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Inverse Problems & Imaging

2014 , Volume 8 , Issue 4

Special issue on complex geometrical optics (CGO) solutions

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Foreword
Sarah Hamilton, Kim Knudsen, Samuli Siltanen and Gunther Uhlmann
2014, 8(4): i-ii doi: 10.3934/ipi.2014.8.4i +[Abstract](245) +[PDF](106.9KB)
Abstract:
Complex Geometrical Optics (CGO) solutions have, for almost three decades, played a large role in the rigorous analysis of nonlinear inverse problems. They have the added bonus of also being useful in practical reconstruction algorithms. The main benefit of CGO solutions is to provide solutions in the form of almost-exponential functions that can be used in a variety of ways, for example for defining tailor-made nonlinear Fourier transforms to study the unique solvability of a nonlinear inverse problem.

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Stability of the Calderón problem in admissible geometries
Pedro Caro and Mikko Salo
2014, 8(4): 939-957 doi: 10.3934/ipi.2014.8.939 +[Abstract](537) +[PDF](449.3KB)
Abstract:
In this paper we prove log log type stability estimates for inverse boundary value problems on admissible Riemannian manifolds of dimension $n \geq 3$. The stability estimates correspond to the uniqueness results in [13]. These inverse problems arise naturally when studying the anisotropic Calderón problem.
Partial data for the Neumann-Dirichlet magnetic Schrödinger inverse problem
Francis J. Chung
2014, 8(4): 959-989 doi: 10.3934/ipi.2014.8.959 +[Abstract](308) +[PDF](499.4KB)
Abstract:
We show that an electric potential and magnetic field can be uniquely determined by partial boundary measurements of the Neumann-to-Dirichlet map of the associated magnetic Schrödinger operator. This improves upon the results in [4] by including the determination of a magnetic field. The main technical advance is an improvement on the Carleman estimate in [4]. This allows the construction of complex geometrical optics solutions with greater regularity, which are needed to deal with the first order term in the operator. This improved regularity of CGO solutions may have applications in the study of inverse problems in systems of equations with partial boundary data.
Numerical nonlinear complex geometrical optics algorithm for the 3D Calderón problem
Fabrice Delbary and Kim Knudsen
2014, 8(4): 991-1012 doi: 10.3934/ipi.2014.8.991 +[Abstract](486) +[PDF](1600.7KB)
Abstract:
The Calderón problem is the mathematical formulation of the inverse problem in Electrical Impedance Tomography and asks for the uniqueness and reconstruction of an electrical conductivity distribution in a bounded domain from the knowledge of the Dirichlet-to-Neumann map associated to the generalized Laplace equation. The 3D problem was solved in theory in late 1980s using complex geometrical optics solutions and a scattering transform. Several approximations to the reconstruction method have been suggested and implemented numerically in the literature, but here, for the first time, a complete computer implementation of the full nonlinear algorithm is given. First a boundary integral equation is solved by a Nyström method for the traces of the complex geometrical optics solutions, second the scattering transform is computed and inverted using fast Fourier transform, and finally a boundary value problem is solved for the conductivity distribution. To test the performance of the algorithm highly accurate data is required, and to this end a boundary element method is developed and implemented for the forward problem. The numerical reconstruction algorithm is tested on simulated data and compared to the simpler approximations. In addition, convergence of the numerical solution towards the exact solution of the boundary integral equation is proved.
A real-time D-bar algorithm for 2-D electrical impedance tomography data
Melody Dodd and Jennifer L. Mueller
2014, 8(4): 1013-1031 doi: 10.3934/ipi.2014.8.1013 +[Abstract](413) +[PDF](412.9KB)
Abstract:
The aim of this paper is to show the feasibility of the D-bar method for real-time 2-D EIT reconstructions. A fast implementation of the D-bar method for reconstructing conductivity changes on a 2-D chest-shaped domain is described. Cross-sectional difference images from the chest of a healthy human subject are presented, demonstrating what can be achieved in real time. The images constitute the first D-bar images from EIT data on a human subject collected on a pairwise current injection system.
Reconstruction of complex-valued tensors in the Maxwell system from knowledge of internal magnetic fields
Chenxi Guo and Guillaume Bal
2014, 8(4): 1033-1051 doi: 10.3934/ipi.2014.8.1033 +[Abstract](326) +[PDF](492.7KB)
Abstract:
This paper concerns the reconstruction of a complex-valued anisotropic tensor $\gamma = \sigma + \iota\omega\varepsilon$ from knowledge of several internal magnetic fields $H$, where $H$ satisfies the anisotropic Maxwell system on a bounded domain with prescribed boundary conditions. We show that $\gamma$ can be uniquely reconstructed with a loss of two derivatives from errors in the acquisition of $H$. A minimum number of $6$ such functionals is sufficient to obtain a local reconstruction of $\gamma$ in dimension three provided that the electric field satisfies appropriate boundary conditions. When $\gamma$ is close to a scalar tensor, such boundary conditions are shown to exist using the notion of complex geometric optics (CGO) solutions. For arbitrary symmetric tensors $\gamma$, a Runge approximation property is used instead to obtain partial results. This problem finds applications in the medical imaging modalities Current Density Imaging and Magnetic Resonance Electrical Impedance Tomography.
