Conformal mapping for cavity inverse problem: an explicit reconstruction formula

References

• Calderón A-P. On an inverse boundary value problem, in Seminar on numerical analysis and its applications to continuum physics (Rio de Janeiro). Soc. Brasil. Mat. Rio de Janeiro. 1980;1980:65–73.
   Google Scholar
• Friedman A, Vogelius M. Identification of small inhomogeneities of extreme conductivity by boundary measurements: a theorem on continuous dependence. Arch. Rational Mech. Anal. 1989;105:299–326.
   Web of Science ®Google Scholar
• Kress R. Inverse Dirichlet problem and conformal mapping. Math. Comput. Simul. 2004;66:255–265.
   Web of Science ®Google Scholar
• Alessandrini G, Rondi L. Optimal stability for the inverse problemof multiple cavities. J. Differ. Equ. 2001;176:356–386.
   Web of Science ®Google Scholar
• Bacchelli V. Uniqueness for the determination of unknown boundary and impedance with the homogeneous Robin condition. Inverse Prob. 2009;25:015004, 4.
   Web of Science ®Google Scholar
• Mandache N. Exponential instability in an inverse problem for the Schrödinger equation. Inverse Prob. 2001;17:1435–1444.
   Web of Science ®Google Scholar
• Potthast R. A survey on sampling and probe methods for inverse problems. Inverse Prob. 2006;22:R1–R47.
   Web of Science ®Google Scholar
• Borcea L. Electrical impedance tomography. Inverse Prob. 2002;18:R99–R136.
   Web of Science ®Google Scholar
• Capdeboscq Y, Fehrenbach J, de Gournay F, et al. Imaging by modification: numerical reconstruction of local conductivities from corresponding power density measurements. SIAM J. Imaging Sci. 2009;2:1003–1030.
   Web of Science ®Google Scholar
• Kress R, Rundell W. Nonlinear integral equations and the iterative solution for an inverse boundary value problem. Inverse Prob. 2005;21:1207–1223.
   Web of Science ®Google Scholar
• Ivanyshyn O, Kress R. Nonlinear integral equations for solving inverse boundary value problems for inclusions and cracks. J. Integral Equ. Appl. 2006;18:13–38.
   Google Scholar
• Cakoni F, Kress R. Integral equation methods for the inverse obstacle problem with generalized impedance boundary condition. Inverse Prob. 2013;29:015005, 19.
   Web of Science ®Google Scholar
• Bourgeois L, Dardé J. A quasi-reversibility approach to solve the inverse obstacle problem. Inverse Prob. Imaging. 2010;4:351–377.
   Web of Science ®Google Scholar
• Bourgeois L, Dardé J. The “exterior approach” to solve the inverse obstacle problem for the Stokes system. Inverse Prob. Imaging. 2014;8:23–51.
   Web of Science ®Google Scholar
• Akduman I, Kress R. Electrostatic imaging via conformal mapping. Inverse Prob. 2002;18:1659–1672.
   Web of Science ®Google Scholar
• Haddar H, Kress R. Conformal mappings and inverse boundary value problems. Inverse Prob. 2005;21:935–953.
   Web of Science ®Google Scholar
• Haddar H, Kress R. Conformal mapping and an inverse impedance boundary value problem. J. Inverse Ill-Posed Prob. 2006;14:785–804.
   Google Scholar
• Haddar H, Kress R. Conformal mapping and impedance tomography. Inverse Prob. 2010;26:074002, 18.
   Web of Science ®Google Scholar
• Kress R. Inverse problems and conformal mapping. Complex Var. Elliptic Equ. 2012;57:301–316.
   Web of Science ®Google Scholar
• Haddar H, Kress R. A conformal mapping method in inverse obstacle scattering. Complex Var. Elliptic Equ. 2014;59:863–882.
   Web of Science ®Google Scholar
• Ikehata M. Reconstruction of the shape of the inclusion by boundary measurements. Comm. Partial Differ. Equ. 1998;23:1459–1474.
   Web of Science ®Google Scholar
• Ikehata M. Reconstruction of the support function for inclusion from boundary measurements. J. Inverse Ill-Posed Prob. 2000;8:367–378.
   Google Scholar
• Ikehata M. On reconstruction in the inverse conductivity problem with one measurement. Inverse Prob. 2000;16:785–793.
