Integral equations for shape and impedance reconstruction in corrosion detection

In a simply connected planar domain D a pair of Cauchy data of a harmonic function u is given on an accessible part of the boundary curve, and on the non-accessible part u is supposed to satisfy a homogeneous impedance boundary condition. We consider the inverse problems to recover the non-accessible part of the boundary or the impedance function. Our approach extends the method proposed by Kress and Rundell (2005 Inverse Problems 21 1207–23) for the corresponding problem to recover the interior boundary curve of a doubly connected planar domain and can be considered complementary to the potential approach developed by Cakoni and Kress (2007 Inverse Problems Imaging 1 229–45). It is based on a system of nonlinear and ill-posed integral equations which is solved iteratively by linearization. We present the mathematical foundation of the method and, in particular, establish injectivity for the linearized system at the exact solution when the impedance function is known. Numerical reconstructions will show the feasibility of the method.

[1]  Rainer Kress,et al.  Nonlinear integral equations for the inverse electrical impedance problem , 2007 .

[2]  C. Pagani,et al.  Identifiability problems of defects with the Robin condition , 2009 .

[4]  M. Choulli.  An inverse problem in corrosion detection: stability estimates , 2004 .

[5]  Mohamed Jaoua,et al.  Identification of Robin coefficients by the means of boundary measurements , 1999 .

[6]  Recovering nonlinear terms in an inverse boundary value problem for Laplace's equation , 2007 .

[7]  Luca Rondi,et al.  The stability for the Cauchy problem for elliptic equations , 2009, 0907.2882.

[8]  W. McLean.  Strongly Elliptic Systems and Boundary Integral Equations , 2000 .

[9]  V. Bacchelli,et al.  Uniqueness for the determination of unknown boundary and impedance with the homogeneous Robin condition , 2008 .

[10]  A. Vogt.  Analytische und numerische Untersuchung von direkten und inversen Randwertproblemen in Gebieten mit Ecken mittels Integralgleichungsmethoden , 2002 .

[11]  Ivan G. Graham,et al.  An optimal order collocation method for first kind boundary integral equations on polygons , 1995 .

[12]  D. Colton,et al.  The direct and inverse scattering problems for partially coated obstacles , 2001 .

[13]  Fadil Santosa,et al.  A Method for Imaging Corrosion Damage in Thin Plates from Electrostatic Data , 1995 .

[14]  Costabel Martin,et al.  A Singularly mixed boundary value problem , 1996 .

[15]  R. Kress.  Linear Integral Equations , 1989 .

[16]  Ernst P. Stephan,et al.  On the integral equation method for the plane mixed boundary value problem of the Laplacian , 1979 .

[17]  A. Kirsch.  An Introduction to the Mathematical Theory of Inverse Problems , 1996, Applied Mathematical Sciences.

[18]  David Elliott,et al.  Sigmoidal Transformations and the Trapezoidal Rule , 1998 .

[19]  Thouraya Baranger,et al.  Solving Cauchy problems by minimizing an energy-like functional , 2006 .

[20]  Rainer Kress,et al.  Nonlinear Integral Equations for Solving Inverse Boundary Value Problems for Inclusions and Cracks , 2006 .

[21]  Fioralba Cakoni,et al.  Integral equations for inverse problems in corrosion detection from partial Cauchy data , 2007 .

[22]  G. Alessandrini,et al.  Stable determination of corrosion by a single electrostatic boundary measurement , 2003 .

[23]  Roland Potthast,et al.  Frechet differentiability of boundary integral operators in inverse acoustic scattering , 1994 .

[24]  Rainer Kress,et al.  A Nyström method for boundary integral equations in domains with corners , 1990 .

[25]  Ian H. Sloan,et al.  On integral equations of the first kind with logarithmic kernels , 1988 .

[26]  G. Inglese,et al.  An inverse problem in corrosion detection , 1997 .

[27]  R. Kress,et al.  Nonlinear integral equations and the iterative solution for an inverse boundary value problem , 2005 .

[28]  Fadil Santosa,et al.  Nondestructive evaluation of corrosion damage using electrostatic measurements , 1995 .

[29]  W. Rundell.  Recovering an obstacle and its impedance from Cauchy data , 2008 .

[30]  Martin Costabel,et al.  A singularly perturbed mixed boundary value problem , 1996 .