Abstract. This work reformulates the complete electrode model of electrical impedance tomography in order to enable more efficient numerical solution. The model traditionally assumes constant contact conductances on all electrodes, which leads to a discontinuous Robin boundary condition since the gaps between the electrodes can be described by vanishing conductance. As a consequence, the regularity of the electromagnetic potential is limited to less than two square-integrable weak derivatives, which negatively affects the convergence of, e.g., the finite element method. In this paper, a smoothened model for the boundary conductance is proposed, and the unique solvability and improved regularity of the ensuing boundary value problem are proven. Numerical experiments demonstrate that the proposed model is both computationally feasible and also compatible with real-world measurements. In particular, the new model allows faster convergence of the finite element method.

Electrical impedance tomography aims at reconstructing the interior electrical conductivity from surface measurements of currents and voltages. As the current-voltage pairs depend nonlinearly on the conductivity, impedance tomography leads to a nonlinear inverse problem. Often, the forward problem is linearized with respect to the conductivity and the resulting linear inverse problem is regarded as a subproblem in an iterative algorithm or as a simple reconstruction method as such. In this paper, we compare this basic linearization approach to linearizations with respect to the resistivity or the logarithm of the conductivity. It is numerically demonstrated that the conductivity linearization often results in compromised accuracy in both forward and inverse computations. Inspired by these observations, we present and analyze a new linearization technique which is based on the logarithm of the Neumannto-Dirichlet operator. The method is directly applicable to discrete settings, including the complete electrode model. We also consider Fréchet derivatives of the logarithmic operators. Numerical examples indicate that the proposed method is an accurate way of linearizing the problem of electrical impedance tomography.n. hyvönen and l. mustonen 2 respect to the conductivity can be computed explicitly [4,14,18]. By applying the chain rule of Banach spaces, or via direct computation, it is possible to consider derivatives with respect to the resistivity or the logarithm of the conductivity as well. In this paper, we compare the linearization errors resulting from these different input parametrizations of the electrical properties. We also discuss the corresponding methods for the complete electrode model (CEM), which is an accurate model for practical EIT measurements [5,21].As the main novelty, we introduce the logarithm of the Neumann-to-Dirichlet operator and use its differentiability properties to construct a new linearization method for EIT. Loosely speaking, the traditional forward operator is replaced by an operator that maps the logarithm of the conductivity to the logarithm of the boundary potentials, the latter being understood in a sense of linear operators. The corresponding least squares inversion algorithm then fits the computed logarithmic potentials to the logarithm of the measurement matrix. Although this logarithmic forward operator is still nonlinear, numerical experiments show that the resulting linearization errors are in most cases smaller than with any other considered linearization method.This paper is organized as follows. In Section 2, the continuum forward model of EIT and the related Neumann-to-Dirichlet operator are reviewed. We write the dependence of the Neumannto-Dirichlet operator on the electrical properties in three different ways and recall also the corresponding Fréchet derivatives. Section 3 presents the formal definition for the logarithm of the Neumann-to-Dirichlet operator and studies its differentiability and other properties as an unbounded operator on the space of square-integ...

Abstract. This work considers the inclusion detection problem of electrical impedance tomography with stochastic conductivities. It is shown that a conductivity anomaly with a random conductivity can be identified by applying the Factorization Method or the Monotonicity Method to the mean value of the corresponding Neumann-to-Dirichlet map provided that the anomaly has high enough contrast in the sense of expectation. The theoretical results are complemented by numerical examples in two spatial dimensions.

Abstract. The objective of electrical impedance tomography is to reconstruct the internal conductivity of a physical body based on measurements of current and potential at a finite number of electrodes attached to its boundary. Although the conductivity is the quantity of main interest in impedance tomography, a real-world measurement configuration includes other unknown parameters as well: the information on the contact resistances, electrode positions and body shape is almost always incomplete. In this work, the dependence of the electrode measurements on all aforementioned model properties is parametrized via polynomial collocation. The availability of such a parametrization enables efficient simultaneous reconstruction of the conductivity and other unknowns by a Newton-type output least squares algorithm, which is demonstrated by two-dimensional numerical experiments based on both noisy simulated data and experimental data from two water tanks.

We provide a mathematical formulation and develop a computational framework for identifying multiple strains of microorganisms from mixed samples of DNA. Our method is applicable in public health domains where efficient identification of pathogens is paramount, e.g., for the monitoring of disease outbreaks. We formulate strain identification as an inverse problem that aims at simultaneously estimating a binary matrix (encoding presence or absence of mutations in each strain) and a real-valued vector (representing the mixture of strains) such that their product is approximately equal to the measured data vector. The problem at hand has a similar structure to blind deconvolution, except for the presence of binary constraints, which we enforce in our approach. Following a Bayesian approach, we derive a posterior density. We present two computational methods for solving the non-convex maximum a posteriori estimation problem. The first one is a local optimization method that is made efficient and scalable by decoupling the problem into smaller independent subproblems, whereas the second one yields a global minimizer by converting the problem into a convex mixed-integer quadratic programming problem. The decoupling approach also provides an efficient way to integrate over the posterior. This provides useful information about the ambiguity of the underdetermined problem and, thus, the uncertainty associated with numerical solutions. We evaluate the potential and limitations of our framework in silico using synthetic and experimental data with available ground truths.