Reconstruction of boundary data for a class of nonlinear inverse problems

Essaouini, M.; Nachaoui, Abdeljalil; Hajji, Saïd El

doi:10.1515/1569394042248238

Cited by 10 publications

(5 citation statements)

References 6 publications

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“…Then it was implemented and improved by relaxation schemes in [27,28,29]. After that, different studies have been done using these algorithms for solving ill-posed problems governed by partial differential Equations [1,2,6,7,9,11,12,13,41].…”

Section: Description Of Algorithmsmentioning

confidence: 99%

An effective relaxed alternating procedure for Cauchy problem connected with Helmholtz Equation

Berdawood

Nachaoui

et al. 2021

Numerical Methods Partial

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This paper is concerned with the Cauchy problem for the Helmholtz equation. Recently, some new works asked the convergence of the well‐known alternating iterative method. Our main result is to propose a new alternating algorithm based on relaxation technique. In contrast to the existing results, the proposed algorithm is simple to implement, converges for all choice of wave number, and it can be used as an acceleration of convergence in the case where the classical alternating algorithm converges. We present theoretical results of the convergence of our algorithm. The numerical results obtained using our relaxed algorithm and the finite element approximation show the numerical stability, consistency and convergence of this algorithm. This confirms the efficiency of the proposed method.

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Section: Description Of Algorithmsmentioning

confidence: 99%

An effective relaxed alternating procedure for Cauchy problem connected with Helmholtz Equation

Berdawood

Nachaoui

et al. 2021

Numerical Methods Partial

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“…These problems involve the determination of unknown parameters or fields inside a domain based on measurements or observations made on the boundary. A such class of problems is the nonlinear inverse boundary problem for the Poisson equation, which aims to recover the temperature distribution within a domain [33][34][35].…”

Section: Introductionmentioning

confidence: 99%

“…It involves finding a solution to the Poisson equation inside a domain, given values on a subset of the boundary. Various numerical methods have been developed to tackle this problem, such as the boundary element method [33,34,[40][41][42], finite element method [18,34,[43][44][45], and finite difference method [28,46]. These methods typically rely on mesh-based discretization techniques and have proven to be effective in solving the linear Cauchy problem.…”

Section: Introductionmentioning

confidence: 99%

Polynomial Approximation of a Nonlinear Inverse Cauchy Problem

2023

acad. sci. j.

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In this paper, a class of nonlinear inverse boundary problem in the context of heat transfer is considered. We consider a class of nonlinear inverse boundary problems in the context of heat transfer. The problem involves determining the temperature distribution within a domain subject to a Cauchy boundary condition on a part of its boundary. We introduce a transformed variable, which allows us to reformulate the problem as a linear Cauchy problem followed by a series of nonlinear equations. We propose a polynomial expansion method to solve the linear Cauchy problem for the Laplace equation, and we employ the Newton method to solve the resulting nonlinear equations. Importantly, our approach does not rely on mesh-based discretization, allowing for parallel computation and preserving the mesh-free nature of the problem. We present numerical results obtained using our methodology and discuss the effectiveness of the proposed approach. The results show that the method provides a robust and efficient framework for solving nonlinear inverse boundary problems in heat transfer, with potential applications in various engineering and scientific fields.

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“…The main advantages of this method consist in: firstly, the computational schemes can be implemented easily. Secondly, the schemes for the problems with linear and nonlinear operators are similar [47,23,22]. Finally, it does not require any chosen regularization parameter as in the previous methods.…”

Section: Introductionmentioning

confidence: 99%

An accelerated alternating iterative algorithm for data completion problems connected with Helmholtz equation

Berdawood

Nachaoui

2023

Stat., optim. inf. comput.

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This paper deals with an inverse problem governed by the Helmholtz equation. It consists in recovering lackingdata on a part of the boundary based on the Cauchy data on the other part. We propose an optimal choice of the relaxationparameter calculated dynamically at each iteration. This choice of relaxation parameter ensures convergence without priordetermination of the interval of the relaxation factor required in our previous work. The numerous numerical example showsthat the number of iterations is drastically reduced and thus, our new relaxed algorithm guarantees the convergence for allwavenumber k and gives an automatic acceleration without any intervention of the user.

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Reconstruction of boundary data for a class of nonlinear inverse problems

Cited by 10 publications

References 6 publications

An effective relaxed alternating procedure for Cauchy problem connected with Helmholtz Equation

An effective relaxed alternating procedure for Cauchy problem connected with Helmholtz Equation

Polynomial Approximation of a Nonlinear Inverse Cauchy Problem

An accelerated alternating iterative algorithm for data completion problems connected with Helmholtz equation

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