A Partially Reflecting Random Walk on Spheres Algorithm for Electrical Impedance Tomography

TL;DR

This paper introduces a novel probabilistic Monte Carlo estimator for electrical impedance tomography that efficiently computes voltage-to-current maps using a partially reflecting random walk on spheres, with variance reduction techniques.

Contribution

The paper presents a new partially reflecting random walk on spheres algorithm for EIT, combining probabilistic estimators with variance reduction for improved computational efficiency.

Findings

The estimator accurately computes voltage-to-current maps in EIT.

Variance reduction significantly improves estimator performance.

The method is validated both theoretically and experimentally.

Abstract

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.