieval with and without regularization. In both settings, we focused on relaxing the Lipschitz-smoothness assumption generally required by first-order splitting algorithms, invalid for phase retrieval cast as a least-square problem. The key idea here is to replace the Euclidean geometry by a non-Euclidean Bregman divergence associated to an appropriate kernel. We use a Bregman gradient/mirror descent algorithm with this divergence to solve the phase retrieval problem without regularization, and we show exact recovery both in a deterministic setting and with high probability for a sufficient number of random measurements. Furthermore, we establish the robustness of this approach against small additive noise. Shifting to regularized phase retrieval, we first develop and analyze an Inertial Bregman Proximal Gradient algorithm for minimizing the sum of two functions in finite dimension, one of which is convex and possibly nonsmooth and the second is relatively smooth in the Bregman geometry. We provide both global and local convergence guarantees for this algorithm. Finally, we study noiseless and stable recovery of low complexity regularized phase retrieval. For this, we formulate the problem as the minimization of an objective functional involving a nonconvex smooth data fidelity term and a convex regularizer promoting solutions conforming to some notion of low-complexity related to their nonsmoothness points. We establish conditions for exact and stable recovery and provide sample complexity bounds for random measurements to ensure that these conditions hold. These sample bounds depend on the low complexity of the signals to be recovered. Our new results allow to go far beyond the case of sparse phase retrieval.

We developed a novel computational approach to optimal transport. In a project title: ”A new transportation distance with bulk/interface interactions and flux penalization: Numericals aspect ” I have implemented a type of Monge/Kanterovich type distance also known as Wasserstein distance on the space of measures. This method incorporates the borders of our domains which enables modeling complex scenarios in optimal transport problems.