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17
On optimal finitedifference approximation of PML
 SIAM J. Numer. Anal
"... Abstract. A technique derived from two related methods suggested earlier by some of the authors for optimization of finitedifference grids and absorbing boundary conditions is applied to discretization of perfectly matched layer (PML) absorbing boundary conditions for wave equations in Cartesian co ..."
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Abstract. A technique derived from two related methods suggested earlier by some of the authors for optimization of finitedifference grids and absorbing boundary conditions is applied to discretization of perfectly matched layer (PML) absorbing boundary conditions for wave equations in Cartesian coordinates. We formulate simple sufficient conditions for optimality and implement them. It is found that the minimal error can be achieved using pure imaginary coordinate stretching. As such, the PML discretization is algebraically equivalent to the rational approximation of the square root on [0, 1] conventionally used for approximate absorbing boundary conditions. We present optimal solutions for two cost functions, with exponential (and exponential of the square root) rates of convergence with respect to the number of the discrete PML layers using a second order finitedifference scheme with optimal grids. Results of numerical calculations are presented.
Circular resistor networks for electrical impedance tomography with partial boundary measurements.
"... 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 ..."
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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 socalled 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 tradeoff 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.
Pyramidal resistor networks for electrical impedance tomography with partial boundary measurements
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THREEPOINT FINITEDIFFERENCE SCHEMES, PADÉ AND THE SPECTRAL GALERKIN METHOD. I. ONESIDED IMPEDANCE APPROXIMATION
"... Abstract. A method for calculating special grid placement for threepoint 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 metho ..."
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Abstract. A method for calculating special grid placement for threepoint 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 GalerkinPetrov method. In fact we show that for every stable GalerkinPetrov 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 twodimensional optimal grid with a spectral Galerkin method. 1.
Resistor network approaches to electrical impedance tomography
 Inside Out, Mathematical Sciences Research Institute Publications
, 2011
"... We review a resistor network approach to the numerical solution of the inverse problem of electrical impedance tomography (EIT). The networks arise in the context of finite volume discretizations of the elliptic equation for the electric potential, on sparse and adaptively refined grids that we call ..."
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Cited by 6 (2 self)
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We review a resistor network approach to the numerical solution of the inverse problem of electrical impedance tomography (EIT). The networks arise in the context of finite volume discretizations of the elliptic equation for the electric potential, on sparse and adaptively refined grids that we call optimal. The name refers to the fact that the grids give spectrally accurate approximations of the Dirichlet to Neumann map, the data in EIT. The fundamental feature of the optimal grids in inversion is that they connect the discrete inverse problem for resistor networks to the continuum EIT problem. 1.
Optimal grids for anisotropic problems
 Electronic Transactions on Numerical Analysis
"... Abstract. 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 isotrop ..."
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Cited by 4 (1 self)
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Abstract. 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 semiinfinite 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 2D anisotropic media.
Resistor network approaches to the numerical solution of electrical impedance tomography with partial boundary measurements
 Rice University
, 2009
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STUDY OF NOISE EFFECTS IN ELECTRICAL IMPEDANCE TOMOGRAPHY WITH RESISTOR NETWORKS
, 1105
"... Abstract. We present a study of the numerical solution of the two dimensional electrical impedance tomography problem, with noisy measurements of the Dirichlet to Neumann map. The inversion uses parametrizations of the conductivity on optimal grids. The grids are optimal in the sense that finite vol ..."
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Abstract. We present a study of the numerical solution of the two dimensional electrical impedance tomography problem, with noisy measurements of the Dirichlet to Neumann map. The inversion uses parametrizations of the conductivity on optimal grids. The grids are optimal in the sense that finite volume discretizations on them give spectrally accurate approximations of the Dirichlet to Neumann map. The approximations are Dirichlet to Neumann maps of special resistor networks, that are uniquely recoverable from the measurements. Inversion on optimal grids has been proposed and analyzed recently, but the study of noise effects on the inversion has not been carried out. In this paper we present a numerical study of both the linearized and the nonlinear inverse problem. We take three different parametrizations of the unknown conductivity, with the same number of degrees of freedom. We obtain that the parametrization induced by the inversion on optimal grids is the most efficient of the three, because it gives the smallest standard deviation of the maximum a posteriori estimates of the conductivity, uniformly in the domain. For the nonlinear problem we compute the mean and variance of the maximum aposterioriestimatesoftheconductivity,onoptimalgrids. Forsmallnoise, we obtain that the estimates are unbiased and their variance is very close to the optimal one, given by the CramérRao bound. For larger noise we use regularization and quantify the tradeoff between reducing the variance and introducing bias in the solution. Both the full and partial measurement setups are considered. 1.
A MODEL REDUCTION APPROACH TO NUMERICAL INVERSION FOR A PARABOLIC PARTIAL DIFFERENTIAL EQUATION
, 1210
"... Abstract. We propose a novel numerical inversion algorithm for parabolic partial differential equations, based on model reduction. The study is motivated by the application of controlled source electromagnetic exploration, where the unknown is the subsurface electrical resistivity and the data are t ..."
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Abstract. We propose a novel numerical inversion algorithm for parabolic partial differential equations, based on model reduction. The study is motivated by the application of controlled source electromagnetic exploration, where the unknown is the subsurface electrical resistivity and the data are time resolved surface measurements of the magnetic field. The algorithm presented in this paper considers a layered medium. The reduced model is obtained with rational interpolation in the frequency (Laplace) domain and a rational Krylov subspace projection method. It amounts to a nonlinear mapping from the function space of the unknown resistivity to the small dimensional space of the parameters of the reduced model. We use this mapping as a nonlinear preconditioner for the GaussNewton iterative solution of the inverse problem. The advantage of the inversion algorithm is twofold. First, the nonlinear preconditioner resolves most of the nonlinearity of the problem. Thus the iterations are less likely to get stuck in local minima and the convergence is fast. Second, the inversion is computationally efficient because it avoids repeated computations of the time domain solutions of the forward problem. We study the stability of the inversion algorithm for various rational Krylov subspaces, and assess its performance with numerical experiments. Key words. Inverse problem, parabolic, CSEM, model reduction, rational Krylov subspace projection
Passive Synthesis of Compact FrequencyDependent Interconnect Models via Quadrature Spectral Rules
"... In this paper, we present a reduced order modeling methodology, based on the utilization of optimal nonuniform grids generated by Gaussian spectral rules, for the direct passive synthesis of SPICEcompatible modeling of multiconductor interconnect structures. The algorithm is based on a PadéCheby ..."
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In this paper, we present a reduced order modeling methodology, based on the utilization of optimal nonuniform grids generated by Gaussian spectral rules, for the direct passive synthesis of SPICEcompatible modeling of multiconductor interconnect structures. The algorithm is based on a PadéChebyshev approximation of the frequencydependent input impedance matrix of the passive interconnect system. The synthesized circuit is represented as the concatenation of a number of nonuniform sections of passive lumped coupled circuits. However, contrary to the popular uniform segmentationbased distributed circuit models for interconnects, where 10 to 15 segments per minimum wavelength are needed for multiGHz accuracy, the proposed model is “optimal ” in the sense that highlyaccurate 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 superexponential convergence of the PadéChebyshev approximation of the input impedance of the transmissionline 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 multiport models is guaranteed through the use of passive equivalent circuits for the representation of the frequencydependent, perunitlength series impedance and shunt admittance matrices of the interconnect. Keywords: Transmissionline modeling of interconnects; passive reduced order synthesis; interconnects with frequencydependent losses.