Variationally constrained numerical solution of electrical impedance tomography

Borcea, Liliana; Gray, Genetha Anne.; Zhang, Yin

doi:10.1088/0266-5611/19/5/309

Cited by 29 publications

(40 citation statements)

References 90 publications

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“…The regularization parameter(s) are indicated below these codes. The first refers to β (for Tikhonov) or β 0 (for dynamic regularization), the second, only used for Tikhonov, is α (see (6)). We also indicate the misfit, defined by (28), followed in parenthesis by its theoretical expected value, which is controlled by an input noise level.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Several applications give rise to such a problem formulation. These include DC resistivity [35], linear potential problems [23,7], magnetotelluric inversion [30], diffraction tomography [12], oil reservoir simulation [14] aquifer calibration [17], electrical impedance tomography (EIT) [5,10,6], and Maxwell's equations in low frequencies [26,27,20,21].…”

Section: Introductionmentioning

confidence: 99%

“…It has been suggested by various authors (e.g., [16,9]) to add to this regularization operatorR 2 a term whose gradient depends on curvature, as in (6). In particular, the (modified) total variation term (7) penalizes the length of the level set interface.…”

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confidence: 99%

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On level set regularization for highly ill-posed distributed parameter estimation problems

Doel

Ascher

2006

Journal of Computational Physics

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The recovery of a distributed parameter function with discontinuities from inverse problems with elliptic forward PDEs is fraught with theoretical and practical difficulties. Better results are obtained for problems where the solution may take on at each point only one of two values, thus yielding a shape recovery problem.This article considers level set regularization for such problems. However, rather than explicitly integrating a time embedded PDE to steady state, which typically requires thousands of iterations, methods based on Gauss-Newton are applied more directly. One of these can be viewed as damped GaussNewton utilized to approximate the steady state equations which in turn are viewed as the necessary conditions of a Tikhonov-type regularization with a sharpening sub-step at each iteration. In practice this method is eclipsed, however, by a special "finite time" or Levenberg-Marquardt-type method which we call dynamic regularization applied to the output least squares formulation. Our stopping criterion for the iteration does not involve knowledge of the true solution.The regularization functional is applied to the (smooth) level set function rather than to the discontinuous function to be recovered, and the second focus of this article is on selecting this functional. Typical choices may lead to flat level sets that in turn cause ill-conditioning. We propose a new, quartic, nonlocal regularization term that penalizes flatness and produces a smooth level set evolution, and compare its performance to more usual choices.Two numerical test cases are considered: a potential problem and the classical EIT/DC resistivity problem.

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Section: Numerical Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On level set regularization for highly ill-posed distributed parameter estimation problems

Doel

Ascher

2006

Journal of Computational Physics

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“…This makes the inverse problem highly ill-posed. Applications include linear potential problems [45], DC resistivity [72], magnetotelluric inversion [63], diffraction tomography [31], oil reservoir simulation [34], aquifer calibration [40], electrical impedance tomography (EIT) [13,24,14] and low frequency electromagnetic data inversion [43].…”

Section: Level Sets For Shape Optimizationmentioning

confidence: 99%

Artificial time integration

2007

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Abstract.Many recent algorithmic approaches involve the construction of a differential equation model for computational purposes, typically by introducing an artificial time variable. The actual computational model involves a discretization of the now time-dependent differential system, usually employing forward Euler. The resulting dynamics of such an algorithm is then a discrete dynamics, and it is expected to be "close enough" to the dynamics of the continuous system (which is typically easier to analyze) provided that small -hence many -time steps, or iterations, are taken. Indeed, recent papers in inverse problems and image processing routinely report results requiring thousands of iterations to converge. This makes one wonder if and how the computational modeling process can be improved to better reflect the actual properties sought.In this article we elaborate on several problem instances that illustrate the above observations. Algorithms may often lend themselves to a dual interpretation, in terms of a simply discretized differential equation with artificial time and in terms of a simple optimization algorithm; such a dual interpretation can be advantageous. We show how a broader computational modeling approach may possibly lead to algorithms with improved efficiency.

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“…It is well-know that EIT is a very challenging problem due to its nonlinear and highly ill-posed properties [3,4]. Various methods have been proposed to solve EIT problems such as factorization method [5], variationally constrained numerical method [6], and nonlinear modified gradient-like method [7]. Recently, subspace-based optimization method (SOM) is proposed to solve electrical impedance tomography problems [8].…”

Section: Introductionmentioning

confidence: 99%

Two FFT Subspace-Based Optimization Methods for Electrical Impedance Tomography

Wei¹,

Chen²,

Zhao³

et al. 2016

PIER

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Abstract-Two numerical methods are proposed to solve the electric impedance tomography (EIT) problem in a domain with arbitrary boundary shape. The first is the new fast Fourier transform subspace-based optimization method (NFFT-SOM). Instead of implementing optimization within the subspace spanned by smaller singular vectors in subspace-based optimization method (SOM), a space spanned by complete Fourier bases is used in the proposed NFFT-SOM. We discuss the advantages and disadvantages of the proposed method through numerical simulations and comparisons with traditional SOM. The second is the low frequency subspace optimized method (LF-SOM), in which we replace the deterministic current and noise subspace in SOM with low frequency current and space spanned by discrete Fourier bases, respectively. We give a detailed analysis of strengths and weaknesses of LF-SOM through comparisons with mentioned SOM and NFFT-SOM in solving EIT problem in a domain with arbitrary boundary shape.

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Variationally constrained numerical solution of electrical impedance tomography

Cited by 29 publications

References 90 publications

On level set regularization for highly ill-posed distributed parameter estimation problems

On level set regularization for highly ill-posed distributed parameter estimation problems

Artificial time integration

Two FFT Subspace-Based Optimization Methods for Electrical Impedance Tomography

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