Adaptive spectral decompositions for inverse medium problems

TL;DR

This paper introduces an adaptive spectral inversion method that reduces the computational complexity of inverse medium problems by dynamically adjusting the eigenfunction basis during the reconstruction process.

Contribution

It combines adaptive spectral decompositions with Newton-type methods to efficiently solve inverse scattering problems, with proven estimates and demonstrated numerical success.

Findings

Significant reduction in search space dimension during optimization

High accuracy in reconstructing complex media like salt domes

Efficient convergence demonstrated on geophysical models

Abstract

Inverse medium problems involve the reconstruction of a spatially varying unknown medium from available observations by exploring a restricted search space of possible solutions. Standard grid-based representations are very general but all too often computationally prohibitive due to the high dimension of the search space. Adaptive spectral (AS) decompositions instead expand the unknown medium in a basis of eigenfunctions of a judicious elliptic operator, which depends itself on the medium. Here the AS decomposition is combined with a standard inexact Newton-type method for the solution of time-harmonic scattering problems governed by the Helmholtz equation. By repeatedly adapting both the eigenfunction basis and its dimension, the resulting adaptive spectral inversion (ASI) method substantially reduces the dimension of the search space during the nonlinear optimization. Rigorous estimates of the AS decomposition are proved for a general piecewise constant medium. Numerical results illustrate the accuracy and efficiency of the ASI method for time-harmonic inverse scattering problems, including a salt dome model from geophysics.