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Inverse Problems & Imaging

2015 , Volume 9 , Issue 4

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Boundary and scattering rigidity problems in the presence of a magnetic field and a potential
Yernat M. Assylbekov and  Hanming Zhou
2015, 9(4): 935-950 doi: 10.3934/ipi.2015.9.935 +[Abstract](43) +[PDF](397.0KB)
In this paper, we consider a compact Riemannian manifold with boundary, endowed with a magnetic potential $\alpha$ and a potential $U$. For brevity, this type of systems are called $\mathcal{MP}$-systems. On simple $\mathcal{MP}$-systems, we consider both the boundary rigidity problem and scattering rigidity problem. Unlike the cases of geodesic or magnetic systems, knowing boundary action functions or scattering relations for only one energy level is insufficient to uniquely determine a simple $\mathcal{MP}$-system up to natural obstructions, even under the assumption that the boundary restriction of the system is given, and we provide some counterexamples. By reducing an $\mathcal{MP}$-system to the corresponding magnetic system and applying the results of [6] on simple magnetic systems, we prove rigidity results for metrics in a given conformal class, for simple real analytic $\mathcal{MP}$-systems and for simple two-dimensional $\mathcal{MP}$-systems.
The inverse electromagnetic scattering problem by a mixed impedance screen in chiral media
Christodoulos E. Athanasiadis , Vassilios Sevroglou and  Konstantinos I. Skourogiannis
2015, 9(4): 951-970 doi: 10.3934/ipi.2015.9.951 +[Abstract](41) +[PDF](475.4KB)
In this paper the problem of scattering of time-harmonic electromagnetic waves by a mixed impedance scatterer in chiral media is considered. Our scatterer is a partially coated chiral screen, for which an impedance boundary condition on one side of its boundary, and a perfectly conducting boundary condition on the other side, is satisfied. The direct scattering problem for the modified Maxwell equations is formulated and the appropriate Sobolev space setting is considered. Issues of solvability due to uniqueness and existence are discussed. The corresponding inverse scattering problem is studied and uniqueness results concerning the mixed impedance screen are proved. Further, the shape reconstruction of the boundary of the partially coated screen is established. In particular, a chiral far-field operator is introduced and new results concerning its properties are proved. A modified linear sampling method based on a factorization of the chiral far-field operator, in order to reconstruct the screen is also presented. We end up with useful conclusions and remarks for screens in chiral media.
Application of mixed formulations of quasi-reversibility to solve ill-posed problems for heat and wave equations: The 1D case
Eliane Bécache , Laurent Bourgeois , Lucas Franceschini and  Jérémi Dardé
2015, 9(4): 971-1002 doi: 10.3934/ipi.2015.9.971 +[Abstract](51) +[PDF](1846.4KB)
In this paper we address some ill-posed problems involving the heat or the wave equation in one dimension, in particular the backward heat equation and the heat/wave equation with lateral Cauchy data. The main objective is to introduce some variational mixed formulations of quasi-reversibility which enable us to solve these ill-posed problems by using some classical Lagrange finite elements. The inverse obstacle problems with initial condition and lateral Cauchy data for heat/wave equation are also considered, by using an elementary level set method combined with the quasi-reversibility method. Some numerical experiments are presented to illustrate the feasibility for our strategy in all those situations.
Stabilized BFGS approximate Kalman filter
Alexander Bibov , Heikki Haario and  Antti Solonen
2015, 9(4): 1003-1024 doi: 10.3934/ipi.2015.9.1003 +[Abstract](50) +[PDF](504.4KB)
The Kalman filter (KF) and Extended Kalman filter (EKF) are well-known tools for assimilating data and model predictions. The filters require storage and multiplication of $n\times n$ and $n\times m$ matrices and inversion of $m\times m$ matrices, where $n$ is the dimension of the state space and $m$ is dimension of the observation space. Therefore, implementation of KF or EKF becomes impractical when dimensions increase. The earlier works provide optimization-based approximative low-memory approaches that enable filtering in high dimensions. However, these versions ignore numerical issues that deteriorate performance of the approximations: accumulating errors may cause the covariance approximations to lose non-negative definiteness, and approximative inversion of large close-to-singular covariances gets tedious. Here we introduce a formulation that avoids these problems. We employ L-BFGS formula to get low-memory representations of the large matrices that appear in EKF, but inject a stabilizing correction to ensure that the resulting approximative representations remain non-negative definite. The correction applies to any symmetric covariance approximation, and can be seen as a generalization of the Joseph covariance update.
    We prove that the stabilizing correction enhances convergence rate of the covariance approximations. Moreover, we generalize the idea by the means of Newton-Schultz matrix inversion formulae, which allows to employ them and their generalizations as stabilizing corrections.
Homogenization of the transmission eigenvalue problem for periodic media and application to the inverse problem
Fioralba Cakoni , Houssem Haddar and  Isaac Harris
2015, 9(4): 1025-1049 doi: 10.3934/ipi.2015.9.1025 +[Abstract](49) +[PDF](741.7KB)
We consider the interior transmission problem associated with the scattering by an inhomogeneous (possibly anisotropic) highly oscillating periodic media. We show that, under appropriate assumptions, the solution of the interior transmission problem converges to the solution of a homogenized problem as the period goes to zero. Furthermore, we prove that the associated real transmission eigenvalues converge to transmission eigenvalues of the homogenized problem. Finally we show how to use the first transmission eigenvalue of the period media, which is measurable from the scattering data, to obtain information about constant effective material properties of the periodic media. The convergence results presented here are not optimal. Such results with rate of convergence involve the analysis of the boundary correction and will be subject of a forthcoming paper.
