Piecewise Constant Conductivity Research Articles

Determining an anisotropic conductivity by boundary measurements: Stability at the boundary

We consider the inverse problem of determining, the possibly anisotropic, conductivity of a body Ω⊂Rn, n≥3, by means of the so-called local Neumann-to-Dirichlet map on a curved portion Σ of its boundary ∂Ω. Motivated by the uniqueness result for piecewise constant anisotropic conductivities proved in Inverse Problems 33 (2018), 125013, we provide a Hölder stability estimate on Σ when the conductivity is a-priori known to be a constant matrix near Σ.

Local recovery of a piecewise constant anisotropic conductivity in EIT on domains with exposed corners

We study the local recovery of an unknown piecewise constant anisotropic conductivity in electric impedance tomography on certain bounded Lipschitz domains Ω in with corners. The measurement is conducted on a connected open subset of the boundary of Ω containing corners and is given as a localized Neumann-to-Dirichlet map. The above unknown conductivity is defined via a decomposition of Ω into polygonal cells. Specifically, we consider a parallelogram-based decomposition and a trapezoid-based decomposition. We assume that the decomposition is known, but the conductivity on each cell is unknown. We prove that the local recovery is almost surely true near a known piecewise constant anisotropic conductivity γ 0. We do so by proving that the injectivity of the Fréchet derivative of the forward map F, say, at γ 0 is almost surely true. The proof presented, here, involves defining different classes of decompositions for γ 0 and a perturbation or contrast H in a proper way so that we can find in the interior of a cell for γ 0 exposed single or double corners of a cell of suppH for the former decomposition and latter decomposition, respectively. Then, by adapting the usual proof near such corners, we establish the aforementioned injectivity.

A Bernstein–von-Mises theorem for the Calderón problem with piecewise constant conductivities

This note considers a finite dimensional statistical model for the Calderón problem with piecewise constant conductivities. In this setting it is shown that injectivity of the forward map and its linearisation suffice to prove the invertibility of the information operator, resulting in a Bernstein–von-Mises theorem and optimality guarantees for estimation by Bayesian posterior means.

Inverse problems on low-dimensional manifolds

We consider abstract inverse problems between infinite-dimensional Banach spaces. These inverse problems are typically nonlinear and ill-posed, making the inversion with limited and noisy measurements a delicate process. In this work, we assume that the unknown belongs to a finite-dimensional manifold: this assumption arises in many real-world scenarios where natural objects have a low intrinsic dimension and belong to a certain submanifold of a much larger ambient space. We prove uniqueness and Hölder and Lipschitz stability results in this general setting, also in the case when only a finite discretization of the measurements is available. Then, a Landweber-type reconstruction algorithm from a finite number of measurements is proposed, for which we prove global convergence, thanks to a new criterion for finding a suitable initial guess.These general results are then applied to several examples, including two classical nonlinear ill-posed inverse boundary value problems. The first is Calderón’s inverse conductivity problem, for which we prove a Lipschitz stability estimate from a finite number of measurements for piece-wise constant conductivities with discontinuities on an unknown triangle. A similar stability result is then obtained for Gel’fand-Calderón’s problem for the Schrödinger equation, in the case of piece-wise constant potentials with discontinuities on a finite number of non-intersecting balls.

Simplified reconstruction of layered materials in EIT

This short note considerably simplifies a reconstruction method by the author (Garde, 2020), for reconstructing piecewise constant layered conductivities (PCLC) from partial boundary measurements in electrical impedance tomography. Theory from monotonicity-based reconstruction of extreme inclusions eliminates most of the bookkeeping related to multiple components of each layer, and also simplifies the involved test operators. Moreover, the method no longer requires a priori lower and upper bounds to the unknown conductivity values.

A Numerical Method for Determining the Inhomogeneity Boundary in the Electrical Impedance Tomography Problem in the Case of Piecewise-Constant Conductivity

A Numerical Method for Determining the Inhomogeneity Boundary in the Electrical Impedance Tomography Problem in the Case of Piecewise-Constant Conductivity

CORRECT DETERMINATION OF INTEGRAL CONDUCTIVITY IN SOLVING INCORRECT INVERSE PROBLEMS OF ELECTROMAGNEIC LOGGING

The paper suggests and investigates a problem statement of well-logging inverse problem that is based on the integral conductivity parameter to describe a geoelectric section. Approach was introduced for a layered cylindrical model with radially heterogeneous continuous distribution of electric properties that parametrize the problem with a function of total longitudinal conductivity. The results of hydrodynamic modeling for oil/fresh water- and brine-based drilling muds were used to study multiple propagation resistivity tool signal equivalency for two classes of models with continuous and piece-wise constant conductivity distribution. Physically based algorithm enabling one to convert one model class to the other, preserving the signal equivalency was proposed. It was demonstrated that the radial models with different radial conductivity distribution and similar integral conductivity curves are equivalent. This fact lays a rationale of using the integral conductivity parameters along with conductivity while inversion. The integral conductivity parameter can be used to build the functionals, whose minimization improves algorithm stability and enables determining functional parameters in hydrodynamic filtration models.

