The present study aims to introduce the linear sampling method for cavity/inclusion detection in a two-dimensional Laplace problem by measuring data on the boundary. This method is categorized as a qualitative approach to image the geometrical features of unknown targets. There has been no specific attempt to apply this method to the identification of cavities/inclusions in the Laplace equation although it has been used in the context of inverse problems such as acoustics, electromagnetism and elastostatics. Therefore, the implementation of the linear sampling method, coupled with the finite element method, is emphasized in this study. A set of numerical simulations on two-dimensional problems is presented to highlight many effective features of the proposed qualitative identification method.
BakushinskyABKokurinMY.Iterative methods for approximate solution of inverse problems, vol. 577. New York, NY: Springer Science & Business Media, 2005.
2.
TanakaMMasudaY.Boundary element method applied to some inverse problems. Eng Anal Bound Elem1986; 3(3): 138–143.
3.
GallegoRSuárezJ.Solution of inverse problems by boundary integral equations without residual minimization. Int J Solid Struct2000; 37(40): 5629–5652.
4.
KaramiGHematiyanM.A boundary element method of inverse non-linear heat conduction analysis with point and line heat sources. Abbrev Tit Commun Numer Meth Eng2000; 16(3): 191–203.
5.
SumbatyanMTibulloVZampoliV.On the theory of direct and inverse problems in the elastic rectangle: Antiplane case. Res Nondestr Eval2008; 19(1): 20–43.
6.
KołodziejJAMierzwiczakMCiałkowskiM.Application of the method of fundamental solutions and radial basis functions for inverse heat source problem in case of steady-state. Int Commun Heat Mass Transf2010; 37(2): 121–124.
7.
YoungDTsaiCChenC. The method of fundamental solutions and condition number analysis for inverse problems of Laplace equation. Comput Math Appl2008; 55(6): 1189–1200.
8.
JinBMarinL.The method of fundamental solutions for inverse source problems associated with the steady-state heat conduction. Int J Numer Method Eng2007; 69(8): 1570–1589.
9.
YangFYanLWeiT.Reconstruction of part of a boundary for the Laplace equation by using a regularized method of fundamental solutions. Invers Prob Sci Eng2009; 17(8): 1113–1128.
10.
MeraNElliottLInghamD.Boundary identification for a 3D Laplace equation using a genetic algorithm. Comput Meth Appl Mech Eng2005; 194(42): 4411–4424.
11.
CakoniFColtonDMonkP.The linear sampling method in inverse electromagnetic scattering, vol. 80. Phildelphia, PA: SIAM, 2011.
12.
CakoniFColtonD.Qualitative methods in inverse scattering theory: An introduction. New York, NY: Springer Science & Business Media, 2005.
