Current-voltage measurements in electrical impedance tomography can be partially ordered with respect to definiteness of the associated self-adjoint Neumann-to-Dirichlet operators (NtD). With this ordering, a point-wise larger conductivity leads to smaller current-voltage measurements, and smaller conductivities lead to larger measurements.We present a converse of this simple monotonicity relation and use it to solve the shape reconstruction (aka inclusion detection) problem in EIT. The outer shape of a region where the conductivity differs from a known background conductivity can be found by simply comparing the measurements to that of smaller or larger test regions. † Birth name: Bastian Gebauer,

The most accurate model for real-life electrical impedance tomography is the complete electrode model, which takes into account electrode shapes and (usually unknown) contact impedances at electrode-object interfaces. When the electrodes are small, however, it is tempting to formally replace them by point sources. This simplifies the model considerably and completely eliminates the effect of contact impedance. In this work we rigorously justify such a point electrode model for the important case of having difference measurements ("relative data") as data for the reconstruction problem. We do this by deriving the asymptotic limit of the complete model for vanishing electrode size. This is supplemented by an analogous result for the case that the distance between two adjacent electrodes also tends to zero, thus providing a physical interpretation and justification of the so-called backscattering data introduced by two of the authors.

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A prominent result of Arridge and Lionheart (1998 Opt. Lett. 23 882-4) demonstrates that it is in general not possible to simultaneously recover both the diffusion (aka scattering) and the absorption coefficient in steadystate (dc) diffusion-based optical tomography. In this work we show that it suffices to restrict ourselves to piecewise constant diffusion and piecewise analytic absorption coefficients to regain uniqueness. Under this condition both parameters can simultaneously be determined from complete measurement data on an arbitrarily small part of the boundary.

This work extends monotonicity-based methods in inverse problems to the case of the Helmholtz (or stationary Schrödinger) equation (∆ + k 2 q)u = 0 in a bounded domain for fixed non-resonance frequency k > 0 and real-valued scattering coefficient function q. We show a monotonicity relation between the scattering coefficient q and the local Neumann-Dirichlet operator that holds up to finitely many eigenvalues. Combining this with the method of localized potentials, or Runge approximation, adapted to the case where finitely many constraints are present, we derive a constructive monotonicity-based characterization of scatterers from partial boundary data. We also obtain the local uniqueness result that two coefficient functions q 1 and q 2 can be distinguished by partial boundary data if there is a neighborhood of the boundary part where q 1 ≥ q 2 and q 1 ≡ q 2 . and the local (or partial) Neumann-to-Dirichlet (NtD) operatorwhere u ∈ H 1 (Ω) solves (1.1) with Neumann data ∂ ν u| ∂Ω = g on Σ, 0 else.Here Σ ⊆ ∂Ω is assumed to be an arbitrary non-empty relatively open subset of ∂Ω.Since k is a non-resonance frequency, Λ(q) is well defined and is easily shown to be a self-adjoint compact operator.We will show that2. The Helmholtz equation in a bounded domain. We start by summarizing some properties of the Neumann-to-Dirichlet-operators, discuss well-posedness and the role of resonance frequencies, and state a unique continuation result for the Helmholtz equation in a bounded domain.

We discuss a time-harmonic inverse scattering problem for the Helmholtz equation with compactly supported penetrable and possibly inhomogeneous scattering objects in an unbounded homogeneous background medium, and we develop a monotonicity relation for the far field operator that maps superpositions of incident plane waves to the far field patterns of the corresponding scattered waves. We utilize this monotonicity relation to establish novel characterizations of the support of the scattering objects in terms of the far field operator. These are related to and extend corresponding results known from factorization and linear sampling methods to determine the support of unknown scattering objects from far field observations of scattered fields. An attraction of the new characterizations is that they only require the refractive index of the scattering objects to be above or below the refractive index of the background medium locally and near the boundary of the scatterers. An important tool to prove these results are so-called localized wave functions that have arbitrarily large norm in some prescribed region while at the same time having arbitrarily small norm in some other prescribed region. We present numerical examples to illustrate our theoretical findings. Mathematics subject classifications (MSC2010): 35R30, (65N21)

For the linearized reconstruction problem in Electrical Impedance Tomography (EIT) with the Complete Electrode Model (CEM), Lechleiter and Rieder (2008 Inverse Problems 24 065009) have shown that a piecewise polynomial conductivity on a fixed partition is uniquely determined if enough electrodes are being used. We extend their result to the full non-linear case and show that measurements on a sufficiently high number of electrodes uniquely determine a conductivity in any finite-dimensional subset of piecewise-analytic functions. We also prove Lipschitz stability, and derive analogue results for the continuum model, where finitely many measurements determine a finite-dimensional Galerkin projection of the Neumann-to-Dirichlet operator on a boundary part.

We consider an inverse problem for the fractional Schrödinger equation by using monotonicity formulas. We provide if-and-only-if monotonicity relations between positive bounded potentials and their associated nonlocal Dirichlet-to-Neumann maps. Based on the monotonicity relation, we can prove uniqueness for the nonlocal Calderón problem in a constructive manner. Secondly, we offer a reconstruction method for an unknown obstacles in a given domain. Our method is independent of the dimension and only requires the background solution of the fractional Schrödinger equation.