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# Communications in Mathematical Sciences

## Volume 9 (2011)

### Number 3

### Discovery of point sources in the Helmholtz equation posed in unknown domains with obstacles

Pages: 903 – 928

DOI: http://dx.doi.org/10.4310/CMS.2011.v9.n3.a11

#### Authors

#### Abstract

We consider an inverse source problem for the Helmholtz equation in domains with possibly unknown obstacles, which are much larger than the wavelength. The inverse problem is to determine the number and locations of point sources in the domain based on sparse measurements of the wave field. Our proposed strategy relies on solving a local inverse scattering problem to obtain the incoming directions of waves passing through an observation location. We formulate the problem as an L1 constrained minimization problem and solve it using the Bregman iterative procedure. The wave direction rays are then traced back and sources are uniquely determined at the intersection of the rays from several observing locations. We present examples in 2D, however all of the formulas and methods used have direct analogues in 3D.

#### Keywords

Inverse scattering, Helmholtz equation, geometric optics, microlocal analysi

#### 2010 Mathematics Subject Classification

34L25, 35J05, 78A05