In this thesis we present a Krylov subspace technique and use residual minimization to efficiently solve parametric electromagnetic inversion problems. We exploit the shift-invariance property of Krylov subspaces to compute total fields inside a homogeneous object for a whole range of contrast values. As soon as these fields are found, we can determine the corresponding scattered fields in a straightforward manner. This approach allows us to solve the inverse problem by simply inspecting an objective function which measures the discrepancy between the measured and modeled scattered field data.