Topological and shape gradient strategy for solving geometrical inverse problems

Chaabane, Slim; Masmoudi, Mohamed; Meftahi, Houcine

doi:10.1016/j.jmaa.2012.11.044

Cited by 20 publications

(37 citation statements)

References 22 publications

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“…Following the approach of [8], [14], but taking into account the time-dependence of the problem, we introduce the mismatch functional…”

Section: A Reconstruction Algorithmmentioning

confidence: 99%

“…This slight loss in accuracy seems to be an intrinsic limit of the topological gradient strategy applied to the considered problem. We point out that the reconstruction is performed by relying on a single measurement acquired on the boundary, which is indeed a constraint imposed by the In [3] and many related works the authors introduce a cost functional inherited from imaging techniques, whereas in [14], [35] different strategies involving the KohnVogelius functional or similar ones are explored. Nevertheless, the nonlinearity of the direct Figure 3: Reconstruction: value of the topological gradient in different configurations problem considered in this work prevents the possibility to apply these techniques, since the analitycal expressions of the fundamental solution, single and double layer potentials would not be available.…”

Section: Reconstruction Of Small Inclusionsmentioning

confidence: 99%

“…We use these results to set a reconstruction procedure for detecting the inclusion. To this aim, as in [8], we propose a reconstruction algorithm based on topological optimization, where a suitable quadratic functional is minimized to detect the position and the size of the inclusion (see also [14]). Numerical results obtained on an idealized left ventricle geometry assess the feasibility of the proposed procedure.…”

Section: Introductionmentioning

confidence: 99%

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An inverse problem for a semilinear parabolic equation arising from cardiac electrophysiology

Beretta¹,

Cavaterra²,

Cerutti³

et al. 2017

Inverse Problems

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In this paper we develop theoretical analysis and numerical reconstruction techniques for the solution of an inverse boundary value problem dealing with the nonlinear, timedependent monodomain equation, which models the evolution of the electric potential in the myocardial tissue. The goal is the detection of an inhomogeneity ω (where the coefficients of the equation are altered) located inside a domain Ω starting from observations of the potential on the boundary ∂Ω. Such a problem is related to the detection of myocardial ischemic regions, characterized by severely reduced blood perfusion and consequent lack of electric conductivity. In the first part of the paper we provide an asymptotic formula for electric potential perturbations caused by internal conductivity inhomogeneities of low volume fraction, extending the results published in [7] to the case of three-dimensional, parabolic problems. In the second part we implement a reconstruction procedure based on the topological gradient of a suitable cost functional. Numerical results obtained on an idealized three-dimensional left ventricle geometry for different measurement settings assess the feasibility and robustness of the algorithm.

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“…Following the approach of [8], [14], but taking into account the time-dependence of the problem, we introduce the mismatch functional…”

Section: A Reconstruction Algorithmmentioning

confidence: 99%

Section: Reconstruction Of Small Inclusionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

An inverse problem for a semilinear parabolic equation arising from cardiac electrophysiology

Beretta¹,

Cavaterra²,

Cerutti³

et al. 2017

Inverse Problems

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show abstract

“…Topological asymptotic expansions of the form (1) have been derived for many different problems constrained by linear PDEs. We refer the interested reader to [14,16,18,30,5,4,10,11,12] as well as the monograph [32]. Besides the field of shape and topology optimization, topological derivatives are also used in applications from mathematical imaging, such as image segmentation [25] or electric impedance tomography [26,28], or other geometric inverse problems such as the detection of obstacles, of cracks or of impurities of a material, see e.g., [16,23] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Topological derivative for the nonlinear magnetostatic problem

Gangl¹,

Amstutz²

2019

etna

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The topological derivative represents the sensitivity of a domain-dependent functional with respect to a local perturbation of the domain and is a valuable tool in topology optimization. Motivated by an application from electrical engineering, we derive the topological derivative for an optimization problem which is constrained by the quasilinear equation of two-dimensional magnetostatics. Here, the main ingredient is to establish a sufficiently fast decay of the variation of the direct state at scale 1 as |x| → ∞. In order to apply the method in a bi-directional topology optimization algorithm, we derive both the sensitivity for introducing air inside ferromagnetic material and the sensitivity for introducing material inside an air region. We explicitly compute the arising polarization matrices and introduce a way to efficiently evaluate the obtained formulas. Finally, we employ the derived formulas in a level-set based topology optimization algorithm and apply it to the design optimization of an electric motor.

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“…The so-called Kohn-Vogelius criterion [19] (13) has already been investigated in the topological gradient context for the detection of cracks [20] in the steady-state heat equation, for the localization of small cavities in Stokes flow [12] and for the shape reconstruction of inclusions in an inverse conductivity problem [21].…”

Section: Application To Cavities Identificationmentioning

confidence: 99%

Cavities Identification from Partially Overdetermined Boundary Data in Linear Elasticity

AB¹

2016

J Appl Computat Math

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Topological and shape gradient strategy for solving geometrical inverse problems

Cited by 20 publications

References 22 publications

An inverse problem for a semilinear parabolic equation arising from cardiac electrophysiology

An inverse problem for a semilinear parabolic equation arising from cardiac electrophysiology

Topological derivative for the nonlinear magnetostatic problem

Cavities Identification from Partially Overdetermined Boundary Data in Linear Elasticity

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