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High contrast impedance tomography
 INVERSE PROBLEMS
, 1996
"... We introduce an output leastsquares method for impedance tomography problems that have regions of high conductivity surrounded by regions of lower conductivity. The high conductivity is modeled on network approximation results from an asymptotic analysis and its recovery is based on this model. The ..."
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Cited by 44 (6 self)
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We introduce an output leastsquares method for impedance tomography problems that have regions of high conductivity surrounded by regions of lower conductivity. The high conductivity is modeled on network approximation results from an asymptotic analysis and its recovery is based on this model. The smoothly varying part of the conductivity is recovered by a linearization process as is usual. We present the results of several numerical experiments that illustrate
On effective methods for implicit piecewise smooth surface recovery
 SIAM J. Scient. Comput
"... Abstract. This paper considers the problem of reconstructing a piecewise smooth model function from given, measured data. The data are compared to a field which is given as a possibly nonlinear function of the model. A regularization functional is added which incorporates the a priori knowledge that ..."
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Cited by 32 (24 self)
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Abstract. This paper considers the problem of reconstructing a piecewise smooth model function from given, measured data. The data are compared to a field which is given as a possibly nonlinear function of the model. A regularization functional is added which incorporates the a priori knowledge that the model function is piecewise smooth and may contain jump discontinuities. Regularization operators related to total variation (TV) are therefore employed. Two popular methods are modified TV and Huber’s function. Both contain a parameter which must be selected. The Huber variant provides a more natural approach for selecting its parameter, and we use this to propose a scheme for both methods. Our selected parameter depends both on the resolution and on the model average roughness; thus, it is determined adaptively. Its variation from one iteration to the next yields additional information about the progress of the regularization process. The modified TV operator has a smoother generating function; nonetheless we obtain a Huber variant with comparable, and occasionally better, performance. For large problems (e.g., high resolution) the resulting reconstruction algorithms can be tediously slow. We propose two mechanisms to improve efficiency. The first is a multilevel continuation approach aimed mainly at obtaining a cheap yet good estimate for the regularization parameter and the solution. The second is a special multigrid preconditioner for the conjugate gradient algorithm used to solve the linearized systems encountered in the procedures for recovering the model function.
Preconditioned AllAtOnce Methods for Large, Sparse Parameter Estimation Problems
, 2000
"... The problem of recovering a parameter function based on measurements of solutions of a system of partial differential equations in several space variables leads to a number of computational challenges. Upon discretization of a regularized formulation a large, sparse constrained optimization prob ..."
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Cited by 24 (4 self)
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The problem of recovering a parameter function based on measurements of solutions of a system of partial differential equations in several space variables leads to a number of computational challenges. Upon discretization of a regularized formulation a large, sparse constrained optimization problem is obtained. Typically in the literature, the constraints are eliminated and the resulting unconstrained formulation is solved by some variant of Newton's method, usually the GaussNewton method. A preconditioned conjugate gradient algorithm is applied at each iteration for the resulting reduced Hessian system. In this paper we apply instead a preconditioned Krylov method directly to the KKT system arising from a Newtontype method for the constrained formulation (an "allatonce" approach). A variant of symmetric QMR is employed, and an effective preconditioner is obtained by solving the reduced Hessian system approximately. Since the reduced Hessian system presents significa...
Inversion of 3D electromagnetic data in frequency and time domain using an inexact allatonce approach
 Geophys
"... approach ..."
A SURVEY ON MULTIPLE LEVEL SET METHODS WITH APPLICATIONS FOR IDENTIFYING PIECEWISE CONSTANT FUNCTIONS
, 2004
"... We try to give a brief survey about using multiple level set methods for identifying piecewise constant or piecewise smooth functions. A general framework is presented. Application using this general framework for different practical problems are shown. We try to show some details in applying the g ..."
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Cited by 20 (7 self)
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We try to give a brief survey about using multiple level set methods for identifying piecewise constant or piecewise smooth functions. A general framework is presented. Application using this general framework for different practical problems are shown. We try to show some details in applying the general approach for applications to: image segmentation, optimal shape design, elliptic inverse coefficient identification, electricall impedance tomography and positron emission tomography. Numerical experiments are also presented for some of the problems.