A shape optimization approach for electrical impedance tomography with point measurements

Working within the class of piecewise constant conductivities, the inverse problem of electrical impedance tomography can be recast as a shape optimization problem where the discontinuity interface is the unknown. Using Gröger’s -estimates for mixed boundary value problems, the averaged adjoint method is extended to the case of Banach spaces, which allows one to compute the derivative of shape functionals involving point evaluations. We compute the corresponding distributed expression of the shape derivative and show that it may contain Dirac measures in addition to the usual domain integrals. We use this distributed shape derivative to devise a numerical algorithm, show various numerical results supporting the method, and based on these results we discuss the influence of the point measurements patterns on the quality of the reconstructions.

Reconstruction of piecewise constant layered conductivities in electrical impedance tomography

This work presents a new constructive uniqueness proof for Calderón’s inverse problem of electrical impedance tomography, subject to local Cauchy data, for a large class of piecewise constant conductivities that we call piecewise constant layered conductivities (PCLC). The resulting reconstruction method only relies on the physically intuitive monotonicity principles of the local Neumann-to-Dirichlet map, and therefore the method lends itself well to efficient numerical implementation and generalization to electrode models. Several direct reconstruction methods exist for the related problem of inclusion detection, however they share the property that “holes in inclusions” or “inclusions-within-inclusions” cannot be determined. One such method is the monotonicity method of Harrach, Seo, and Ullrich, and in fact the method presented here is a modified variant of the monotonicity method which overcomes this problem. More precisely, the presented method abuses that a PCLC type conductivity can be decomposed into nested layers of positive and/or negative perturbations that, layer-by-layer, can be determined via the monotonicity method. The conductivity values on each layer are found via basic one-dimensional optimization problems constrained by monotonicity relations.

Adaptive reconstruction for electrical impedance tomography with a piecewise constant conductivity

In this work we propose and analyze a numerical method for electrical impedance tomography to recover a piecewise constant conductivity from boundary voltage measurements. It is based on standard Tikhonov regularization with a Modica–Mortola penalty functional and adaptive mesh refinement using suitable a posteriori error estimators of residual type that involve the state, adjoint and variational inequality in the necessary optimality condition and a separate marking strategy. We prove the convergence of the adaptive algorithm in the following sense: the sequence of discrete solutions contains a subsequence convergent to a solution of the continuous necessary optimality system. Several numerical examples are presented to illustrate the convergence behavior of the algorithm.

On mild and weak solutions for stochastic heat equations with piecewise-constant conductivity

We investigate a stochastic partial differential equation with second order elliptic operator in divergence form, having a piecewise constant diffusion coefficient, and driven by a space–time white noise. We introduce a notion of weak solution of this equation and prove its equivalence to the already known notion of mild solution.

Boosting the Maxwell double layer potential using a right spin factor

We construct new spin singular integral equations for solving scattering problems for Maxwell’s equations, both against perfect conductors and in media with piecewise constant permittivity, permeability and conductivity, improving and extending earlier formulations by the author. These differ in a fundamental way from classical integral equations, which use double layer potential operators, and have the advantage of having a better condition number, in particular in Fredholm sense and on Lipschitz regular interfaces, and do not suffer from spurious resonances. The construction of the integral equations builds on the observation that the double layer potential factorises into a boundary value problem and an ansatz. We modify the ansatz, inspired by a non-selfadjoint local elliptic boundary condition for Dirac equations.

Impact of spatially varying hydraulic conductivities on tumor interstitial fluid pressure distribution

Abstract This work deals with numerical computation of the interstitial fluid pressure distribution for various synthetic tumor configurations in two spatial dimensions. In order to attend a reasonable degree of realism, we also consider asymmetric configurations and use unstructured triangular grids to accurately discretize the computational domains. A multi point flux approximation method, previously used for porous medium flow simulations, is utilized for solving the corresponding Poisson equation for the pressure. We are particularly interested in the effects of spatial variation of the hydraulic conductivity as well as vessel hydraulic conductivities on the interstitial fluid pressure. Spatially varying hydraulic conductivity was first studied by Liu and Schlesinger [17] in some special situations using analytical methods. Here we complement their work by performing a set of relatively comprehensive numerical investigations. A considerable difference in the computed pressure profiles is found when comparing the results with the more classical approach using piecewise constant hydraulic conductivities. We also explore the response of the numerical model when varying i.e. size of necrotic core and vessel hydraulic conductivities. The pressure profiles display a striking dependence on the arterial hydraulic conductivity in the peripheral region. A strong dependence on the arterial hydraulic conductivity is also observed when introducing so called “vascular hot-spots” in the model as well as looking at non-symmetric geometries. Finally, we discuss our numerical results in light of other numerical studies of TIFP distribution and experimental studies of tumor biology. It is concluded that the computed results are in accordance with the general understanding of TIFP. Although we have worked on a simplified system, our results highlight the importance of describing hydraulic conductivity as a spatially varying parameter.

