arXiv subjects

Lekbir Afraites

Publications and source records attributed to Lekbir Afraites.

9 recordsLinked to original sources

arXiv2026

Cavity shape reconstruction with a homogeneous Robin condition via a constrained coupled complex boundary method with ADMM

We revisit the problem of identifying an unknown portion of a boundary subject to a Robin condition based on a pair of Cauchy data on the accessible part of the boundary. It is known that a single measurement may correspond to infinitely many admissible domains. Nonetheless, numerical strategies based on shape optimization have been shown to yield reasonable reconstructions of the unknown boundary. In this study, we propose a new application of the coupled complex boundary method to address this class of inverse boundary identification problems. The overdetermined problem is reformulated as a complex boundary value problem with a complex Robin condition that couples the Cauchy data on the accessible boundary. The reconstruction is achieved by minimizing a cost functional constructed from the imaginary part of the complex-valued solution. To improve stability with respect to noisy data and initialization, we augment the formulation with inequality constraints through prior admissible bounds on the state, leading to a constrained shape optimization problem. The shape derivative of the complex state and the corresponding shape gradient of the cost functional are derived, and the resulting problem is solved using an alternating direction method of multipliers (ADMM) framework. The proposed approach is implemented using the finite element method and validated through various numerical experiments.

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arXiv2026

Enhanced shape recovery in advection--diffusion problems via a novel ADMM-based CCBM optimization

This work proposes a novel shape optimization framework for geometric inverse problems governed by the advection--diffusion equation, based on the coupled complex boundary method (CCBM). Building on recent developments [Afr22, Rab23, Rab25, RAN25, RN24], we aim to recover the shape of an unknown inclusion via shape optimization driven by a cost functional constructed from the imaginary part of the complex-valued state variable over the entire domain. We rigorously derive the associated shape derivative in variational form and provide explicit expressions for the gradient and second-order information. Optimization is carried out using a Sobolev gradient method within a finite element framework. To address difficulties in reconstructing obstacles with concave boundaries, particularly under measurement noise and the combined effects of advection and diffusion, we introduce a state-of-the-art numerical scheme inspired by the Alternating Direction Method of Multipliers (ADMM). In addition to implementing this non-conventional approach, we demonstrate how the adjoint method can be efficiently applied and utilize partial gradients todevelop a more efficient CCBM-ADMM scheme. The accuracy and robustness of the proposed computational approach are validated through various numerical experiments.

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arXiv2026

Statistical Topological Gradient and Shape Optimization for Robust Metal--Semiconductor Contact Reconstruction

We develop a statistically robust framework for reconstructing metal--semiconductor contact regions using topological gradients. The inverse problem is formulated as the identification of an unknown contact region from boundary measurements governed by an elliptic model with piecewise coefficients. Deterministic stability of the topological gradient with respect to measurement noise is established, and the analysis is extended to a statistical setting with multiple independent observations. A central limit theorem in a separable Hilbert space is proved for the empirical topological gradient, yielding optimal $n^{-1/2}$ convergence and enabling the construction of confidence intervals and hypothesis tests for contact detection. To further refine the reconstruction, a shape optimization procedure is employed, where the free parameter $β$ in the CCBM formulation plays a crucial role in controlling interface sensitivity. While $β$ affects both topological and shape reconstructions, its influence is particularly pronounced in the shape optimization stage, allowing more accurate estimation of the size and geometry of the contact subregion. The proposed approach provides a rigorous criterion for distinguishing true structural features from noise-induced artifacts, and numerical experiments demonstrate the robustness, precision, and enhanced performance of the combined statistical, topological, and $β$-informed shape-based reconstruction.

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arXiv2025

Simultaneous recovery of a corroded boundary and admittance using the Kohn-Vogelius method

We address the problem of identifying an unknown portion $Γ$ of the boundary of a $d$-dimensional ($d \in \{1, 2\}$) domain $Ω$ and its associated Robin admittance coefficient, using two sets of boundary Cauchy data $(f, g)$--representing boundary temperature and heat flux--measured on the accessible portion $Σ$ of the boundary. Identifiability results \cite{Bacchelli2009,PaganiPierotti2009} indicate that a single measurement on $Σ$ is insufficient to uniquely determine both $Γ$ and $α$, but two independent inputs yielding distinct solutions ensure the uniqueness of the pair $Γ$ and $α$. In this paper, we propose a cost function based on the energy-gap of two auxiliary problems. We derive the variational derivatives of this objective functional with respect to both the Robin boundary $Γ$ and the admittance coefficient $α$. These derivatives are utilized to develop a nonlinear gradient-based iterative scheme for the simultaneous numerical reconstruction of $Γ$ and $α$. Numerical experiments are presented to demonstrate the effectiveness and practicality of the proposed method.

