Adaptive eigenspace for multi-parameter inverse scattering problems. (English) Zbl 1442.49032
Summary: A nonlinear optimization method is proposed for inverse scattering problems in the frequency domain, when the unknown medium is characterized by one or several spatially varying parameters. The time-harmonic inverse medium problem is formulated as a PDE-constrained optimization problem and solved by an inexact truncated Newton-type method combined with frequency stepping. Instead of a grid-based discrete representation, each parameter is projected to a separate finite-dimensional subspace, which is iteratively adapted during the optimization. Each subspace is spanned by the first few eigenfunctions of a linearized regularization penalty functional chosen a priori. The (small and slowly increasing) finite number of eigenfunctions effectively introduces regularization into the inversion and thus avoids the need for standard Tikhonov-type regularization. Numerical results illustrate the accuracy and efficiency of the resulting adaptive eigenspace regularization for single and multi-parameter problems, including the well-known Marmousi model from geosciences.
MSC:
| 49K45 | Optimality conditions for problems involving randomness |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35P25 | Scattering theory for PDEs |
| 35R30 | Inverse problems for PDEs |
| 78A46 | Inverse problems (including inverse scattering) in optics and electromagnetic theory |
Keywords:
inverse medium problem; Helmholtz equation; full waveform inversion; PDE constrained optimization; multi-parameter estimation; nonlinear spectral representationReferences:
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