Power series analytical solution for 2D quasi-Laplace equation with piecewise constant conductivities

In this paper, the analytical solution for 2D quasi-Laplace equation with piecewise constant conductivities is provided. The analytical solution can be expressed as an infinite power series with a group of intrinsic non-integer power exponents around each singular point. Combined with the given boundary conditions, the coefficient of each term can be determined by numerical methods.

Finite Element Approximation of Electromagnetic Fields Using Nonfitting Meshes for Geophysics

We analyze the use of non-fitting meshes for simulating the propagation of electromagnetic waves inside the Earth. In order to take into account subcell conductivity variations, we propose a simple exact integration technique that avoids the use of parameter homogenization and employs standard edge finite element basis functions. For our geophysical applications, we consider a 3D Maxwell's system with piecewise constant conductivity and globally constant permittivity and permeability. The model is analysed and discretized using both the E and H-formulations. Our main contribution is to develop a sharp error estimate for both the electric and magnetic fields. In the presence of singularities, our estimate shows that the magnetic field approximation is converging faster than the electric field approximation. As a result, we conclude that error estimates available in the literature for Nedelec's elements with fitting meshes are sharp with respect to the electric field error, but provide pessimistic convergence rates for the magnetic field in our geophysical applications. Another surprising consequence of our analysis is that non-fitting meshes deliver the same convergence rate than fitting meshes to approximate the magnetic field. Our theoretical results are numerically illustrated via 2D experiments. For the analyzed cases, the accuracy loss due to the use of non-fitting meshes is limited, even for high conductivity contrasts.

Uniqueness on recovery of piecewise constant conductivity and inner core with one measurement

We consider the recovery of piecewise constant conductivity and an unknown inner core in inverse conductivity problem. We first show the unique recovery of the conductivity in a one layer structure without inner core by one measurement on any surface enclosing the unknown medium. Then we recover the unknown inner core in a one layer structure. We then show that in a two layer structure, the conductivity can be uniquely recovered by using one measurement.

Uniqueness for the electrostatic inverse boundary value problem with piecewise constant anisotropic conductivities

We discuss the inverse problem of determining the, possibly anisotropic, conductivity of a body when the so-called Neumann-to-Dirichlet map is locally given on a non-empty curved portion Σ of the boundary . We prove that anisotropic conductivities that are a priori known to be piecewise constant matrices on a given partition of Ω with curved interfaces can be uniquely determined in the interior from the knowledge of the local Neumann-to-Dirichlet map.

Reconstruction of a piecewise constant conductivity on a polygonal partition via shape optimization in EIT

In this paper, we develop a shape optimization-based algorithm for the electrical impedance tomography (EIT) problem of determining a piecewise constant conductivity on a polygonal partition from boundary measurements. The key tool is to use a distributed shape derivative of a suitable cost functional with respect to movements of the partition. Numerical simulations showing the robustness and accuracy of the method are presented for simulated test cases in two dimensions.

Critical Points for Elliptic Equations with Prescribed Boundary Conditions

This paper concerns the existence of critical points for solutions to second order elliptic equations of the form $\nabla\cdot \sigma(x)\nabla u=0$ posed on a bounded domain $X$ with prescribed boundary conditions. In spatial dimension $n=2$, it is known that the number of critical points (where $\nabla u=0$) is related to the number of oscillations of the boundary condition independently of the (positive) coefficient $\sigma$. We show that the situation is different in dimension $n\geq3$. More precisely, we obtain that for any fixed (Dirichlet or Neumann) boundary condition for $u$ on $\partial X$, there exists an open set of smooth coefficients $\sigma(x)$ such that $\nabla u$ vanishes at least at one point in $X$. By using estimates related to the Laplacian with mixed boundary conditions, the result is first obtained for a piecewise constant conductivity with infinite contrast, a problem of independent interest. A second step shows that the topology of the vector field $\nabla u$ on a subdomain is not modified for appropriate bounded, sufficiently high-contrast, smooth coefficients $\sigma(x)$. These results find applications in the class of hybrid inverse problems, where optimal stability estimates for parameter reconstruction are obtained in the absence of critical points. Our results show that for any (finite number of) prescribed boundary conditions, there are coefficients $\sigma(x)$ for which the stability of the reconstructions will inevitably degrade.

TOPOLOGICAL ALGORITHMS TO SOLVE INVERSE PROBLEM IN ELECTRICAL TOMOGRAPHY

In this paper, there were investigated topological algorithms to solve the inverse problem in electrical tomography. The level set method, material derivative, shape derivative and topological derivative are based on shape and topology optimization approach to electrical impedance tomography problems with piecewise constant conductivities. The cost of the numerical algorithm is enough good, because the shape is captured on a fixed grid. The proposed solution is initialized by using topological sensitivity analysis. Shape derivative and material derivative (or topological derivative) have been incorporated with level set methods to investigate shape optimization problems.