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145
Formulae and equations for finding scattering data from the DirichlettoNeumann map with nonzero background potential
 Inverse Problems
"... For the Schrödinger equation at fixed energy with a potential supported in a bounded domain we give formulas and equations for finding scattering data from the DirichlettoNeumann map with nonzero background potential. For the case of zero background potential these results were obtained in [R.G.Nov ..."
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Cited by 14 (8 self)
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For the Schrödinger equation at fixed energy with a potential supported in a bounded domain we give formulas and equations for finding scattering data from the DirichlettoNeumann map with nonzero background potential. For the case of zero background potential these results were obtained in [R.G.Novikov, Multidimensional inverse spectral problem for the equation −∆ψ + (v(x) − Eu(x))ψ = 0, Funkt. Anal. i Ego Prilozhen
Enhanced Electrical Impedance Tomography via the MumfordShah Functional
 ESAIM: Control, Optimization and Calculus of Variations
, 2001
"... We consider the problem of electrical impedance tomography where conductivity distribution in a domain is to be reconstructed from boundary measurements of voltage and currents. It is wellknown that this problem is highly illposed. In this work, we propose the use of the MumfordShah functional, dev ..."
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Cited by 14 (0 self)
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We consider the problem of electrical impedance tomography where conductivity distribution in a domain is to be reconstructed from boundary measurements of voltage and currents. It is wellknown that this problem is highly illposed. In this work, we propose the use of the MumfordShah functional, developed for segmentation and denoising of images, as a regularization. After establishing existence properties of the resulting variational problem, we proceed by demonstrating the approach in several numerical examples. Our results indicate that this is an eective approach for overcoming the illposedness. Moreover, it has the capability of enhancing the reconstruction while at the same time segmenting the conductivity image. 1 Introduction and formulation of the problem The purpose of this work is to demonstrate that the MumfordShah functional from image processing can be used eectively to regularize the classical problem of electrical impedance tomography. In electrical impedance tomogr...
Inverse problems for the anisotropic Maxwell equations (preprint). 17 R. Leis, Initial boundary value problems in mathematical physics
, 1986
"... Abstract. We prove that the electromagnetic material parameters are uniquely determined by boundary measurements for the timeharmonic Maxwell equations in certain anisotropic settings. We give a uniqueness result in the inverse problem for Maxwell equations on an admissible Riemannian manifold, and ..."
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Cited by 13 (12 self)
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Abstract. We prove that the electromagnetic material parameters are uniquely determined by boundary measurements for the timeharmonic Maxwell equations in certain anisotropic settings. We give a uniqueness result in the inverse problem for Maxwell equations on an admissible Riemannian manifold, and a uniqueness result for Maxwell equations in Euclidean space with admissible matrix coefficients. The proofs are based on a new Fourier analytic construction of complex geometrical optics solutions on admissible manifolds, and involve a proper notion of uniqueness for such solutions. 1.
DETERMINING NONSMOOTH FIRST ORDER TERMS FROM PARTIAL BOUNDARY MEASUREMENTS
, 2006
"... Abstract. We extend results of Dos Santos FerreiraKenigSjöstrandUhlmann (arXiv:math.AP/0601466) to less smooth coefficients, and we show that measurements on part of the boundary for the magnetic Schrödinger operator determine uniquely the magnetic field related to a Hölder continuous potential. ..."
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Cited by 12 (6 self)
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Abstract. We extend results of Dos Santos FerreiraKenigSjöstrandUhlmann (arXiv:math.AP/0601466) to less smooth coefficients, and we show that measurements on part of the boundary for the magnetic Schrödinger operator determine uniquely the magnetic field related to a Hölder continuous potential. We give a similar result for determining a convection term. The proofs involve Carleman estimates, a smoothing procedure, and an extension of the NakamuraUhlmann pseudodifferential conjugation method to logarithmic Carleman weights. 1.
MODEL PROBLEMS FOR THE MULTIGRID OPTIMIZATION OF SYSTEMS GOVERNED BY DIFFERENTIAL EQUATIONS
, 2005
"... We discuss a multigrid approach to the optimization of systems governed by differential equations. Such optimization problems appear in many applications and are of a different nature than systems of equations. Our approach uses an optimizationbased multigrid algorithm in which the multigrid algori ..."
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Cited by 12 (1 self)
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We discuss a multigrid approach to the optimization of systems governed by differential equations. Such optimization problems appear in many applications and are of a different nature than systems of equations. Our approach uses an optimizationbased multigrid algorithm in which the multigrid algorithm relies explicitly on nonlinear optimization models as subproblems on coarser grids. Our goal is not to argue for a particular optimizationbased multigrid algorithm, but instead to demonstrate how multigrid can be used to accelerate nonlinear programming algorithms. Furthermore, using several model problems we give evidence (both theoretical and numerical) that the optimization setting is well suited to multigrid algorithms. Some of the model problems show that the optimization problem may be more amenable to multigrid than the governing differential equation. In addition, we relate the multigrid approach to more traditional optimization methods as further justification for the use of an optimizationbased multigrid algorithm.
