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Volume 1, 2007

Inverse Problems & Imaging

2008 , Volume 2 , Issue 1

The editors of the Journal of Modern Dynamics are happy to dedicate this issue to Gregory Margulis, who, over the last four decades, has influenced dynamical systems as deeply as few others have, and who has blazed broad trails in the application of dynamical systems to other fields of core mathematics. For the full preface, please click the "full text" button above. Additional editors: Leonid Polterovich, Ralf Spatzier, Amie Wilkinson, and Anton Zorich.

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Inverse problems for quantum trees
Sergei Avdonin and Pavel Kurasov
2008, 2(1): 1-21 doi: 10.3934/ipi.2008.2.1 +[Abstract](219) +[PDF](281.4KB)
Abstract:
Three different inverse problems for the Schrödinger operator on a metric tree are considered, so far with standard boundary conditions at the vertices. These inverse problems are connected with the matrix Titchmarsh-Weyl function, response operator (dynamic Dirichlet-to-Neumann map) and scattering matrix. Our approach is based on the boundary control (BC) method and in particular on the study of the response operator. It is proven that the response operator determines the quantum tree completely, i.e. its connectivity, lengths of the edges and potentials on them. The same holds if the response operator is known for all but one boundary points, as well as for the Titchmarsh-Weyl function and scattering matrix. If the connectivity of the graph is known, then the lengths of the edges and the corresponding potentials are determined by just the diagonal terms of the data.
Inverse transport with isotropic sources and angularly averaged measurements
Guillaume Bal, Ian Langmore and François Monard
2008, 2(1): 23-42 doi: 10.3934/ipi.2008.2.23 +[Abstract](181) +[PDF](265.6KB)
Abstract:
We consider the reconstruction of a spatially-dependent scattering coefficient in a linear transport equation from diffusion-type measurements. In this setup, the contribution to the measurement is an integral of the scattering kernel against a product of harmonic functions, plus an additional term that is small when absorption and scattering are small. The linearized problem is severely ill-posed. We construct a regularized inverse that allows for reconstruction of the low frequency content of the scattering kernel, up to quadratic error, from the nonlinear map. An iterative scheme is used to improve this error so that it is small when the high frequency content of the scattering kernel is small.
2D EIT reconstructions using Calderon's method
Jutta Bikowski and Jennifer L. Mueller
2008, 2(1): 43-61 doi: 10.3934/ipi.2008.2.43 +[Abstract](266) +[PDF](468.3KB)
Abstract:
The pioneering work ''On an inverse boundary value problem'' by A. Calderón has inspired a multitude of research, both theoretical and numerical, on the inverse conductivity problem (ICP). The problem has an important application in a medical imaging technique known as electrical impedance tomography (EIT) in which currents are applied on electrodes on the surface of a body, the resulting voltages are measured, and the ICP is solved to determine the conductivity distribution in the interior of the body, which is then displayed to form an image. In this article, the reconstruction method proposed by Calderón is implemented in 2D for both simulated and experimental data including perfusion data collected on a human chest.
Iterative time-reversal control for inverse problems
Kenrick Bingham, Yaroslav Kurylev, Matti Lassas and Samuli Siltanen
2008, 2(1): 63-81 doi: 10.3934/ipi.2008.2.63 +[Abstract](247) +[PDF](277.2KB)
Abstract:
A novel method to solve inverse problems for the wave equation is introduced. The method is a combination of the boundary control method and an iterative time reversal scheme, leading to adaptive imaging of coefficient functions of the wave equation using focusing waves in unknown medium. The approach is computationally effective since the iteration lets the medium do most of the processing of the data.
    The iterative time reversal scheme also gives an algorithm for approximating a given wave in a subset of the domain without knowing the coefficients of the wave equation.
An inverse problem for fluid-solid interaction
Johannes Elschner, George C. Hsiao and Andreas Rathsfeld
2008, 2(1): 83-120 doi: 10.3934/ipi.2008.2.83 +[Abstract](216) +[PDF](1287.3KB)
Abstract:
Any acoustic plane wave incident to an elastic obstacle results in a scattered field with a corresponding far field pattern. Mathematically, the scattered field is the solution of a transmission problem coupling the reduced elastodynamic equations over the obstacle with the Helmholtz equation in the exterior. The inverse problem is to reconstruct the elastic body represented by a parametrization of its boundary.
    We define an objective functional depending on a non-negative regularization parameter such that, for any positive regularization parameter, there exists a regularized solution minimizing the functional. Moreover, for the regularization parameter tending to zero, these regularized solutions converge to the solution of the inverse problem provided the latter is uniquely determined by the given far field patterns. The whole approach is based on the variational form of the partial differential operators involved. Hence, numerical approximations can be found applying finite element discretization. Note that, though the transmission problem may have non-unique solutions for domains with so-called Jones frequencies, the scattered field and its far field pattern is unique and depend continuously on the shape of the obstacle.
An inverse boundary value problem for a nonlinear wave equation
Gen Nakamura and Michiyuki Watanabe
2008, 2(1): 121-131 doi: 10.3934/ipi.2008.2.121 +[Abstract](266) +[PDF](169.3KB)
Abstract:
An inverse boundary value problem for nonlinear wave equation of divergence form in one space dimension is considered. By assuming the nonlinear term is unknown, we show the linear and quadratic part of this term can be identified from the Dirichlet to Neumann map. Here, the nonlinearity is only in terms of the first derivative with respect to the space variable, and the linear and quadratic parts are defined in terms of this derivative. The identification not only gives the uniqueness but also the reconstruction.
Analytical bounds on the minimizers of (nonconvex) regularized least-squares
Mila Nikolova
2008, 2(1): 133-149 doi: 10.3934/ipi.2008.2.133 +[Abstract](182) +[PDF](248.8KB)
Abstract:
This is a theoretical study on the minimizers of cost-functions composed of an l 2 data-fidelity term and a possibly nonsmooth or nonconvex regularization term acting on the differences or the discrete gradients of the image or the signal to restore. More precisely, we derive general nonasymptotic analytical bounds characterizing the local and the global minimizers of these cost-functions. We first derive bounds that compare the restored data with the noisy data. For edge-preserving regularization, we exhibit a tight data-independent bound on the l norm of the residual (the estimate of the noise), even if its l 2 norm is being minimized. Then we focus on the smoothing incurred by the (local) minimizers in terms of the differences or the discrete gradient of the restored image (or signal).
On uniqueness of obstacles and boundary conditions from restricted dynamical and scattering data
Victor Isakov
2008, 2(1): 151-165 doi: 10.3934/ipi.2008.2.151 +[Abstract](163) +[PDF](210.3KB)
Abstract:
We show uniqueness of a (time independent) domain $D$ and of an impedance type boundary condition from either dynamical or scattering data at many frequencies. We assume that the additonal boundary (scattering) data are given for one set of boundary data or for one incident direction. In case of general domain and finite (sharp) observation time we assume Neumann boundary condition on $\partial D$ and for polygonal $D$ we can handle more general case. If the data are available for all times we show uniqueness of the most general impedance boundary condition by using continuation of the corresponding scattering data into complex domain and modifying the Schiffer's argument.

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