One-iteration reconstruction algorithm for geometric inverse source problem

Hrizi, Mourad; Hassine, Maatoug

doi:10.1007/s41808-018-0015-4

Cited by 10 publications

(9 citation statements)

References 31 publications

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“…This result was proved in [25] to describe the variation of a Kohn-Vogelius type functional with respect to a single small topological perturbation of sources.…”

Section: Asymptotic Expansionmentioning

confidence: 89%

“…This concept was originally introduced by Sokolowski and Zochowski [38]. Since then, this concept has been successfully applied to many relevant scientific and engineering problems such as geometry inverse problems [5,25,30,37], topology optimization [3,6,34,35], structural mechanics [19,41], image processing [24,29], and damage evolution modeling [4], and many other applications.…”

Section: The Kohn-vogelius Formulationmentioning

confidence: 99%

“…From (25) and (34) one can deduce that the adjoint state θ N [ f ] ∈ H 1 (Ω) solution to (16) can be written as…”

Section: Asymptotic Expansionmentioning

confidence: 99%

“…In this paper, we follow the approach introduced in [26] (see also [25]) and we propose a non-iterative algorithm for the reconstruction of the source term f * from a single Cauchy data, but without using the Newtonian potential to complement the unavailable information about the hidden boundary as presented in [12]. The proposed approach is based on the advantage of the Kohn-Vogelius formulation [31] and the topological sensitivity analysis method [3,19].…”

Section: Introductionmentioning

confidence: 99%

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A non-iterative reconstruction method for bioluminescence tomography

Hrizi

2020

Filomat

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In this paper, we study an inverse source problem of the bioluminescence tomography in three dimensional case. Our aim is to reconstruct a bioluminescent source distribution within a body from the knowledge of the boundary measurements. The inverse source problem is reformulated as a topology optimization one minimizing an energy like type functional. It measures the difference between the solutions of two auxiliary boundary value problems. An asymptotic expansion of the considered functional with respect to a set of ball-shaped anomalies is computed using the topological sensitivity analysis method. The obtained theoretical result leads to build a non-iterative reconstruction algorithm. Finally, some numerical examples in 3D are presented in order to show the effectiveness of the devised reconstruction algorithm.

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“…This result was proved in [25] to describe the variation of a Kohn-Vogelius type functional with respect to a single small topological perturbation of sources.…”

Section: Asymptotic Expansionmentioning

confidence: 89%

Section: The Kohn-vogelius Formulationmentioning

confidence: 99%

“…From (25) and (34) one can deduce that the adjoint state θ N [ f ] ∈ H 1 (Ω) solution to (16) can be written as…”

Section: Asymptotic Expansionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A non-iterative reconstruction method for bioluminescence tomography

Hrizi

2020

Filomat

Self Cite

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“…In [29] and [30], formula (11) was first derived. We think it is useful to show how, using the classical methods of shape optimization, we can prove both existence of the derivative and problem (11).…”

Section: 2mentioning

confidence: 99%

Shape optimization method for an inverse geometric source problem and stability at critical shape

Afraites¹,

Masnaoui²,

Nachaoui³

2022

DCDS-S

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This work deals with a geometric inverse source problem. It consists in recovering inclusion in a fixed domain based on boundary measurements. The inverse problem is solved via a shape optimization formulation. Two cost functions are investigated, namely, the least squares fitting, and the Kohn-Vogelius function. In this case, the existence of the shape derivative is given via the first order material derivative of the state problems. Furthermore, using the adjoint approach, the shape gradient of both cost functions is characterized. Then, the stability is investigated by proving the compactness of the Hessian at the critical shape for both considered cases. Finally, based on the gradient method, a steepest descent algorithm is developed, and some numerical experiments for non-parametric shapes are discussed.

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Reconstruction and stability analysis of potential appearing in time‐fractional subdiffusion

Kahlaoui,

Hrizi

2023

Math Methods in App Sciences

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In this work, we establish a local stability estimate for a problem involving the reconstruction of a domain in the time‐fractional diffusion equation with an order of : from the knowledge of a Neumann additional data on a part of the boundary. This problem is motivated by several applications in anomalous diffusion phenomena. The stability estimate is obtained through a straightforward approach utilizing shape optimization techniques. To reconstruct the domain , we formulate the inverse problem as a shape optimization one by minimizing a least‐squares cost function. Using the topological derivative method, we perform a noniterative algorithm for numerically recovering the location and shape of the unknown domain. Finally, we demonstrate the effectiveness of our approach in reconstructing multiple subdomains of varying shapes and sizes, considering noisy data, through numerical experiments.

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One-iteration reconstruction algorithm for geometric inverse source problem

Cited by 10 publications

References 31 publications

A non-iterative reconstruction method for bioluminescence tomography

A non-iterative reconstruction method for bioluminescence tomography

Shape optimization method for an inverse geometric source problem and stability at critical shape

Reconstruction and stability analysis of potential appearing in time‐fractional subdiffusion

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