Adaptive spectral decompositions for inverse medium problems

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