  The acoustic inverse obstacle scattering problem consists of determining the
shape of a domain from measurements of the scattered far field due to some set
of incident fields (probes). For a penetrable object with known sound speed,
this can be accomplished by treating the boundary alone as an unknown curve.
Alternatively, one can treat the entire object as unknown and use a more
general volumetric representation, without making use of the known sound speed.
Both lead to strongly nonlinear and nonconvex optimization problems for which
recursive linearization provides a useful framework for numerical analysis.
After extending our shape optimization approach developed earlier for
impenetrable bodies, we carry out a systematic study of both methods and
compare their performance on a variety of examples. Our findings indicate that
the volumetric approach is more robust, even though the number of degrees of
freedom is significantly larger. We conclude with a discussion of this
phenomenon and potential directions for further research.
