Bayesian anomaly detection in heterogeneous media with applications to geophysical tomography

Abstract

In this paper, we consider the problem of detecting a parameterized anomaly in an isotropic, stationary and ergodic conductivity random field via electrical impedance tomography. A homogenization result for a stochastic forward problem built on the complete electrode model is derived, which serves as the basis for a two-stage numerical method in the framework of Bayesian inverse problems. The novelty of this method lies in the introduction of an enhanced error model accounting for the approximation errors that result from reducing the full forward model to a homogenized one. In the first stage, a MAP estimate for the reduced forward model equipped with the enhanced error model is computed. Then, in the second stage, a bootstrap prior based on the first stage results is defined and the resulting posterior distribution is sampled via Markov chain Monte Carlo. We provide the theoretical foundation of the proposed method, discuss different aspects of a numerical implementation and present numerical experiments to support our findings.

Anomaly Detection in Random Heterogeneous Media

“…The novelty of this method lies in the introduction of an enhanced error model accounting for the approximation errors that result from reducing the full forward model to a homogenized one. We proceed along the lines of the recent article [148] by the author: In the first stage, a MAP estimate for the reduced forward model equipped with the enhanced error model is computed. Then, in the second stage, a bootstrap prior based on the first stage results is defined and the resulting posterior distribution is sampled via Markov chain Monte Carlo.…”

From Feynman–Kac formulae to numerical stochastic homogenization in electrical impedance tomography

“…Knowledge of u yields the corresponding electrode current vector J ∈ R N via (9) a setting is found for instance in geophysical applications, where measurements can only be taken on the surface, cf., e.g., [55]. Let (Γ, G, P) be a probability space and let Θ : Γ → Γ denote an ergodic ddimensional dynamical system, i.e., a family of automorphisms {Θ x , x ∈ R d } which satisfies the following conditions:…”

“…In this subsection, we provide the theoretical foundation for the convergence analysis of numerical homogenization methods based on simulation of the underlying diffusion process. More precisely, a rigorous convergence analysis of such a method requires a quantitative estimate that is stronger than the qualitative result (55), which was obtained in [42] using merely the central limit theorem for martingales. We provide such a quantitative result in the following theorem by generalizing a classical argument due to Kipnis and Varadhan [40].…”

“…Recently, the second author has proposed a novel method for the detection of conductivity anomalies in a random background conductivity which is based on homogenization of the underlying stochastic boundary value problem, cf. [55]. Although the homogenization theory for elliptic divergence form operators is welldeveloped, cf., e.g., [6,47,48,64], the numerical approximation of the effective conductivity in the random setting still poses major challenges.…”

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

“…from boundary measurements of the electric potential u and the corresponding current on the boundary of a bounded, convex domain D ⊂ R d , d = 2, 3, with piecewise smooth boundary ∂D and connected complement. Due to the limited capabilities of static EIT, many practical applications focus on the detection of conductivity anomalies in a known background conductivity rather than conductivity imaging, cf., e.g., Pursiainen [38] and the recent work [42] by the second author. In this work, we consider such an anomaly detection problem, where a perfectly conducting inclusion occupies a region T inside the domain D. A possible practical application modeled by this setting is breast cancer detection, where the electric conductivity of highwater-content tissue, such as malignant tumors, is approximately one order of magnitude higher than the conductivity of low-water-content tissue, such as fat, which is the main component of healthy breast tissue, cf.…”