This work considers properties of the Neumann-to-Dirichlet map for the conductivity equation under the assumption that the conductivity is identically one close to the boundary of the examined smooth, bounded and simply connected domain. It is demonstrated that the so-called bisweep data, i.e., the (relative) potential differences between two boundary points when delta currents of opposite signs are applied at the very same points, uniquely determine the whole Neumann-to-Dirichlet map. In two dimensions, the bisweep data extend as a holomorphic function of two variables to some (interior) neighborhood of the product boundary. It follows that the whole Neumann-to-Dirichlet map is characterized by the derivatives of the bisweep data at an arbitrary point. On the diagonal of the product boundary, these derivatives can be given with the help of the derivatives of the (relative) boundary potentials at some fixed point caused by the distributional current densities supported at the same point, and thus such point measurements uniquely define the Neumann-to-Dirichlet map. This observation also leads to a new, truly local uniqueness result for the so-called Calderón inverse conductivity problem.2010 Mathematics Subject Classification. 35R30, 35Q60.
This paper considers detection of conductivity inhomogeneities inside an otherwise homogeneous object by electrical impedance tomography using only two electrodes: one of the electrodes is held fixed, while the other moves around the examined object. Unit current is maintained between the electrodes, and the corresponding (relative) potential difference is measured as a function of the position of the dynamic electrode, thus producing so-called sweep data. In two dimensions and with point-like electrodes, the sweep data have previously been shown to extend as a holomorphic function to the exterior of the inhomogeneities. We derive a holomorphic asymptotic expansion for the (extended) sweep data with respect to the size of Lipschitz inclusions with constant conductivity levels. Based on this result, we subsequently introduce a numerical algorithm that locates the inclusions and estimates their strengths by considering the poles and corresponding residues of suitable Laurent-Padé approximants of the sweep data. The functionality of the reconstruction technique is demonstrated via numerical experiments, some of which are three dimensional and/or based on simulated complete electrode model measurements.
A conductivity equation is studied in piecewise smooth plane domains and with measure-valued current patterns (Neumann boundary values). This allows one to extend the recently introduced concept of bisweep data to piecewise smooth domains, which yields a new partial data result for the Calderón inverse conductivity problem. It is also shown that bisweep data are (up to a constant scaling factor) the Schwartz kernel of the relative Neumann-to-Dirichlet map. A numerical method for reconstructing the supports of inclusions from discrete bisweep data is also presented.
Abstract. Conditioning of a nonsingular matrix subspace is addressed in terms of its best conditioned elements. Computationally the problem is seemingly challenging. By associating with the task an intersection problem with unitary matrices leads to a more accessible approach. A resulting matrix nearness problem can be viewed to generalize the so-called Löwdin problem in quantum chemistry. For critical points in the Frobenius norm, a differential equation on the manifold of unitary matrices is derived. Another resulting matrix nearness problem allows locating points of optimality more directly, once formulated as a problem in computational algebraic geometry.Key words. conditioning, matrix intersection problem, matrix nearness problem, Löwdin's problem, generalized eigenvalue problem AMS subject classifications. 15A12, 65F351. Introduction. This paper is concerned with the problem of conditioning of a nonsingular matrix subspace V of C n×n over C (or R). Matrix subspaces typically appear in large scale numerical linear algebra problems where assuming additional structure is quite unrealistic. Nonsingularity means that there exists invertible elements in V. The conditioning of V is then defined in terms of its best conditioned elements. In the applications that we have in mind, typically dim V ≪ n 2 . For example, in the generalized eigenvalue problem dim V = 2 only. In this paper the task of assessing conditioning is formulated as a matrix intersection problem for V and the set of unitary matrices.1 Since this can be done in many ways, the interpretation is amenable to computations through matrix nearness problems and versatile enough in view of addressing operator theoretic problems more generally.Denote by U (n) the set of unitary matrices in C n×n . The intersection problem for V and U (n), which are both smooth submanifolds of C n×n , can be formulated as a matrix nearness problem
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