  We present a two-stage least-squares method to inverse medium problems of
reconstructing multiple unknown coefficients simultaneously from noisy data. A
direct sampling method is applied to detect the location of the inhomogeneity
in the first stage, while a total least-squares method with mixed
regularization is used to recover the medium profile in the second stage. The
total least-squares method is designed to minimize the residual of the model
equation and the data fitting, along with an appropriate regularization, in an
attempt to significantly improve the accuracy of the approximation obtained
from the first stage. We shall also present an analysis on the well-posedness
and convergence of this algorithm. Numerical experiments are carried out to
verify the accuracies and robustness of this novel two-stage least-squares
algorithm, with great tolerance of noise.
