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

Laura Bazahica, Vesa Kaarnioja, Lassi Roininen

TL;DR

The paper addresses uncertainty quantification for electrical impedance tomography (EIT) under a Bayesian framework with a parametric complete electrode model (pCEM). It develops theory for randomly shifted rank-1 lattice quasi-Monte Carlo rules to approximate posterior expectations, deriving parametric regularity results under a Gevrey class and providing a CBC-based QMC estimator. An error analysis yields dimension-robust convergence rates, and the authors validate the approach with simulated boundary measurements, reporting credible envelopes for reconstructions. The work demonstrates that QMC can outperform Monte Carlo in both accuracy and efficiency for realistic EIT, and it outlines potential enhancements such as importance sampling and joint recovery of additional unknowns.

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.

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

TL;DR

The paper addresses uncertainty quantification for electrical impedance tomography (EIT) under a Bayesian framework with a parametric complete electrode model (pCEM). It develops theory for randomly shifted rank-1 lattice quasi-Monte Carlo rules to approximate posterior expectations, deriving parametric regularity results under a Gevrey class and providing a CBC-based QMC estimator. An error analysis yields dimension-robust convergence rates, and the authors validate the approach with simulated boundary measurements, reporting credible envelopes for reconstructions. The work demonstrates that QMC can outperform Monte Carlo in both accuracy and efficiency for realistic EIT, and it outlines potential enhancements such as importance sampling and joint recovery of additional unknowns.

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.

Paper Structure

This paper contains 10 sections, 12 theorems, 68 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Parametric complete electrode model
  4. Parametric regularity
  5. Forward problem
  6. Bayesian inverse problem
  7. Quasi-Monte Carlo
  8. Error analysis
  9. Numerical experiments
  10. Conclusions

Key Result

Lemma 3.1

Let Assumption assumption hold and let $(u(\cdot,{\boldsymbol{y}}),U({\boldsymbol{y}}))\in\mathcal{H}$ be the solution to eq:variational. Then $\partial_{y_j}(u(\cdot,{\boldsymbol{y}}),U({\boldsymbol{y}}))$ exists and belongs to $\mathcal{H}$ for all $j\geq1$ and ${\boldsymbol{y}}\in \Upsilon$.

Figures (3)

  • Figure 1: Two randomly generated ground truth conductivity fields which were used to simulate the noisy electrode measurements for numerical inversion.
  • Figure 3: The credible margins of the approximate 95% credible envelopes corresponding to the reconstructions in Figure \ref{['fig:2a']} and Figure \ref{['fig:2b']}.
  • Figure 4: The R.M.S. errors for reconstructions corresponding to the reconstructions illustrated in Figures \ref{['fig:2a']} and \ref{['fig:2b']} with $R=16$ random shifts plotted alongside the corresponding least squares fits.

Theorems & Definitions (18)

  • Lemma 3.1
  • proof
  • Corollary 3.2
  • Lemma 3.3: Recursive bound
  • proof
  • Lemma 3.4: cf. gk24plus
  • Lemma 3.5: Inductive bound
  • Lemma 3.6
  • proof
  • Lemma 3.7
  • ...and 8 more