Electrical resistance tomography: complementarity and quadratic models

Electrical resistance tomography, the nonlinear inverse problem of retrieving the resistivity of an unknown body from steady-state boundary measurements, is addressed by a quadratic model that approximates the operator mapping the unknown parameters into boundary data. The forward problem, i.e. the computation of the measurements for a given resistivity, is solved by averaging the solutions, computed by a finite element method, of the two complementary formulations. Numerical results show that the complementary solutions, for the cases under investigation, are affected by approximately opposite numerical errors. Therefore, their average gives a good estimate of the solution even for coarse meshes. The overhead resulting from the need to compute two solutions is balanced by the selection of a coarse mesh. In particular, upper and lower bounds to the discretization errors are computed by the complementary solutions and used to optimize the finite element mesh. The retrieval of the resistivity is stated as the search for the minimizer of an error functional related to the quadratic model. The algebraic properties of quadratic models are exploited to estimate the number of linearly independent equations and to obtain an upper bound to the maximum number of unknown parameters, which guarantee an error functional without local minima.