We consider the problem of imaging the conductivity from knowledge of one current and corresponding voltage on a part of the boundary of an inhomogeneous isotropic object and of the magnitude |J(x)| of the current density inside. The internal data are obtained from magnetic resonance measurements. The problem is reduced to a boundary value problem with partial data for the equation ∇ ⋅ |J(x)||∇u|−1∇u = 0. We show that equipotential surfaces are minimal surfaces in the conformal metric |J|2/(n−1)I. In two dimensions, we solve the Cauchy problem with partial data and show that the conductivity is uniquely determined in the region spanned by the characteristics originating from the part of the boundary where measurements are available. We formulate sufficient conditions on the Dirichlet data to guarantee the unique recovery of the conductivity throughout the domain. The proof of uniqueness is constructive and yields an efficient algorithm for conductivity imaging. The computational feasibility of this algorithm is demonstrated in numerical experiments.
We consider the problem of recovering the conductivity of an object from knowledge of the magnitude of one current density field in its interior. A known voltage potential is assumed imposed at the boundary. We prove identifiability and propose an iterative reconstruction procedure. The computational feasibility of this procedure is demonstrated in some numerical experiments.
A sequential minimisation algorithm for the numerical solution of inverse problems of frequency sounding is presented. The algorithm is based on the concept of convexification of a multiextremal objective function proposed recently by the authors. A key point in the sequential minimisation algorithm is that unlike conventional layerstripping algorithms, it provides the stable approximate solution via minmisation of a finite sequence of strictly convex objective functions resulted from applying the nonlinear weighet least squares method with Carleman's weight functions. Another advantage of the proposed algorithm is that the starting vectors for the descent methods of minimisation are directly determined from the data eliminating the uncertainty inherent to the local methods, such as the gradient or Newton-like methods. The 1-D inverse model of frequency sounding is selected to demonstrate its computational feasibility. Based on the localising property of Carleman's weight functions, it is proven that the distance between the approximate and "exact" solutions is small if the approximation error is small. The computational experiments with several realistic and synthetic marine shallow water configurations are presented to demonstrate the computational feasibility of the proposed algorithm.
We present a unified framework for constructing the globally convergent algorithms for a broad class of multidimensional coefficient inverse problems arising in natural science and industry. Based on the convexification approach, the unified framework substantiates the numerical solution of the corresponding problem of nonconvex optimization. A globally convergent iterative algorithm for an inverse problem of diffuse optical mammography is constructed. It utilizes the contraction property of a nonlinear operator resulting from applying the convexification approach. The effectiveness of this algorithm is demonstrated in computational experiments.
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