On the inverse scattering problem for the Helmholtz equation in one dimension

Abstract

Interest in the numerical solution of acoustic inverse scattering problems arises in a number of areas. Examples include medical diagnostics, non-destructive industrial testing, geophysical prospecting for petroleum and minerals, and detection of earthquakes. The highly nonlinear and oscillatory nature of the probltm is one of the major difficulties one encounters in the construction of effective inversion algorithms. Schemes based on global or local linearization methods, or nonlinear optimization techniques, tend to work only when the index of refraction is almost constant. They develop serious convergence problems whenever the perturbation of the index of refraction increases. Limited successes in the solution of the inverse problems have been achieved only in one dimensional cases (Gelfand-Levitan and layer striping methods are among the most notable). These methods are generally unstable numerically since the procedures used to calculate the index of refraction are ill-conditioned. We present a method for the solution of inverse problems for the one dimensional Helmholtz equation. The scheme is based on a combination of the standard Riccati equation for the impedance function with a new trace formula for the derivative of the index of refraction, and can be viewed as a frequency domain version of the layer-stripping approach. The principal advantage of the procedure is that if the scatterer to be reconstructed has m > 1 continuous derivatives, the accuracy of the reconstruction is proportional to 1/am, where a is the highest frequency for which scattering data axe available. Thus, a smooth scatterer is reconstructed very accurately from a limited amount of available data. The scheme has an asymptotic cost 0(n 2 ), where n is the number of features to be recovered (as do classical layer-stripping algorithms), and is stable with respect to perturbations of the scattering data. The performance of the algorithm is illustrated by several numerical examples. Generalizations of this approach in two dimensions are discussed.

Short-Term Recurrence Krylov Subspace Methods for Nearly Hermitian Matrices

“…and m is a function of the refractive index [3]. In the case of constant m, the skew-adjoint part of the operator (3.2) has a two-dimensional range [28].…”

“…Once a candidate value for W has been found, one can solve the small s × s system (4.6) for y ∈ C s , where s n. 3 With y in hand, one can check if W satisfies the stopping criterion; if not, conduct further MINRES iterations to refine W, and test the criterion again with the updated y. Adapting notation slightly, let W and W denote two approximate solutions to HW = F with corresponding solutions y and y to (4.6).…”

Some new results in 1D inverse scattering

“…We follow the approach in [6], see [I], [2], and [3] for other approaches to layer stripping. We consider the Helmholtz equation (1) ^+c.V(y)u=0…”

Reconstruction of steplike potentials

“…Direct solution methods could also be based on the trace formula of [4], see e.g. [13] for examples of their use in a slightly different situation. Other methods, based on specific types of approximation of the potential are discussed in [3], [14], although the latter reference seems mostly concerned with potentials having bound states.…”