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15
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 ..."
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Cited by 35 (5 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.
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 14 (6 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.
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 9 (5 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 8 (4 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 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.
Inverse boundary problems for systems in two dimensions
 Ann. Henri Poincare
"... Abstract. We prove identification of coefficients up to gauge by Cauchy data at the boundary for elliptic systems on oriented compact surfaces with boundary or domains of C. In the geometric setting, we fix a Riemann surface with boundary, and consider both a Diractype operator plus potential acti ..."
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Cited by 6 (3 self)
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Abstract. We prove identification of coefficients up to gauge by Cauchy data at the boundary for elliptic systems on oriented compact surfaces with boundary or domains of C. In the geometric setting, we fix a Riemann surface with boundary, and consider both a Diractype operator plus potential acting on sections of a Clifford bundle and a connection Laplacian plus potential (i.e. Schrödinger Laplacian with external YangMills field) acting on sections of a Hermitian bundle. In either case we show that the Cauchy data determines both the connection and the potential up to a natural gauge transformation: conjugation by an endomorphism of the bundle which is the identity at the boundary. For domains of C, we recover zeroth order terms up to gauge from Cauchy data at the boundary in first order elliptic systems.
Identification of a connection from Cauchy data on a Riemann surface with boundary, Geom. Funct
 Anal
"... Abstract. We consider a connection ∇X on a complex line bundle over a Riemann surface with boundary M0, with connection 1form X. We show that the Cauchy data space of the connection Laplacian (also called magnetic Laplacian) L: = ∇X X + q, with q a complex valued potential, uniquely determines the ..."
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Cited by 6 (1 self)
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Abstract. We consider a connection ∇X on a complex line bundle over a Riemann surface with boundary M0, with connection 1form X. We show that the Cauchy data space of the connection Laplacian (also called magnetic Laplacian) L: = ∇X X + q, with q a complex valued potential, uniquely determines the connection up to gauge isomorphism, and the potential q. 1.
Resistor network approaches to the numerical solution of electrical impedance tomography with partial boundary measurements
 Rice University
, 2009
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INVERSE SCATTERING AT FIXED ENERGY ON SURFACES WITH EUCLIDEAN ENDS
"... Abstract. On a fixed Riemann surface (M0, g0) with N Euclidean ends and genus g, we show that, under a topological condition, the scattering matrix SV (λ) at frequency λ> 0 for the operator ∆ + V determines the potential V if V ∈ C 1,α j −γd(·,z0) (M0) ∩ e L ∞ (M0) for all γ> 0 and for some j ..."
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Cited by 3 (0 self)
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Abstract. On a fixed Riemann surface (M0, g0) with N Euclidean ends and genus g, we show that, under a topological condition, the scattering matrix SV (λ) at frequency λ> 0 for the operator ∆ + V determines the potential V if V ∈ C 1,α j −γd(·,z0) (M0) ∩ e L ∞ (M0) for all γ> 0 and for some j ∈ {1, 2}, where d(z, z0) denotes the distance from z to a fixed point z0 ∈ M0. The topological condition is given by N ≥ max(2g + 1, 2) for j = 1 and by N ≥ g + 1 if j = 2. In R 2 this implies that the operator SV (λ) determines any C 1,α potential V such that V (z) = O(e −γz2) for all γ> 0. 1.