A new Kohn-Vogelius type formulation for inverse source problems
Xiaoliang Cheng , Rongfang Gong and  Weimin Han
2015, 9(4): 1051-1067 doi: 10.3934/ipi.2015.9.1051 +[Abstract](102) +[PDF](527.8KB)
In this paper we propose a Kohn-Vogelius type formulation for an inverse source problem of partial differential equations. The unknown source term is to be determined from both Dirichlet and Neumann boundary conditions. We introduce two different boundary value problems, which depend on two different positive real numbers $\alpha$ and $\beta$, and both of them incorporate the Dirichlet and Neumann data into a single Robin boundary condition. This allows noise in both boundary data. By using the Kohn-Vogelius type Tikhonov regularization, data to be fitted is transferred from boundary into the whole domain, making the problem resolution more robust. More importantly, with the formulation proposed here, satisfactory reconstruction could be achieved for rather small regularization parameter through choosing properly the values of $\alpha$ and $\beta$. This is a desirable property to have since a smaller regularization parameter implies a more accurate approximation of the regularized problem to the original one. The proposed method is studied theoretically. Two numerical examples are provided to show the usefulness of the proposed method.
A parallel space-time domain decomposition method for unsteady source inversion problems
Xiaomao Deng , Xiao-Chuan Cai and  Jun Zou
2015, 9(4): 1069-1091 doi: 10.3934/ipi.2015.9.1069 +[Abstract](53) +[PDF](595.9KB)
In this paper, we propose a parallel space-time domain decomposition method for solving an unsteady source identification problem governed by the linear convection-diffusion equation. Traditional approaches require to solve repeatedly a forward parabolic system, an adjoint system and a system with respect to the unknown sources. The three systems have to be solved one after another. These sequential steps are not desirable for large scale parallel computing. A space-time restrictive additive Schwarz method is proposed for a fully implicit space-time coupled discretization scheme to recover the time-dependent pollutant source intensity functions. We show with numerical experiments that the scheme works well with noise in the observation data. More importantly it is demonstrated that the parallel space-time Schwarz preconditioner is scalable on a supercomputer with over $10^3$ processors, thus promising for large scale applications.
Locally sparse reconstruction using the $l^{1,\infty}$-norm
Pia Heins , Michael Moeller and  Martin Burger
2015, 9(4): 1093-1137 doi: 10.3934/ipi.2015.9.1093 +[Abstract](80) +[PDF](766.8KB)
This paper discusses the incorporation of local sparsity information, e.g. in each pixel of an image, via minimization of the $\ell^{1,\infty}$-norm. We discuss the basic properties of this norm when used as a regularization functional and associated optimization problems, for which we derive equivalent reformulations either more amenable to theory or to numerical computation. Further focus of the analysis is put on the locally 1-sparse case, which is well motivated by some biomedical imaging applications.
    Our computational approaches are based on alternating direction methods of multipliers (ADMM) and appropriate splittings with augmented Lagrangians. Those are tested for a model scenario related to dynamic positron emission tomography (PET), which is a functional imaging technique in nuclear medicine.
    The results of this paper provide insight into the potential impact of regularization with the $\ell^{1,\infty}$-norm for local sparsity in appropriate settings. However, it also indicates several shortcomings, possibly related to the non-tightness of the functional as a relaxation of the $\ell^{0,\infty}$-norm.
Bilevel optimization for calibrating point spread functions in blind deconvolution
Michael Hintermüller and  Tao Wu
2015, 9(4): 1139-1169 doi: 10.3934/ipi.2015.9.1139 +[Abstract](67) +[PDF](1364.5KB)
Blind deconvolution problems arise in many imaging modalities, where both the underlying point spread function, which parameterizes the convolution operator, and the source image need to be identified. In this work, a novel bilevel optimization approach to blind deconvolution is proposed. The lower-level problem refers to the minimization of a total-variation model, as is typically done in non-blind image deconvolution. The upper-level objective takes into account additional statistical information depending on the particular imaging modality. Bilevel problems of such type are investigated systematically. Analytical properties of the lower-level solution mapping are established based on Robinson's strong regularity condition. Furthermore, several stationarity conditions are derived from the variational geometry induced by the lower-level problem. Numerically, a projected-gradient-type method is employed to obtain a Clarke-type stationary point and its convergence properties are analyzed. We also implement an efficient version of the proposed algorithm and test it through the experiments on point spread function calibration and multiframe blind deconvolution.
Iterative choice of the optimal regularization parameter in TV image restoration
Alina Toma , Bruno Sixou and  Françoise Peyrin
2015, 9(4): 1171-1191 doi: 10.3934/ipi.2015.9.1171 +[Abstract](231) +[PDF](2108.1KB)
We present iterative methods for choosing the optimal regularization parameter for linear inverse problems with Total Variation regularization. This approach is based on the Morozov discrepancy principle or on a damped version of this principle and on an approximating model function for the data term. The theoretical convergence of the method of choice of the regularization parameter is demonstrated. The choice of the optimal parameter is refined with a Newton method. The efficiency of the method is illustrated on deconvolution and super-resolution experiments on different types of images. Results are provided for different levels of blur, noise and loss of spatial resolution. The damped Morozov discrepancy principle often outerperforms the approaches based on the classical Morozov principle and on the Unbiased Predictive Risk Estimator. Moreover, the proposed methods are fast schemes to select the best parameter for TV regularization.

2016  Impact Factor: 1.094




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