  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 multifrequency
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.

Analysis of PDEs