  We propose domain decomposition preconditioners for the solution of an
integral equation formulation of forward and inverse acoustic scattering
problems with point scatterers. We study both forward and inverse problems and
propose preconditioning techniques to accelerate the iterative solvers. For the
forward scattering problem, we extend the domain decomposition based
preconditioning techniques presented for partial differential equations in {\em
"A restricted additive Schwarz preconditioner for general sparse linear
systems", SIAM Journal on Scientific Computing, 21 (1999), pp. 792--797}, to
integral equations. We combine this domain decomposition preconditioner with a
low-rank correction, which is easy to construct, forming a new preconditioner.
For the inverse scattering problem, we use the forward problem preconditioner
as a building block for constructing a preconditioner for the Gauss-Newton
Hessian. We present numerical results that demonstrate the performance of both
preconditioning strategies.
