The reciprocity gap functional method for the inverse scattering problem for cavities

Abstract

In this paper, we concern with determining the shape of a perfectly conducting cavity from the Cauchy data on a curve inside the cavity. The near-field linear sampling method (LSM), i.e. the reciprocity gap (RG) functional method, is employed to reconstruct the shape of the cavity. The equivalence of the RG method and the linear sampling method with mere the scattered field is established. But from the examples, we can see that the reconstructions are as satisfactory as the exterior scattering problems. I think this behavior is due to our reconstruction method since this method is due to the Cauchy data, but the LSM with mere the scattered field is used. Numerical tests show that the methods can provide qualitative information on the cavity. The numerical influence of the proposed method with respect to the wave numbers, the curve for the Cauchy data on which are measured, and the curve which is used to construct the single-layer potential function, respectively, are also analyzed with some examples. In particular, we give the examples of determining the cavity from the Cauchy data measured on a portion of the curve inside the cavity.

Keywords:

• Helmholtz equation
• Cauchy data
• inverse scattering problem
• reciprocity gap functional method
• cavity

AMS Subject Classifications:

• 35R30
• 65J22
• 78A46

Acknowledgements

We would like to thank the referees for their careful reading and valuable comments which improved the quality of our submitted manuscript, and we also would like to thank Xiaodong Liu at Chinese Academy of Sciences for the valuable discussions for this work.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research was supported by the Natural Science Foundation of China [No. 91230119], [No. 11371172]; the fundamental Research Funds for the Central Universities [No. 3122015L007].