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Calderón inverse problem with partial data on Riemann surfaces
 Duke Math. J
"... Abstract. On a fixed smooth compact Riemann surface with boundary (M0, g), we show that for the Schrödinger operator ∆ + V with potential V ∈ C 1,α (M0) for some α> 0, the DirichlettoNeumann map NΓ measured on an open set Γ ⊂ ∂M0 determines uniquely the potential V. We also discuss briefly the co ..."
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Cited by 7 (2 self)
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Abstract. On a fixed smooth compact Riemann surface with boundary (M0, g), we show that for the Schrödinger operator ∆ + V with potential V ∈ C 1,α (M0) for some α> 0, the DirichlettoNeumann map NΓ measured on an open set Γ ⊂ ∂M0 determines uniquely the potential V. We also discuss briefly the corresponding consequences for potential scattering at 0 frequency on Riemann surfaces with asymptotically Euclidean or asymptotically hyperbolic ends. 1.
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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Cited by 5 (4 self)
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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.
Calderón inverse problem for Schrodinger operator on Riemann surfaces. To appear
 in Proceedings of the Centre for Mathematics and its Applications ANU
, 2010
"... Abstract. On a fixed smooth compact Riemann surface with boundary (M0, g), we show that the Cauchy data space (or DirichlettoNeumann map N) of the Schrödinger operator ∆ + V with V ∈ C 2 (M0) determines uniquely the potential V. We also discuss briefly the corresponding consequences for potential ..."
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Cited by 4 (2 self)
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Abstract. On a fixed smooth compact Riemann surface with boundary (M0, g), we show that the Cauchy data space (or DirichlettoNeumann map N) of the Schrödinger operator ∆ + V with V ∈ C 2 (M0) determines uniquely the potential V. We also discuss briefly the corresponding consequences for potential scattering at 0 frequency on Riemann surfaces with asymptotically Euclidean or asymptotically hyperbolic ends. 1.
Pyramidal resistor networks for electrical impedance tomography with partial boundary measurements
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Resistor network approaches to the numerical solution of electrical impedance tomography with partial boundary measurements
 Rice University
, 2009
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Inverse problems with partial data for a Dirac system: a Carleman estimate approach
, 2009
"... Abstract. We prove that the material parameters in a Dirac system with magnetic and electric potentials are uniquely determined by measurements made on a possibly small subset of the boundary. The proof is based on a combination of Carleman estimates for first and second order systems, and involves ..."
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Cited by 3 (3 self)
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Abstract. We prove that the material parameters in a Dirac system with magnetic and electric potentials are uniquely determined by measurements made on a possibly small subset of the boundary. The proof is based on a combination of Carleman estimates for first and second order systems, and involves a reduction of the boundary measurements to the second order case. For this reduction a certain amount of decoupling is required. To effectively make use of the decoupling, the Carleman estimates are established for coefficients which may become singular in the asymptotic limit. 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 2 (1 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.
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.
NOTES ON THE CALDERÓN PROBLEM WITH PARTIAL DATA MICHAEL VANVALKENBURGH Abstract. These notes are my commentary on the paper “The Calderón Problem with Partial Data ” by Kenig, Sjöstrand, and Uhlmann.
"... In the following pages, I give a nearly linebyline discussion of the paper “The Calderón Problem with Partial Data ” by Kenig, Sjöstrand, and Uhlmann [14]. But before I begin, it is best to reflect on Alberto P. Calderón himself, and the way he worked. We recall that “one of Calderón’s main charac ..."
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In the following pages, I give a nearly linebyline discussion of the paper “The Calderón Problem with Partial Data ” by Kenig, Sjöstrand, and Uhlmann [14]. But before I begin, it is best to reflect on Alberto P. Calderón himself, and the way he worked. We recall that “one of Calderón’s main characteristics [is that] he always sought his own proofs, developed his own methods. From the start, Calderón worked in mathematics that way: he rarely read the work of others farther than the statements of theorems, and after grasping the general nature of the problem, went ahead by himself. In this process, Calderón not only rediscovered results, but added new insights to the subject ” [4]. For those who wish to consider the problem themselves before reading about someone else’s methods, in this brief section I only state the main problem. Let Ω ⊂ ⊂ R n be a bounded open connected set with, say, C ∞ boundary. For q ∈ L ∞ (Ω) we consider the operator − ∆ + q: L 2 (Ω) → L 2 (Ω) with domain H 2 (Ω) ∩ H 1 0(Ω), and we assume that 0 is not an eigenvalue of − ∆ + q: H 2 (Ω) ∩ H 1 0(Ω) → L 2 (Ω). Under this assumption, we have a welldefined DirichlettoNeumann (DN) map Nq: H 1/2 (∂Ω) ∋ v ↦ → ∂νu∂Ω ∈ H −1/2 (∂Ω), where ν denotes the exterior unit normal and u is the unique solution in of the problem H∆(Ω): = {u ∈ H 1 (Ω); ∆u ∈ L 2 (Ω)} (− ∆ + q)u = 0 in Ω, u∂Ω = v. Question: Let q1, q2 be two functions as above. Given two subsets Γ1, Γ2 ⊂ ∂Ω, we would like to say that if Nq1u = Nq2u in Γ1, for all u ∈ H 1/2 (∂Ω) ∩ E ′ (Γ2), then q1 = q2. What are conditions on Γ1 and Γ2 such that this is true?