   Web of Science ®Google Scholar
• Ikehata M, Siltanen S. Numerical method for finding the convex hull of an inclusion in conductivity from boundary measurements. Inverse Prob. 2000;16:1043.
   Web of Science ®Google Scholar
• Erhard K, Potthast R. A numerical study of the probe method. SIAM J. Sci. Comput. 2006;28:1597–1612. electronic.
   Web of Science ®Google Scholar
• Brühl M, Hanke M. Numerical implementation of two noniterative methods for locating inclusions by impedance tomography. Inverse Prob. 2000;16:1029–1042.
   Web of Science ®Google Scholar
• Hanke M, Brühl M. Recent progress in electrical impedance tomography. Inverse Prob. 2003;19:S65–S90.
   Web of Science ®Google Scholar
• Kirsch A. The factorization method for a class of inverse elliptic problems. Math. Nachr. 2005;278:258–277.
   Web of Science ®Google Scholar
• Ammari H, Kang H. Reconstruction of small inhomogeneities from boundary measurements. Vol. 1846. Lecture Notes in Mathematics: Springer-Verlag, Berlin; 2004.
   Google Scholar
• Ammari H, Kang H. Generalized polarization tensors, inverse conductivity problems, and dilute composite materials: a review. Inverse problems, multi-scale analysis and effective medium theory. Vol. 408, Contemporary Mathematics. Providence (RI): American Mathematical Society; 2006. pp. 1–67.
   Google Scholar
• Ammari H, Kang H. Polarization and moment tensors. Vol. 162. Springer, New York (NY): Applied mathematical sciences; 2007.
   Google Scholar
• Ammari H, Garnier J, Kang H, et al. Generalized polarization tensors for shape description. Numer. Math. 2014;126:199–224.
   Web of Science ®Google Scholar
• Kang H, Lee H, Lim M. Construction of conformal mappings by generalized polarization tensors. Math. Methods Appl. Sci. 2015;38:1847–1854.
   Web of Science ®Google Scholar
• McLean W. Strongly elliptic systems and boundary integral equations. Cambridge: Cambridge University Press; 2000.
   Google Scholar
• Steinbach O. Numerical approximation methods for elliptic boundary value problems. New York (NY): Springer; 2008.
   Google Scholar
• Hsiao GC, Wendland WL. Boundary integral equations. Vol. 164, Applied mathematical sciences, Berlin: Springer-Verlag; 2008.
   Google Scholar
• Amrouche C, Girault V, Giroire J. Dirichlet and Neumann exterior problems for the n-dimensional Laplace operator: an approach in weighted Sobolev spaces. J. Math. Pures Appl. 1997;76:55–81.
   Web of Science ®Google Scholar
• Girault V, Giroire J, Sequeira A. A stream-function-vorticity variational formulation for the exterior Stokes problem in weighted Sobolev spaces. Math. Methods Appl. Sci. 1992;15:345–363.
   Web of Science ®Google Scholar
• Costabel M, Stephan E. A direct boundary integral equation method for transmission problems. J. Math. Anal. Appl. 1985;106:367–413.
   Web of Science ®Google Scholar
• Tucsnak M, Weiss G. Observation and control for operator semigroups. Birkhäuser Advanced Texts: Basler Lehrbücher [Birkhäuser advanced texts: Basel textbooks]. Basel: Birkhäuser Verlag; 2009.
   Google Scholar
• Ammari H, Deng Y, Kang H, et al. Reconstruction of inhomogeneous conductivities via the concept of generalized polarization tensors. Ann. Inst. H. Poincaré Anal. Non Linéaire. 2014;31:877–897.
   Web of Science ®Google Scholar
• Pommerenke C. Boundary behaviour of conformal maps. Vol. 299, Grundlehren der Mathematischen Wissenschaften [Fundamental principles of mathematical sciences]. Berlin: Springer-Verlag; 1992.
   Google Scholar
• Hille E. Analytic function theory. Vol. II, Introductions to Higher Mathematics, Boston: Ginn and Co.; 1962.
   Google Scholar
• Alessandrini G, Vessella S. Lipschitz stability for the inverse conductivity problem. Adv. Appl. Math. 2005;35:207–241.
   Web of Science ®Google Scholar