Lipschitz stability for the electrical impedance tomography problem: the complex case

In this paper we investigate the boundary value problem {div(\gamma\nabla u)=0 in \Omega, u=f on \partial\Omega where $\gamma$ is a complex valued $L^\infty$ coefficient, satisfying a strong ellipticity condition. In Electrical Impedance Tomography, $\gamma$ represents the admittance of a conducting body. An interesting issue is the one of determining $\gamma$ uniquely and in a stable way from the knowledge of the Dirichlet-to-Neumann map $\Lambda_\gamma$. Under the above general assumptions this problem is an open issue. In this paper we prove that, if we assume a priori that $\gamma$ is piecewise constant with a bounded known number of unknown values, then Lipschitz continuity of $\gamma$ from $\Lambda_\gamma$ holds

Uniqueness and Lipschitz stability for the identification of Lam\'e parameters from boundary measurements

In this paper we consider the problem of determining an unknown pair $\lambda$, $\mu$ of piecewise constant Lam\'{e} parameters inside a three dimensional body from the Dirichlet to Neumann map. We prove uniqueness and Lipschitz continuous dependence of $\lambda$ and $\mu$ from the Dirichlet to Neumann map

A transmission problem on a polygonal partition: regularity and shape differentiability

We consider a transmission problem on a polygonal partition for the two-dimensional conductivity equation. For suitable classes of partitions we establish the exact behaviour of the gradient of solutions in a neighbourhood of the vertexes of the partition. This allows to prove shape differentiability of solutions and to establish an explicit formula for the shape derivative

Algorithms for Social Good: A Study of Fairness and Bias in Automated Data-Driven Decision-Making Systems

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A variational method for quantitative photoacoustic tomography with piecewise constant coefficients

In this article, we consider the inverse problem of determining spatially heterogeneous absorption and diffusion coefficients from a single measurement of the absorbed energy (in the steady-state diffusion approximation of light transfer). This problem, which is central in quantitative photoacoustic tomography, is in general ill-posed since it admits an infinite number of solution pairs. We show that when the coefficients are known to be piecewise constant functions, a unique solution can be obtained. For the numerical determination of the coefficients, we suggest a variational method based based on an Ambrosio-Tortorelli-approximation of a Mumford-Shah-like functional, which we implemented numerically and tested on simulated two-dimensional data

Lipschitz stability of an inverse boundary value problem for a Schr\"{o}dinger type equation

In this paper we study the inverse boundary value problem of determining the potential in the Schr\"{o}dinger equation from the knowledge of the Dirichlet-to-Neumann map, which is commonly accepted as an ill-posed problem in the sense that, under general settings, the optimal stability estimate is of logarithmic type. In this work, a Lipschitz type stability is established assuming a priori that the potential is piecewise constant with a bounded known number of unknown values

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