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We develop a boundary integral equation-based numerical method to solve forthe electrostatic potential in two dimensions, inside a medium with piecewiseconstant conductivity, where the boundary condition is given by the completeelectrode model (CEM). The CEM is seen as the most accurate model of thephysical setting where electrodes are placed on the surface of an electricallyconductive body, and currents are injected through the electrodes and theresulting voltages are measured again on these same electrodes. The integralequation formulation is based on expressing the electrostatic potential as thesolution to a finite number of Laplace equations which are coupled throughboundary matching conditions. This allows us to re-express the solution interms of single layer potentials; the problem is thus re-cast as a system ofintegral equations on a finite number of smooth curves. We discuss an adaptivemethod for the solution of the resulting system of mildly singular integralequations. This solver is both fast and accurate. We then present a numericalinverse solver for electrical impedance tomography (EIT) which uses our forwardsolver at its core. To demonstrate the applicability of our results we test ournumerical methods on an open electrical impedance tomography data set providedby the Finnish Inverse Problems Society.

A boundary integral equation method for the complete electrode model in electrical impedance tomography with tests on real-world data