Uncertainty quantification for electrical impedance tomography using quasi-Monte Carlo methods

TL;DR

This paper applies quasi-Monte Carlo methods with rank-1 lattice rules to improve uncertainty quantification in electrical impedance tomography, demonstrating faster convergence and effective reconstruction of interior conductivity from boundary measurements.

Contribution

It provides a theoretical framework for using QMC methods in EIT inverse problems, showing dimension-independent convergence rates and validating with numerical experiments.

Findings

QMC methods achieve faster convergence than Monte Carlo in EIT

Theoretical proof of dimension-independent convergence rates

Numerical results confirm improved reconstruction accuracy

Abstract

The theoretical development of quasi-Monte Carlo (QMC) methods for uncertainty quantification of partial differential equations (PDEs) is typically centered around simplified model problems such as elliptic PDEs subject to homogeneous zero Dirichlet boundary conditions. In this paper, we present a theoretical treatment of the application of randomly shifted rank-1 lattice rules to electrical impedance tomography (EIT). EIT is an imaging modality, where the goal is to reconstruct the interior conductivity of an object based on electrode measurements of current and voltage taken at the boundary of the object. This is an inverse problem, which we tackle using the Bayesian statistical inversion paradigm. As the reconstruction, we consider QMC integration to approximate the unknown conductivity given current and voltage measurements. We prove under moderate assumptions placed on the parameterization of the unknown conductivity that the QMC approximation of the reconstructed estimate has a dimension-independent, faster-than-Monte Carlo cubature convergence rate. Finally, we present numerical results for examples computed using simulated measurement data.