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arXiv2025

Numerical solution by shape optimization method to an inverse shape problem in multi-dimensional advection-diffusion problem with space dependent coefficients

This work focuses on numerically solving a shape identification problem related to advection-diffusion processes with space-dependent coefficients using shape optimization techniques. Two boundary-type cost functionals are considered, and their corresponding variations with respect to shapes are derived using the adjoint method, employing the chain rule approach. This involves firstly utilizing the material derivative of the state system and secondly using its shape derivative. Subsequently, an alternating direction method of multipliers (ADMM) combined with the Sobolev-gradient-descent algorithm is applied to stably solve the shape reconstruction problem. Numerical experiments in two and three dimensions are conducted to demonstrate the feasibility of the methods.

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arXiv2024

Boundary shape reconstruction with Robin condition: existence result, stability analysis, and inversion via multiple measurements

This study revisits the problem of identifying the unknown interior Robin boundary of a connected domain using Cauchy data from the exterior region of a harmonic function. It investigates two shape optimization reformulations employing least-squares boundary-data-tracking cost functionals. Firstly, it rigorously addresses the existence of optimal shape solutions, thus filling a gap in the literature. The argumentation utilized in the proof strategy is contingent upon the specific formulation under consideration. Secondly, it demonstrates the ill-posed nature of the two shape optimization formulations by establishing the compactness of the Riesz operator associated with the quadratic shape Hessian corresponding to each cost functional. Lastly, the study employs multiple sets of Cauchy data to address the difficulty of detecting concavities in the unknown boundary. Numerical experiments in two and three dimensions illustrate the numerical procedure relying on Sobolev gradients proposed herein.

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arXiv2024

Detecting immersed obstacle in Stokes fluid flow using the coupled complex boundary method

A non-conventional shape optimization approach is introduced to address the identification of an obstacle immersed in a fluid described by the Stokes equation within a larger bounded domain, relying on boundary measurements on the accessible surface. The approach employs tools from shape optimization, utilizing the coupled complex boundary method to transform the over-specified problem into a complex boundary value problem by incorporating a complex Robin boundary condition. This condition is derived by coupling the Dirichlet and Neumann boundary conditions along the accessible boundary. The identification of the obstacle involves optimizing a cost function constructed based on the imaginary part of the solution across the entire domain. The subsequent calculation of the shape gradient of this cost function, rigorously performed via the rearrangement method, enables the iterative solution of the optimization problem using a Sobolev gradient descent algorithm. The feasibility of the method is illustrated through numerical experiments in both two and three spatial dimensions, demonstrating its effectiveness in reconstructing obstacles with pronounced concavities under high-level noise-contaminated data, all without perimeter or volume functional penalization.

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arXiv2024

A robust alternating direction numerical scheme in a shape optimization setting for solving geometric inverse problems

The alternating direction method of multipliers within a shape optimization framework is developed for solving geometric inverse problems, focusing on a cavity identification problem from the perspective of non-destructive testing and evaluation techniques. The rationale behind this method is to achieve more accurate detection of unknown inclusions with pronounced concavities, emphasizing the aspect of shape optimization. Several numerical results to illustrate the applicability and efficiency of the method are presented for various shape detection problems. These numerical experiments are conducted in both two- and three-dimensional settings, with a focus on cases involving noise-contaminated data. The main finding of the study is that the proposed method significantly outperforms conventional shape optimization methods in reconstructing unknown cavity shapes.

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arXiv2007

On second order shape optimization methods for electrical impedance tomography

This paper is devoted to the analysis of a second order method for recovering the \emph{a priori} unknown shape of an inclusion $ω$ inside a body $Ω$ from boundary measurement. This inverse problem - known as electrical impedance tomography - has many important practical applications and hence has focussed much attention during the last years. However, to our best knowledge, no work has yet considered a second order approach for this problem. This paper aims to fill that void: we investigate the existence of second order derivative of the state $u$ with respect to perturbations of the shape of the interface $\partialω$, then we choose a cost function in order to recover the geometry of $\partial ω$ and derive the expression of the derivatives needed to implement the corresponding Newton method. We then investigate the stability of the process and explain why this inverse problem is severely ill-posed by proving the compactness of the Hessian at the global minimizer.

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