Identifying conductivity in electrical impedance tomography with total variation regularization

Hinze, Michael; Kaltenbacher, Barbara; Quyen, Tran Nhan Tam

doi:10.1007/s00211-017-0920-8

Cited by 35 publications

(25 citation statements)

References 45 publications

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“…Further, the sequence of minimizers \{ q \ast h \} h>0 contains a subsequence that converges in H 1 (\Omega ) to a minimizer to problem (3.1)--(3.2). The proof follows by a standard argument from calculus of variation and the density of the space V h in H 1 (\Omega ) and thus is omitted; see [21,37] for related analysis.…”

Section: \Biggl\{mentioning

confidence: 99%

Error Analysis of Finite Element Approximations of Diffusion Coefficient Identification for Elliptic and Parabolic Problems

Jin¹

2021

SIAM J. Numer. Anal.

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\bfA \bfb \bfs \bft \bfr \bfa \bfc \bft . In this work, we present a novel error analysis for recovering a spatially dependent diffusion coefficient in an elliptic or parabolic problem. It is based on the standard regularized output least-squares formulation with an H 1 (\Omega ) seminorm penalty and then discretized using the Galerkin finite element method with conforming piecewise linear finite elements for both state and coefficient and backward Euler in time in the parabolic case. We derive a priori weighted L 2 (\Omega ) estimates where the constants depend only on the given problem data for both elliptic and parabolic cases. Further, these estimates also allow deriving standard L 2 (\Omega ) error estimates under a positivity condition that can be verified for certain problem data. Numerical experiments are provided to complement the error analysis.\bfK \bfe \bfy \bfw \bfo \bfr \bfd \bfs . parameter identification, finite element approximation, error estimate, Tikhonov regularization \bfA \bfM \bfS \bfs \bfu \bfb \bfj \bfe \bfc \bft \bfc \bfl \bfa \bfs \bfs \bfi fi\bfc \bfa \bft \bfi \bfo \bfn \bfs . 65M30, 65M15, 65M60 \bfD \bfO

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Section: \Biggl\{mentioning

confidence: 99%

Error Analysis of Finite Element Approximations of Diffusion Coefficient Identification for Elliptic and Parabolic Problems

Jin¹

2021

SIAM J. Numer. Anal.

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“…From (8), g can be calculated if b and A are known. Generally, b and A are acquired by calculating the forward problem (Hinze et al, 2018; Liu et al, 2019). In this work, a novel two-stage image reconstruction strategy that combines TGV method and tight wavelet approach (TGV-TW) is proposed.…”

Section: A Two-stage Image Reconstruction Strategymentioning

confidence: 99%

A two-stage image reconstruction strategy for electrical impedance tomography

Shi

Kong

Wang

et al. 2021

Transactions of the Institute of Measurement and Control

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Electrical impedance tomography (EIT) is a potential and promising tomographic technique. Based on a reconstruction strategy, conductivity distribution can be imaged by processing boundary measurements. It should be noticed that the process of image reconstruction involves the solution of a nonlinear ill-posed inverse problem. To tackle this problem, a novel two-stage image reconstruction strategy is proposed in this work. It combines the advantages of total generalized variation regularization method and tight wavelet approach. The solution of the proposed method is acquired by employing alternating minimization algorithm and spilt Bregman algorithm. In the numerical simulation, reconstruction of five models is studied. Aside from the visual observation, we have also validated the proposed method with quantitative comparison. Meanwhile, the impact of noise on the reconstruction is considered. Furthermore, the proposed method is evaluated by phantom experimental data. The simulation and experimental results have demonstrated the superior performance of the proposed method in visualizing conductivity distribution.

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“…Proof. Let (p n , q n ) be solution sequence of ( 7)- (9). Using the estimate (10) we guaranty the existence of p, q ∈ H 1 (Ω) such that p n and q n converge weakly to p and q in H 1 (Ω).…”

mentioning

confidence: 98%

“…By the Poincaré inequality A is coercive. Hence the variational problem (7) has a unique solution, and with the Riesz representation theorem it is easy to show the existence of q ∈ W solution of the problem (9) we refer to [9]. we recall the following result: Proposition 1.…”

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A new hybrid method for shape optimization with application to semiconductor equations

Yazidi¹,

Ellabib²

2022

NACO

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<p style='text-indent:20px;'>The aim of this work is to reconstruct the depletion region in pn junction. Starting with famous drift diffusion model, we establish the simplified equation for the considered semiconductor. There we call the shape optimization technique to formulate a minimization problem from the inverse problem at hand. The existence of an optimal solution of the optimization problem is proved. The proposed numerical algorithm is a combined Domain Decomposition method with an efficient hybrid conjugate gradient guided by differential evolution heuristic algorithm, the finite element method is used to discretize the state equation. At the end we establish several numerical examples, to prove the validity of theoretical results using the proposed algorithm, in addition we show some simulation of the depletion region approximation under two different functioning modes.</p>

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Identifying conductivity in electrical impedance tomography with total variation regularization

Cited by 35 publications

References 45 publications

Error Analysis of Finite Element Approximations of Diffusion Coefficient Identification for Elliptic and Parabolic Problems

Error Analysis of Finite Element Approximations of Diffusion Coefficient Identification for Elliptic and Parabolic Problems

A two-stage image reconstruction strategy for electrical impedance tomography

A new hybrid method for shape optimization with application to semiconductor equations

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