A boundary element method for the numerical inversion of discontinuous anisotropic conductivities

摘要

An inverse problem is considered to identify the geometry of discontinuities in a conductive material Ωis contained in R~d with anisotropic conductivity (I + (K - I)χ_D) from Cauchy data measurements taken on the boundary partial deriv Ω, where D is contained in Ω, K is a symmetric and positive definite tensor not equal to the identity tensor and χ_D is the characteristic function of the domain D. As an example this models the determination of the shape, size and location of the anisotropic inner core of the Earth from measurements taken at its mantle. There are also other applications in electrical impedance tomography or in non-destructive testing of materials using infra-red scanning. We develop an integral representation of the solution and we propose an efficient boundary element method in conjunction with a least-squares constrained minimisation procedure to detect an anisotropic inclusion D, such as a circle, by a single boundary measurement.