A data-driven edge-preserving D-bar method for electrical impedance tomography
Sarah Jane Hamilton, Andreas Hauptmann and Samuli Siltanen
2014, 8(4): 1053-1072 doi: 10.3934/ipi.2014.8.1053 +[Abstract](459) +[PDF](2349.8KB)
Abstract:
In Electrical Impedance Tomography (EIT), the internal conductivity of a body is recovered via current and voltage measurements taken at its surface. The reconstruction task is a highly ill-posed nonlinear inverse problem, which is very sensitive to noise, and requires the use of regularized solution methods, of which D-bar is the only proven method. The resulting EIT images have low spatial resolution due to smoothing caused by low-pass filtered regularization. In many applications, such as medical imaging, it is known a priori that the target contains sharp features such as organ boundaries, as well as approximate ranges for realistic conductivity values. In this paper, we use this information in a new edge-preserving EIT algorithm, based on the original D-bar method coupled with a deblurring flow stopped at a minimal data discrepancy. The method makes heavy use of a novel data fidelity term based on the so-called CGO sinogram. This nonlinear data step provides superior robustness over traditional EIT data formats such as current-to-voltage matrices or Dirichlet-to-Neumann operators, for commonly used current patterns.
An inverse problem for a three-dimensional heat equation in thermal imaging and the enclosure method
Masaru Ikehata and Mishio Kawashita
2014, 8(4): 1073-1116 doi: 10.3934/ipi.2014.8.1073 +[Abstract](333) +[PDF](667.2KB)
Abstract:
This paper studies a prototype of inverse initial boundary value problems whose governing equation is the heat equation in three dimensions. An unknown discontinuity embedded in a three-dimensional heat conductive body is considered. A single set of the temperature and heat flux on the lateral boundary for a fixed observation time is given as an observation datum. It is shown that this datum yields the minimum length of broken paths that start at a given point outside the body, go to a point on the boundary of the unknown discontinuity and return to a point on the boundary of the body under some conditions on the input heat flux, the unknown discontinuity and the body. This is new information obtained by using enclosure method.
Calderón problem for Maxwell's equations in cylindrical domain
Oleg Yu. Imanuvilov and Masahiro Yamamoto
2014, 8(4): 1117-1137 doi: 10.3934/ipi.2014.8.1117 +[Abstract](382) +[PDF](492.8KB)
Abstract:
We prove some uniqueness results in determination of the conductivity, the permeability and the permittivity of Maxwell's equations in a cylindrical domain $\Omega \times (0,L)$ from partial boundary map. More specifically, for an arbitrarily given subboundary $\Gamma_0 \subset \partial\Omega$, we prove that the coefficients of Maxwell's equations can be uniquely determined in the subdomain $(\Omega \setminus$ [the convex hull of $\Gamma_0])$ $ \times (0,L)$ by the boundary map only for inputs vanishing on $\Gamma_0 \times (0,L)$.
Increasing stability for determining the potential in the Schrödinger equation with attenuation from the Dirichlet-to-Neumann map
Victor Isakov and Jenn-Nan Wang
2014, 8(4): 1139-1150 doi: 10.3934/ipi.2014.8.1139 +[Abstract](378) +[PDF](346.5KB)
Abstract:
We derive some bounds which can be viewed as an evidence of increasing stability in the problem of recovering the potential coefficient in the Schrödinger equation from the Dirichlet-to-Neumann map in the presence of attenuation, when energy level/frequency is growing. These bounds hold under certain a-priori regularity constraints on the unknown coefficient. Proofs use complex and bounded complex geometrical optics solutions.
The nonlinear Fourier transform for two-dimensional subcritical potentials
Michael Music
2014, 8(4): 1151-1167 doi: 10.3934/ipi.2014.8.1151 +[Abstract](338) +[PDF](408.6KB)
Abstract:
The inverse scattering method for the Novikov-Veselov equation is studied for a larger class of Schrödinger potentials than could be handled previously. Previous work concerns so-called conductivity type potentials, which have a bounded positive solution at zero energy and are a nowhere dense set of potentials. We relax the conductivity type assumption to include logarithmically growing positive solutions at zero energy. These potentials are stable under perturbations. Assuming only that the potential is subcritical and has two weak derivatives in a weighted Sobolev space, we prove that the associated scattering transform can be inverted, and the original potential is recovered from the scattering data.
An inverse problem for the magnetic Schrödinger operator on a half space with partial data
Valter Pohjola
2014, 8(4): 1169-1189 doi: 10.3934/ipi.2014.8.1169 +[Abstract](424) +[PDF](467.4KB)
Abstract:
In this paper we prove uniqueness for an inverse boundary value problem for the magnetic Schrödinger equation in a half space, with partial data. We prove that the curl of the magnetic potential $A$, when $A\in W_{comp}^{1,\infty}(\overline{\mathbb{R}_{-}^3},\mathbb{R}^3)$, and the electric pontetial $q \in L_{comp}^{\infty}(\overline{\mathbb{R}_{-}^3},\mathbb{C})$ are uniquely determined by the knowledge of the Dirichlet-to-Neumann map on parts of the boundary of the half space.

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