Regularized D-bar method for the inverse conductivity problem

Knudsen, Kim; Lassas, Matti; Mueller, Jennifer L.; Siltanen, Samuli

doi:10.3934/ipi.2009.3.599

Cited by 143 publications

(221 citation statements)

References 78 publications

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“…here equation (17) has been solved in Fourier space to eliminate the function ϕ in (16). Approximating the Fourier transform as in the previous sections via a discrete Fourier transform, one ends up for (97) with a finite dimensional system of ODEs in t which are then numerically integrated with respect to t. The first approach along these lines appears to have been realised in [28].…”

Section: 2mentioning

confidence: 99%

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Spectral Approach to D‐bar Problems

Klein

McLaughlin

2017

Comm Pure Appl Math

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Abstract. We present the first numerical approach to D-bar problems having spectral convergence for real analytic rapidly decreasing potentials. The proposed method starts from a formulation of the problem in terms of an integral equation which is solved with Fourier techniques. The singular integrand is regularized analytically. The resulting integral equation is approximated via a discrete system which is solved with Krylov methods. As an example, the D-bar problem for the Davey-Stewartson II equations is solved. The result is used to test direct numerical solutions of the PDE.

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Section: 2mentioning

confidence: 99%

“…More recently, in [13], fourth order stiff integrators for DS II have been studied. The inversion of the Laplace operator in (17) has introduced the term (ξ…”

Section: 2mentioning

confidence: 99%

Spectral Approach to D‐bar Problems

Klein

McLaughlin

2017

Comm Pure Appl Math

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“…That is, given two sufficiently regular conductivities σ 1 and σ 2 , the following estimate holds 5) where C and α are positive constants. See also the stability estimates in [32], that deal with cases where due to noise, the measured data is no longer a DtN map. These estimates are also of logarithmic type.…”

Section: Introductionmentioning

confidence: 99%

Circular resistor networks for electrical impedance tomography with partial boundary measurements

Borcea¹,

Druskin²,

Mamonov³

2010

Inverse Problems

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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 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.

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“…However, the real difficulty is caused by the exponential ill-posedness of the underlying continuum EIT problem, even in the ideal case of complete knowledge of the DtN map. By exponential instability we mean that the sup norm of perturbations of σ is bounded in terms of the logarithm of the operator norm of perturbations of Λ σ [1,5,33]. The bounds are sharp [38], but the estimates are global and do not give resolution limits of the images of σ(x) as we vary x ∈ Ω.…”

Section: Introductionmentioning

confidence: 99%

Pyramidal resistor networks for electrical impedance tomography with partial boundary measurements

et al. 2010

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Abstract. We introduce an inversion algorithm for electrical impedance tomography (EIT) with partial boundary measurements, in two dimensions. It gives stable and fast reconstructions using sparse parameterizations of the unknown conductivity on optimal grids that are computed as part of the inversion. We follow the approach in [8,27] that connects inverse discrete problems for resistor networks to continuum EIT problems, using optimal grids. The algorithm in [8,27] is based on circular resistor networks, and solves the EIT problem with full boundary measurements. It is extended in [11] to EIT with partial boundary measurements, using extremal quasiconformal mappings that transform the problem to one with full boundary measurements. Here we introduce a different class of optimal grids, based on resistor networks with pyramidal topology, that is better suited for the partial measurements setup. We prove the unique solvability of the discrete inverse problem for these networks, and develop an algorithm for finding them from the measurements of the DtN map. Then, we show how to use the networks to define the optimal grids and to approximate the unknown conductivity. We assess the performance of our approach with numerical simulations and compare the results with those in [11].

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Regularized D-bar method for the inverse conductivity problem

Cited by 143 publications

References 78 publications

Spectral Approach to D‐bar Problems

Spectral Approach to D‐bar Problems

Circular resistor networks for electrical impedance tomography with partial boundary measurements

Pyramidal resistor networks for electrical impedance tomography with partial boundary measurements

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