Phaseless inverse source scattering problem: Phase retrieval, uniqueness and direct sampling methods

Similar to the obstacle or medium scattering problems, an important property of the phaseless far field patterns for source scattering problems is the translation invariance. Thus it is impossible to reconstruct the location of the underlying sources. Furthermore, the phaseless far field pattern is also invariant if the source is multiplied by any complex number with modulus one. Therefore, the source can not be uniquely determined, even the multi-frequency phaseless far field patterns are considered. By adding a reference point source into the model, we propose a simple and stable phase retrieval method and establish several uniqueness results with phaseless far field data. We proceed to introduce a novel direct sampling method for shape and location reconstruction of the source by using broadband sparse phaseless data directly. We also propose a combination method with the novel phase retrieval algorithm and the classical direct sampling methods with phased data. Numerical examples in two dimensions are also presented to demonstrate their feasibility and effectiveness.