Inverse scattering for the one-dimensional Helmholtz equation with piecewise constant wave speed

Abstract

This paper analyzes inverse scattering for the one-dimensional Helmholtz equation in the case where the wave speed is piecewise constant. Scattering data recorded for an arbitrarily small interval of frequencies is shown to determine the wave speed uniquely, and a direct reconstruction algorithm is presented. The algorithm is exact provided data is recorded for a sufficiently wide range of frequencies and the jump points of the wave speed are equally spaced with respect to travel time. Numerical examples show that the algorithm works also in the general case of arbitrary wave speed (either with jumps or continuously varying etc.) giving progressively more accurate approximations as the range of recorded frequencies increases. A key underlying theoretical insight is to associate scattering data to compositions of automorphisms of the unit disk, which are in turn related to orthogonal polynomials on the unit circle. The algorithm exploits the three-term recurrence of orthogonal polynomials to reduce the required computation.

Statistical inverse estimation of source terms in the Helmholtz equation

“…For instance, Pieper et al 5 analyzed the optimality conditions for recovery of a sound source which consists of an unknown number time‐harmonic monopoles in the Helmholtz equation, from pointwise measurements of the acoustic pressure, and solved the problem by sparse optimization problems in measure space in combination with regularized least squares formulation. Bugarija et al 6 analyzed inverse scattering for the one‐dimensional Helmholtz equation in the case where the wave speed is piecewise constant and proposed a direct reconstruction algorithm. Liu and Li 7 focused on the source identification for Helmholtz equation from a single measurement pair of Cauchy data and proposed three stable reconstruction algorithms to detect the number, the location, the size, and the shape of the hidden sources.…”

Convexification for an inverse problem for a 1D wave equation with experimental data

“…The common property of these works and the above cited works on the convexification is that both substantially use Carleman estimates and the resulting numerical methods converge globally in both cases. In addition, we refer to the most recent work of Bugarija, Gibson, Hu, Li and Zhao [8] where a new numerical method is proposed for a CIP for the 1D Helmholtz equation with a piecewise constant unknown coefficient.…”