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37
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 ..."
Abstract

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
An Augmented Lagrangian Method for Identifying Discontinuous Parameters in Elliptic Systems
, 1997
"... . The identification of discontinuous parameters in elliptic systems is formulated as a constrained minimization problem combining the output least squares and the equation error method. The minimization problem is then proved to be equivalent to the saddlepoint problem of an augmented Lagrangian. ..."
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Cited by 39 (17 self)
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. The identification of discontinuous parameters in elliptic systems is formulated as a constrained minimization problem combining the output least squares and the equation error method. The minimization problem is then proved to be equivalent to the saddlepoint problem of an augmented Lagrangian. The finite element method is used to discretize the saddlepoint problem, and the convergence of the discretization is also proved. Finally, an Uzawa algorithm is suggested for solving the discrete saddlepoint problem and is shown to be globally convergent. Key words. parameter identification, elliptic system, augmented Lagrangian, finite element method AMS subject classifications. 65N30, 35R30 PII. S0363012997318602 1. Introduction. The main purpose of this paper is to propose a numerical approach and conduct convergence analyses on each approximation process in the identification of the unknown coe#cient q in the elliptic problem # (q#u) = f in ## u = 0 on #. The identifying process i...
TrustRegion InteriorPoint SQP Algorithms For A Class Of Nonlinear Programming Problems
 SIAM J. CONTROL OPTIM
, 1997
"... In this paper a family of trustregion interiorpoint SQP algorithms for the solution of a class of minimization problems with nonlinear equality constraints and simple bounds on some of the variables is described and analyzed. Such nonlinear programs arise e.g. from the discretization of optimal co ..."
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Cited by 35 (8 self)
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In this paper a family of trustregion interiorpoint SQP algorithms for the solution of a class of minimization problems with nonlinear equality constraints and simple bounds on some of the variables is described and analyzed. Such nonlinear programs arise e.g. from the discretization of optimal control problems. The algorithms treat states and controls as independent variables. They are designed to take advantage of the structure of the problem. In particular they do not rely on matrix factorizations of the linearized constraints, but use solutions of the linearized state equation and the adjoint equation. They are well suited for large scale problems arising from optimal control problems governed by partial differential equations. The algorithms keep strict feasibility with respect to the bound constraints by using an affine scaling method proposed for a different class of problems by Coleman and Li and they exploit trustregion techniques for equalityconstrained optimizatio...
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...
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.
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) =...
Augmented Lagrangian and Total Variation Methods for Recovering Discontinuous Coefficients from Elliptic Equations
 Computational Science for the 21st Century
, 1997
"... : Estimation of coefficients of partial differential equations is illposed. Outputleastsquares method is often used in practice. Convergence of the commonly used minimization algorithms for the inverse problem is often very slow. By using the augmented Lagrangian method, the inverse problem is re ..."
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Cited by 14 (8 self)
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: Estimation of coefficients of partial differential equations is illposed. Outputleastsquares method is often used in practice. Convergence of the commonly used minimization algorithms for the inverse problem is often very slow. By using the augmented Lagrangian method, the inverse problem is reduced to a coupled linear algebraic system, which can be solved efficiently. Total variation techniques have been successfully used in image processing. Here, we use it with the augmented Lagrangian approach to recover discontinuous coefficients. The numerical results show that our approach can recover discontinuous coefficients with large jumps from noisy observations. 1. Introduction Consider the partial differential equation: ae \Gammar \Delta (qru) = f; on\Omega ; u = 0 ; on @\Omega : (1) Our concern is that we know an approximate observation u d for u, i.e. u d ß u, and need to recover the coefficient q. A common approach is the outputleastsquares method, i.e. find the minimize...
Identification Of Discontinuous Coefficients From Elliptic Problems Using Total Variation Regularization
 SIAM J. Sci. Comput
, 1997
"... . We propose several formulations for recovering discontinous coefficient of 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 sol ..."
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Cited by 13 (4 self)
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. We propose several formulations for recovering discontinous coefficient of 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 utilizes the flux. Numerical experiments indicate that the combination of these two techniques enables us to successfully recover highly discontinous coefficient even under observation errors as high as 100% in the L 2 norm. 1. Introduction. Consider the partial differential equa...
Overlapping Domain Decomposition and Multigrid Methods for Inverse Problems
, 1997
"... Introduction This work continues our earlier investigations [2], [3] and [8]. The intention is to develop efficient numerical solvers to recover the diffusion coefficient, using observations of the solution u, from the elliptic equation \Gammar \Delta (q(x)ru) = f(x) in\Omega ; u = 0 on @\Omega ..."
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Cited by 11 (4 self)
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Introduction This work continues our earlier investigations [2], [3] and [8]. The intention is to develop efficient numerical solvers to recover the diffusion coefficient, using observations of the solution u, from the elliptic equation \Gammar \Delta (q(x)ru) = f(x) in\Omega ; u = 0 on @\Omega : (1) Our emphasis is on the numerical treatment of discontinuous coefficient and efficiency of the numerical methods. It is well known that such an inverse problem is illposed. Its numerical solution often suffers from undesirable numerical oscillation and very slow convergence. When the coefficient is smooth, successful numerical methods have been developed in [5] [7]. When the coefficient has large jumps, the numerical problem is much m