  This paper is concerned with uniqueness, phase retrieval and shape
reconstruction methods for inverse elastic scattering problems with phaseless
far field data. Systematically, we study two basic models, i.e., inverse
scattering of plane waves by rigid bodies and inverse scattering of sources
with compact support. For both models, we show that the location of the objects
can not be uniquely recovered by the data. To solve this problem, we consider
simultaneously the incident point sources with one fixed source point and at
most three scattering strengths. We then establish some uniqueness results for
source scattering problem with multi-frequency phaseless far field data.
Furthermore, a fast and stable phase retrieval approach is proposed based on a
simple geometric result which provides a stable reconstruction of a point in
the plane from three distances to given points. Difficulties arise for inverse
scattering by rigid bodies due to the additional unknown far field pattern of
the point sources. To overcome this difficulty, we introduce an artificial
rigid body into the system and show that the underlying rigid bodies can be
uniquely determined by the corresponding phaseless far field data at a fixed
frequency. Noting that the far field pattern of the scattered field
corresponding to point sources is very small if the source point is far away
from the scatterers, we propose an appropriate phase retrieval method for
obstacle scattering problems, without using the artificial rigid body. Finally,
we propose several sampling methods for shape reconstruction with phaseless far
field data. Extended numerical examples in two dimensions are conducted with
noisy data, and the results further verify the effectiveness and robustness of
the proposed phase retrieval techniques and sampling methods.