On level set regularization for highly illposed distributed parameter estimation problems
, 2005
"... ..."
Identification of Discontinuous Coefficients in Elliptic Problems Using Total Variation Regularization
 SIAM J. Sci. Comput
, 2003
"... . We propose several formulations for recovering discontinuous coefficients in elliptic problems by using total variation (TV) regularization. The motivation for using TV is its wellestablished ability to recover sharp discontinuities. We employ an augmented Lagrangian variational formulation for s ..."
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Cited by 17 (8 self)
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. We propose several formulations for recovering discontinuous coefficients in elliptic problems by using total variation (TV) regularization. The motivation for using TV is its wellestablished ability to recover sharp discontinuities. We employ an augmented Lagrangian variational formulation for solving the outputleastsquares inverse problem. In addition to the basic outputleastsquares formulation, we introduce two new techniques to handle large observation errors. First, we use a filtering step to remove as much of the observation error as possible. Second, we introduce two extensions of the outputleastsquares model; one model employs observations of the gradient of the state variable while the other uses the flux. Numerical experiments indicate that the combination of these two techniques enables us to successfully recover discontinuous coefficients even under large observation errors. 1. Introduction. Consider the partial differential equation ae \Gammar \Delta (q(x)ru) =...
Electrical Impedance Tomography Using Level Set Representation and Total Variational Regularization
, 2003
"... In this paper, we propose a numerical scheme for the identification of piecewise constant conductivity coe#cient for a problem arising from electrical impedance tomography. The key feature of the scheme is the use of level set method for the representation of interface between domains with di#er ..."
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Cited by 16 (2 self)
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In this paper, we propose a numerical scheme for the identification of piecewise constant conductivity coe#cient for a problem arising from electrical impedance tomography. The key feature of the scheme is the use of level set method for the representation of interface between domains with di#erent values of coe#cients. Numerical tests show that our method can be able to recover a sharp interface and can tolerate higher level of noise in the observation data. Results concerning the e#ects of number of measurements, noise level in the data as well as the regularization parameters on the accuracy of the scheme are also given.
A Direct Impedance Tomography Algorithm for Locating Small Inhomogeneities
, 2003
"... this paper we consider the case where the goal is to find a number of small objects (inhomogeneities) inside an otherwise known conductor. Taking advantage of the smallness of the inhomogeneities, we can use asymptotic analysis to design a direct (i.e., noniterative) reconstruction algorithm for the ..."
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Cited by 13 (3 self)
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this paper we consider the case where the goal is to find a number of small objects (inhomogeneities) inside an otherwise known conductor. Taking advantage of the smallness of the inhomogeneities, we can use asymptotic analysis to design a direct (i.e., noniterative) reconstruction algorithm for the determination of their locations. The viability of this direct approach is documented by numerical examples. Mathematics Subject Classification (2000): 65N21, 35R30, 35C20 1 Introduction Techniques for recovering the conductivity distribution inside a body from measurements of current flows and voltages on the body's surface go under the heading of electrical impedance tomography (EIT). The vast and growing literature reflects the many possible applications of this method, e.g. for medical diagnosis or nondestructive evaluation of materials. For further details we refer to the recent survey paper [8]. Since the underlying inverse problem is nonlinear and severely illposed it is generally advisable to incorporate all available apriori knowledge ? Supported by the Deutsche Forschungsgemeinschaft (DFG) under grant HA 2121/23 ?? Supported by the National Science Foundation under grant DMS0072556 2 Martin Br uhl, Martin Hanke, Michael S. Vogelius about the unknown conductivity. One such type of knowledge could be that the body consists of a smooth background (of known conductivity) containing a number of unknown, small inclusions with a significantly higher or lower conductivity. This situation arises for example in mine detection, where one tries to locate the position of buried antipersonnel mines from electromagnetic data. The mines have a higher (metal) or lower (plastic) conductivity than the surrounding soil and they are small relative to the area being imaged....