Multifrequency inverse problem for the reduced wave equation with sparse data

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Abstract
The inverse problem for the reduced wave equation Δu+k2n2(x)u=0 is considered where the quantity to be determined is the value of n(x) in a compact domain D in R3. The data consists of a finite set of measurements of the scattered field produced by different incident fields. The measurements are made at various points exterior to D and at possibly different frequencies. The mathematical problem involves solving a system of nonlinear functional equations. Conditions are developed which indicate when the measured body is close to a particular comparison body (a linearized perturbation is valid). A new higher order nonlinear iterative procedure is developed for the full nonlinear problem, which reduces to the usual solution in the linearized region. The method is illustrated by computational results for the one-dimensional case.

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