  The inverse acoustic scattering problems using multi-frequency backscattering
far field patterns at isolated directions are studied. The underlying object
could be point like scatterers, small scatterers, extended inhomogeneities and
obstacles. A fast and robust direct sampling method is proposed for location
and shape reconstruction of the underlying objects. To our best knowledge, this
is the first numerical method for inverse obstacle/medium scattering problems
with the multi-frequency sparse backscattering data. Some approximate
uniqueness results have also been showed based on the weak scattering
approximation and the Kirchhoff approximation. The theoretical basis of the
direct sampling method is initially established based on these two
approximations. Numerical examples in two dimensions are presented to validity
the effectiveness and robustness of the proposed direct sampling method. In
particular, the numerical reconstructions indicate that the direct sampling
method also works very well beyond these two approximations.
