A nonlinear optimization method is proposed for the solution of inverse medium problems with spatially varying properties. The inverse medium problem is formulated as a PDEconstrained optimization problem and solved by a standard gradient based method. To avoid the prohibitively large number of unknown control variables resulting from standard grid-based representations, the misfit is instead minimized in a small subspace spanned by the first few eigenfunctions of a judicious elliptic operator [1, 2], which itself depends on the previous iterate. Hence, the eigenfunctions are recomputed at every iteration and selected according to both their approximation properties and the cost functional’s sensitivities [3]. By repeatedly adapting both the dimension and the basis of the search space, regularization is inherently incorporated at each iteration without the need for extra Tikhonov penalization. Convergence is proved under an angle condition, which is included into the resulting Adaptive Spectral Inversion (ASI) algorithm. The ASI approach compares favorably to standard grid-based inversion using L2-Tikhonov regularization when applied to an elliptic inverse problem. The improved accuracy resulting from the new angle condition is further demonstrated via numerical experiments from time-dependent inverse scattering problems.