  Inverse medium problems involve the reconstruction of a spatially varying
unknown medium from available observations by exploring a restricted search
space of possible solutions. Standard grid-based representations are very
general but all too often computationally prohibitive due to the high dimension
of the search space. Adaptive spectral (AS) decompositions instead expand the
unknown medium in a basis of eigenfunctions of a judicious elliptic operator,
which depends itself on the medium. Here the AS decomposition is combined with
a standard inexact Newton-type method for the solution of time-harmonic
scattering problems governed by the Helmholtz equation. By repeatedly adapting
both the eigenfunction basis and its dimension, the resulting adaptive spectral
inversion (ASI) method substantially reduces the dimension of the search space
during the nonlinear optimization. Rigorous estimates of the AS decomposition
are proved for a general piecewise constant medium. Numerical results
illustrate the accuracy and efficiency of the ASI method for time-harmonic
inverse scattering problems, including a salt dome model from geophysics.
