  We propose a novel numerical inversion algorithm for the coefficients of
parabolic partial differential equations, based on model reduction. The study
is motivated by the application of controlled source electromagnetic
exploration, where the unknown is the subsurface electrical resistivity and the
data are time resolved surface measurements of the magnetic field. The
algorithm presented in this paper considers inversion in one and two
dimensions. The reduced model is obtained with rational interpolation in the
frequency (Laplace) domain and a rational Krylov subspace projection method. It
amounts to a nonlinear mapping from the function space of the unknown
resistivity to the small dimensional space of the parameters of the reduced
model. We use this mapping as a nonlinear preconditioner for the Gauss-Newton
iterative solution of the inverse problem. The advantage of the inversion
algorithm is twofold. First, the nonlinear preconditioner resolves most of the
nonlinearity of the problem. Thus the iterations are less likely to get stuck
in local minima and the convergence is fast. Second, the inversion is
computationally efficient because it avoids repeated accurate simulations of
the time-domain response. We study the stability of the inversion algorithm for
various rational Krylov subspaces, and assess its performance with numerical
experiments.
