  Here we discuss a regularized version of the factorization method for
positive operators acting on a Hilbert Space. The factorization method is a
qualitative reconstruction method that has been used to solve many inverse
shape problems. In general, qualitative methods seek to reconstruct the shape
of an unknown object using little to no a priori information. The regularized
factorization method presented here seeks to avoid numerical instabilities in
the inversion algorithm. This allows one to recover unknown structures in a
computationally simple and analytically rigorous way. We will discuss the
theory and application of the regularized factorization method to examples
coming from acoustic inverse scattering. Numerical examples will also be
presented using synthetic data to show the applicability of the method.
