A Partially Reflecting Random Walk on Spheres Algorithm for Electrical Impedance Tomography
For practitioners of electrical impedance tomography, this work provides a scalable probabilistic method for the forward problem, which is crucial for solving the inverse problem.
The paper introduces a partially reflecting random walk on spheres estimator for the voltage-to-current map in electrical impedance tomography, enabling embarrassingly parallel Monte Carlo computation. The method achieves reduced variance via a control variate conditional sampling technique, with theoretical and experimental analysis of bias and variance.
In this work, we develop a probabilistic estimator for the voltage-to-current map arising in electrical impedance tomography. This novel so-called partially reflecting random walk on spheres estimator enables Monte Carlo methods to compute the voltage-to-current map in an embarrassingly parallel manner, which is an important issue with regard to the corresponding inverse problem. Our method uses the well-known random walk on spheres algorithm inside subdomains where the diffusion coefficient is constant and employs replacement techniques motivated by finite difference discretization to deal with both mixed boundary conditions and interface transmission conditions. We analyze the global bias and the variance of the new estimator both theoretically and experimentally. In a second step, the variance is considerably reduced via a novel control variate conditional sampling technique.