Identification of dipole sources in an elliptic equation from boundary measurements: application to the inverse EEG problem

Badia, Abdellatif El; Farah, Maha

doi:10.1515/156939406777571012

Cited by 18 publications

(22 citation statements)

References 18 publications

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“…−μ) z L 2 0 holds true for all z ∈ L 2 ( ). Now, we conclude from (15) Proof. Note that: (i) the set of optimal solutions of the primal problem is nonempty (consequence of assumption (A4)); (ii) the dualizing parametrization function f (φ, c j ; •) is lsc at z = 0 for every (φ, c j ) (follows from the definition of f ); (iii) the perturbation function ρ is lsc at z = 0 (see remark 4); (iv) There exists an element (λ,μ) ∈ dom(θ ρ ) (see remark 4).…”

Section: Lemmamentioning

confidence: 75%

On piecewise constant level-set (PCLS) methods for the identification of discontinuous parameters in ill-posed problems

Cezaro

Leitão

2012

Inverse Problems

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We investigate level-set-type methods for solving ill-posed problems with discontinuous (piecewise constant) coefficients. The goal is to identify the level sets as well as the level values of an unknown parameter function on a model described by a nonlinear ill-posed operator equation. The PCLS approach is used here to parametrize the solution of a given operator equation in terms of a L 2 level-set function, i.e. the level-set function itself is assumed to be a piecewise constant function. Two distinct methods are proposed for computing stable solutions of the resulting ill-posed problem: the first is based on Tikhonov regularization, while the second is based on the augmented Lagrangian approach with total variation penalization. Classical regularization results (Engl H W et al 1996 Mathematics and its Applications (Dordrecht: Kluwer)) are derived for the Tikhonov method. On the other hand, for the augmented Lagrangian method, we succeed in proving the existence of (generalized) Lagrangian multipliers in the sense of (Rockafellar R T and Wets R J-B 1998 Grundlehren der Mathematischen Wissenschaften (Berlin: Springer)). Numerical experiments are performed for a 2D inverse potential problem (Hettlich F and Rundell W 1996 Inverse Problems 12 251-66), demonstrating the capabilities of both methods for solving this ill-posed problem in a stable way (complicated inclusions are recovered without any a priori geometrical information on the unknown parameter).

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Section: Lemmamentioning

confidence: 75%

On piecewise constant level-set (PCLS) methods for the identification of discontinuous parameters in ill-posed problems

Cezaro

Leitão

2012

Inverse Problems

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“…The analysis of the EEG forward model is an essential preliminary step to the resolution of the corresponding inverse source problem. In the head model of adults, theoretical and numerical results exist [9,13,14,16,29,10,4]. For the neonatal head model, identifiability and stability results for the inverse EEG source problem have been obtained in parallel to the present work [11].…”

Section: Mesh Nodes Tetrahedramentioning

confidence: 84%

EEG in neonates: Forward modeling and sensitivity analysis with respect to variations of the conductivity

Azizollahi

Darbas²,

Diallo³

et al. 2018

Mathematical Biosciences &Amp; Engineering

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The paper is devoted to the analysis of electroencephalography (EEG) in neonates. The goal is to investigate the impact of fontanels on EEG measurements, i.e. on the values of the electric potential on the scalp. In order to answer this clinical issue, a complete mathematical study (modeling, existence and uniqueness result, realistic simulations) is carried out. A model for the forward problem in EEG source localization is proposed. The model is able to take into account the presence and ossification process of fontanels which are characterized by a variable conductivity. From a mathematical point of view, the model consists in solving an elliptic problem with a singular source term in an inhomogeneous medium. A subtraction approach is used to deal with the singularity in the source term, and existence and uniqueness results are proved for the continuous problem. Discretization is performed with 3D Finite Elements of type P1 and error estimates are proved in the energy norm (H 1-norm). Numerical simulations for a three-layer spherical model as well as for a realistic neonatal head model including or not the fontanels have been

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“…The tests we are concerned with here are motivated by the numerical work on the electrical activity of the brain from Electro-Encephalo-Graphs (EEG). For instance, epileptic focii in the brain cortex may be modelled by dipolar sources and an algebraic technique is proposed in [45] to recover these point-wise sources (see also [46]). Roughly said, the potential is recorded at the surface of the scalp, according to a well-established protocol.…”

Section: Numerical Discussionmentioning

confidence: 99%

“…The data which may therefore be polluted are inexact. The other, required by the methodology of the detection process [45], is involved in Cauchy conditions fixed on the internal boundary. They are exact and come from harmonic polynomials.…”

Section: Numerical Discussionmentioning

confidence: 99%

A finite element model for the data completion problem: analysis and assessment

Azaïez

Belgacem

et al. 2011

Inverse Problems in Science and Engineering

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We investigate the computational advantages of the Steklov-Poincare´variational formulation, based on the uniqueness Holmgren theorem, for the badly ill-posed data completion problem. We study the discrete solution with a finite element approximation. The uniqueness issue is dealt with and we check that it is related to a discrete Holmgren result which requires a particular assumption on the mesh. Then, the important point is to show how the finite element variational problem may be recast into a least-squares problem. Lavrentiev's method turns out to be a Tikhonov regularization of an 'underlying' equation that will never be explicited. When employed in conjunction with the Morozov discrepancy principle to select the regularization parameter, the overall mathematical studies realized specifically for the Tikhonov method extends as well to the Lavrentiev method to conclude to a convergent regularization strategy. Similarly, the conjugate gradient method (CGM) may be considered as applied to a normal equation of that same 'hidden underlying' problem. We conduct a brief discussion about this method to explain why it yields a convergent strategy. We close with several numerical simulations for different geometries to assess the Lavrentiev finite element or the PCG finite element solution of the data completion problem.

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Identification of dipole sources in an elliptic equation from boundary measurements: application to the inverse EEG problem

Cited by 18 publications

References 18 publications

On piecewise constant level-set (PCLS) methods for the identification of discontinuous parameters in ill-posed problems

On piecewise constant level-set (PCLS) methods for the identification of discontinuous parameters in ill-posed problems

EEG in neonates: Forward modeling and sensitivity analysis with respect to variations of the conductivity

A finite element model for the data completion problem: analysis and assessment

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