A partially reflecting random walk on spheres algorithm for electrical impedance tomography

Sylvain Maire (1) , Martin Simon (2)

1 TOSCA - TO Simulate and CAlibrate stochastic models
2 MI - Institut für Mathematik

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.

Mots clés: Monte Carlo methods; electrical impedance tomography; random walk on spheres; reflecting Brownian motion; discontinuous diffusion coefficient; random diffusion coefficient; variance reduction

Domaines: Equations aux dérivées partielles [math.AP]

Sylvain Maire, Martin Simon. A partially reflecting random walk on spheres algorithm for electrical impedance tomography. Journal of Computational Physics, 2015, 303, ⟨10.1016/j.jcp.2015.10.005⟩. ⟨hal-01253538⟩