Adaptive spectral inversion for inverse medium problems

Inverse medium problems are typically dedicated to finding the cause behind measured behaviors. Examples include exploring the Earth's interior through seismic or geophysical imaging, using ultrasounds or X-rays for medical imaging, or investigating materials in material science. In each case, our goal is to determine the unknown, also referred to as the medium, responsible for these measured observations by solving a constraint minimization problem for the data misfit. The constraints arise from the physical state, which is modeled and mathematically expressed by a partial differential equation.
In this Thesis, we propose a nonlinear iterative optimization method to solve inverse medium problems. Instead of using a grid based optimization approach, which leads to challenging large scale problems, we iteratively minimize the data misfit within a small finite dimensional subspace spanned by the first few eigenfunctions of a carefully chosen elliptic operator. As the operator depends on the minimizer in the previous search space, so do its eigenfunctions, and consequently the subsequent search space. This approach allows us to incorporate regularization inherently at each iteration without the need for additional penalization, such as Total Variation or Tikhonov regularization.
By introducing a key angle condition, we can prove the convergence of the resulting Adaptive Spectral Inversion (ASI) method and demonstrate its regularizing effect. Through numerical experiments, we illustrate the remarkable accuracy of the ASI, even detecting the smallest inclusions where previous methods failed. Furthermore, we demonstrate that the ASI performs favorably compared to standard grid based inversion using Tikhonov regularization when applied to an elliptic inverse problem.
The choice of the elliptic operator for obtaining the subsequent search space is crucial for achieving accurate reconstructed media. For known piecewise constant media, consisting of a few interior inclusions, we prove that the first few eigenfunctions of the operator, that depend on the medium, effectively approximate the medium and its discontinuities. Then, we validate these analytically proven properties of the operator through various numerical experiments.