Global uniqueness from partial Cauchy data in two dimensions. Arxiv preprint arXiv:0810.2286
, 2008
"... Abstract. We prove for a two dimensional bounded domain that the Cauchy data for the Schrödinger equation measured on an arbitrary open subset of the boundary determines uniquely the potential. This implies, for the conductivity equation, that if we measure the current fluxes at the boundary on an a ..."
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Cited by 12 (1 self)
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Abstract. We prove for a two dimensional bounded domain that the Cauchy data for the Schrödinger equation measured on an arbitrary open subset of the boundary determines uniquely the potential. This implies, for the conductivity equation, that if we measure the current fluxes at the boundary on an arbitrary open subset of the boundary produced by voltage potentials supported in the same subset, we can determine uniquely the conductivity. We use Carleman estimates with degenerate weight functions to construct appropriate complex geometrical optics solutions to prove the results. 1.
Inverse Problems For A Perturbed Dissipative HalfSpace
 INVERSE PROBLEMS
, 1995
"... This paper addresses the scattering of acoustic and electromagnetic waves from a perturbed dissipative halfspace. For simplicity, the perturbation is assumed to have compact support. Section 1 discusses the application that motivated this work and explains how the scalar model used here is related ..."
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Cited by 11 (3 self)
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This paper addresses the scattering of acoustic and electromagnetic waves from a perturbed dissipative halfspace. For simplicity, the perturbation is assumed to have compact support. Section 1 discusses the application that motivated this work and explains how the scalar model used here is related to Maxwell's equations. Section 2 introduces three formulations for direct and inverse problems for the halfspace geometry. Two of these formulations relate to scattering problems, and the third to a boundary value problem. Section 3 shows how the scattering problems can be related to the boundary value problem. This shows that the three inverse problems are equivalent in a certain sense. In section 4, the boundary value problem is used to outline a simple way to formulate a multidimensional layer stripping procedure. This procedure is unstable and does not constitute a practical algorithm for solving the inverse problem. The paper concludes with three appendices, the first two of which car...
The Calderón problem for conormal potentials, I: Global uniqueness and reconstruction
 Comm. Pure Appl. Math
"... The goal of this paper is to establish global uniqueness and obtain reconstruction, in dimensions n ≥ 3, for the Calderón problem in the class of potentials conormal to a smooth submanifold H in R n. In the case of hypersurfaces, the potentials considered here may have any singularity weaker than th ..."
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Cited by 11 (10 self)
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The goal of this paper is to establish global uniqueness and obtain reconstruction, in dimensions n ≥ 3, for the Calderón problem in the class of potentials conormal to a smooth submanifold H in R n. In the case of hypersurfaces, the potentials considered here may have any singularity weaker than that of the delta function
Semiclassical pseudodifferential calculus and the reconstruction of a magnetic field
"... We give a procedure for reconstructing a magnetic field and electric potential from boundary measurements given by the Dirichlet to Neumann map for the magnetic Schrödinger operator in R n, n ≥ 3. The magnetic potential is assumed to be continuous with L ∞ divergence and zero boundary values. The me ..."
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Cited by 11 (5 self)
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We give a procedure for reconstructing a magnetic field and electric potential from boundary measurements given by the Dirichlet to Neumann map for the magnetic Schrödinger operator in R n, n ≥ 3. The magnetic potential is assumed to be continuous with L ∞ divergence and zero boundary values. The method is based on semiclassical pseudodifferential calculus and the construction of complex geometrical optics solutions in weighted Sobolev spaces. 1
Uniqueness in the Inverse Conductivity Problem for Conductivites with 3/2 Derivatives in L^p, p > 2n
"... this paper, we contribute nothing to the analysis of G # . The estimates used are from the paper of Sylvester and Uhlmannn [16]. It is possible that some improvement can be made here. We expect that one should be able to prove uniqueness for conductivities which have 3/2 derivatives in L with p > ..."
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Cited by 9 (0 self)
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this paper, we contribute nothing to the analysis of G # . The estimates used are from the paper of Sylvester and Uhlmannn [16]. It is possible that some improvement can be made here. We expect that one should be able to prove uniqueness for conductivities which have 3/2 derivatives in L with p > 2n/3. However, the straightforward generalization of the argument presented below would require that f # #G # f map functions which are compactly supported to functions which are locally in L with p and r satisfying 1/p 1/r = 1/n. Many such estimates fail, see [2] for further discussion