A truncated-newton algorithm for three-dimensional electrical impedance tomography is presented. Explicit formation of the Hessian, normally a computational bottleneck, is avoided through use of a preconditioned conju...
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A truncated-newton algorithm for three-dimensional electrical impedance tomography is presented. Explicit formation of the Hessian, normally a computational bottleneck, is avoided through use of a preconditioned conjugate gradient (PCG) solution of the Levenberg-Marquardt update. The PCG preconditioner is formed as a product of a sparse approximation of the Jacobian by its transpose.
In this paper we discuss the use of truncated-newton methods, a flexible class of iterative methods, in the solution of large-scale unconstrained minimization problems. At each major iteration, the newton equations ar...
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In this paper we discuss the use of truncated-newton methods, a flexible class of iterative methods, in the solution of large-scale unconstrained minimization problems. At each major iteration, the newton equations are approximately solved by an inner iterative algorithm. The performance of the inner algorithm, and in addition the total method, can be greatly improved by the addition of preconditioning and scaling strategies. Preconditionings can be developed using either the outer nonlinear algorithm or using information computed during the inner iteration. Several preconditioning schemes are derived and tested.
This paper discusses the use of the linear conjugate-gradient method (developed via the Lanczos method) in the solution of large-scale unconstrained minimization problems. At each iteration of a newton-type method, th...
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This paper discusses the use of the linear conjugate-gradient method (developed via the Lanczos method) in the solution of large-scale unconstrained minimization problems. At each iteration of a newton-type method, the direction of search is defined as the solution of a quadratic subproblem. When the number of variables is very large, this subproblem may be solved using the linear conjugate-gradient method of Hestenes and Stiefel. We show how the equivalent Lanczos characterization of the linear conjugate-gradient method may be exploited to define a modified newton method which can be applied to problems that do not necessarily have positive-definite Hessian matrices at all points of the region of interest. This derivation also makes it possible to compute a negative-curvature direction at a stationary point.
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