Abstract. A method for calculating special grid placement for three-point schemes which yields exponential superconvergence of the Neumann to Dirichlet map has been suggested earlier. Here we show that such a grid placement can yield impedance which is equivalent to that of a spectral Galerkin method, or more generally to that of a spectral Galerkin-Petrov method. In fact we show that for every stable Galerkin-Petrov method there is a threepoint scheme which yields the same solution at the boundary. We discuss the application of this result to partial differential equations and give numerical examples. We also show equivalence at one corner of a two-dimensional optimal grid with a spectral Galerkin method. 1.

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Spectral convergence of optimal grids for anisotropic problems is both numerically observed and explained. For elliptic problems, the gridding algorithm is reduced to a Stieltjes rational approximation on an interval of a line in the complex plane instead of the real axis as in the isotropic case. We show rigorously why this occurs for a semi-infinite and bounded interval. We then extend the gridding algorithm to hyperbolic problems on bounded domains. For the propagative modes, the problem is reduced to a rational approximation on an interval of the negative real semiaxis, similarly to in the isotropic case. For the wave problem we present numerical examples in 2-D anisotropic media. 1

Abstract. We introduce an algorithm for the numerical solution of electrical impedance tomography (EIT) in two dimensions, with partial boundary measurements. The algorithm is an extension of the one in [11, 49] for EIT with full boundary measurements. It is based on resistor networks that arise in finite volume discretizations of the elliptic partial differential equation for the potential, on so-called optimal grids that are computed as part of the problem. The grids are adaptively refined near the boundary, where we measure and expect better resolution of the images. They can be used very efficiently in inversion, by defining a reconstruction mapping that is an approximate inverse of the forward map, and acts therefore as a preconditioner in any iterative scheme that solves the inverse problem via optimization. The main result in this paper is the construction of optimal grids for EIT with partial measurements by extremal quasiconformal (Teichmüller) transformations of the optimal grids for EIT with full boundary measurements. We present the algorithm for computing the reconstruction mapping on such grids, and we illustrate its performance with numerical simulations. The results show an interesting trade-off between the resolution of the reconstruction in the domain of the solution and distortions due to artificial anisotropy induced by the distribution of the measurement points on the accessible boundary. 1.

In this paper, we present a reduced order modeling methodology, based on the utilization of optimal non-uniform grids generated by Gaussian spectral rules, for the direct passive synthesis of SPICE-compatible modeling of multi-conductor interconnect structures. The algorithm is based on a Padé-Chebyshev approximation of the frequency-dependent input impedance matrix of the passive interconnect system. The synthesized circuit is represented as the concatenation of a number of non-uniform sections of passive lumped coupled circuits. However, contrary to the popular uniform segmentation-based distributed circuit models for interconnects, where 10 to 15 segments per minimum wavelength are needed for multi-GHz accuracy, the proposed model is “optimal ” in the sense that highly-accurate responses can be obtained with a number of segments per minimum wavelength barely exceeding the Nyquist limit of 2. This high accuracy stems from the super-exponential convergence of the Padé-Chebyshev approximation of the input impedance of the transmission-line model of the interconnect, and results in the synthesis of MNA stamps for the interconnect structure with five to ten times reduction in the number of state variables compared to uniform grids. Moreover, the passivity of the generated SPICEcompatible multi-port models is guaranteed through the use of passive equivalent circuits for the representation of the frequencydependent, per-unit-length series impedance and shunt admittance matrices of the interconnect. Keywords: Transmission-line modeling of interconnects; passive reduced- order synthesis; interconnects with frequencydependent losses.

network approaches to electrical impedance tomography

We consider finite difference approximations of solutions of inverse Sturm-Liouville problems in bounded intervals. Using three-point finite difference schemes, we discretize the equations on so-called optimal grids constructed as follows: For a staggered grid with 2k points, we ask that the finite difference operator (a k × k Jacobi matrix) and the Sturm-Liouville differential operator share the k lowest eigenvalues and the values of the orthonormal eigenfunctions at one end of the interval. This requirement determines uniquely the entries in the Jacobi matrix, which are grid cell averages of the coefficients in the continuum problem. If these coefficients are known, we can find the grid, which we call optimal because it gives, by design, a finite difference operator with a prescribed spectral measure. We focus attention on the inverse problem, where neither the coefficients nor the grid are known. A key question in inversion is how to parametrize the coefficients, i.e., how to choose the grid. It is clear that, to be successful, this grid must be close to the optimal