Adaptive eigenspace for inverse problems in the frequency domain

Inverse scattering problems are used in a vast number of applications, such as geophysical
exploration and medical imaging. The goal is to recover unknown media using wave prop-
agation. The inverse problem is designed to minimize simulated data with observation
data, using partial differential equations (PDE) as constrains. The resulting minimiza-
tion problem is often severely ill-posed and contains a large number of local minima. To
tackle ill-posedness, several optimization and regularization techniques have been explored.
However, the applications are still asking for improvement and stability.
In this thesis, a nonlinear optimization method is proposed for the solution of inverse
scattering problems in the frequency domain, when the scattered field is governed by
the Helmholtz equation. The time-harmonic inverse medium problem is formulated as
a PDE-constrained optimization problem and solved by an inexact truncated Newton-
type method. Instead of a grid-based discrete representation, the unknown wave speed
is projected to a particular finite-dimensional basis, which is iteratively adapted during
the optimization. Truncating the adaptive eigenspace (AE) basis at a (small and slowly
increasing) finite number of eigenfunctions effectively introduces regularization into the
inversion and thus avoids the need for standard Tikhonov-type regularization. We actually
show how to build an AE from the gradients of Tikhonov-regularization functionals.
Both analytical and numerical evidence underpin the accuracy of the AE representation.
Numerical experiments demonstrate the efficiency and robustness to missing or noisy data
of the resulting adaptive eigenspace inversion (AEI) method. We also consider missing
frequency data and apply the AEI to the multi-parameter inverse scattering problem.