Adaptive Eigenspace Method for Inverse Scattering Problems in the Frequency Domain

Marcus J. Grote (1) , Marie Kray (1) , Uri Nahum (1)

1 Department of Mathematics [Basel]

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 iteration. Instead of a grid-based discrete representation, the unknown wave speed is projected to a particular finite-dimensional basis of eigenfunctions, 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. Both analytical and numerical evidence underpins the accuracy of the AE representation. Numerical experiments demonstrate the efficiency and robustness to missing or noisy data of the resulting adaptive eigenspace inversion method.

Mots clés: Helmholtz equation; Inverse medium problem; PDE constrained optimization; full waveform in-version; frequency stepping; parameter estimation; adaptive eigenspace inversion; regularization; sample averaging approximation

Domaines: Analyse numérique [math.NA]; Optimisation et contrôle [math.OC]; Equations aux dérivées partielles [math.AP]

Marcus J. Grote, Marie Kray, Uri Nahum. Adaptive Eigenspace Method for Inverse Scattering Problems in the Frequency Domain. Inverse Problems, 2017, 33 (2), pp.025006. ⟨10.1088/1361-6420/aa5250⟩. ⟨hal-01342694⟩