A distributed source method for inverse acoustic scattering

Abstract

We develop a new algorithm for solving inverse acoustic scattering problems. In particular we show that this algorithm can reproduce scattering shapes efficiently, using synthetic data, from only one incident wave in the acoustically hard case and using at most two incident waves for the acoustically soft problem. In order to test the inversion algorithm we generate synthetic data using a technique which combines the distributed source method and the fundamental solution of the Helmholtz equation in order to calculate the scattered field for each of these problems. Numerical results for three-dimensional axially symmetric shapes are compared with those obtained previously by other authors.

Computational design of metallophone contact sounds

“…", or, do there exist two distinct shapes of membranes that resonate at the same frequencies? While answered "no" by Gordon et al [1992a;1992b], this famous question inspired many inverse acoustic works: given detected sound scattering patterns [Angell et al 1997;Feijóo et al 2004] or room echoes [Dokmanić et al 2013], reconstruct the shape of the structures that affect sound propagation. Our work is also inspired by Kac's question, but has a very different problem formulation: the input specifies not only frequency values but also amplitudes, and our goal is to find a 3D shape composed of elastic materials.…”

Numerical methods in inverse obstacle scattering

“…which is valid for large |x| and where h (1) n denotes the spherical Hankel function of order n and of the first kind. From the asymptotics for the spherical Hankel functions for large |x| it follows that the far field pattern of u s is given by…”

“…Although the methods of Kirsch and Kress and of Angell, Kleinman, and Roach, which were developed in the mid-eighties have been revived through more recent papers (see Angell, Jiang, and Kleinman [1] and Haas and Lehner [8]) they probably will not remain competitive in efficiency with iterative methods such as the regularized Newton method described in the next section. The increase in the computational cost for the Newton method as compared to the methods of Kirsch and Kress and of Angell, Kleinman, and Roach is compensated by notably much more accurate reconstructions.…”

The linear sampling method for three-dimensional inverse scattering problems

“…Although of considerable importance in various areas of science and technology, the mathematical and numerical analysis of such problems is of relatively recent origin. There have been a number of successful reconstruction algorithms proposed for the three dimensional time harmonic inverse scattering problem for both acoustic and electromagnetic waves, all of which are based on some type of nonlinear optimization scheme [4,1,9,12,13,15]. Such schemes are particularly attractive since they are able to treat the nonlinear and improperly posed nature of the inverse scattering problem in a simple and straightforward manner [3,14].…”