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Exact solution at y = b and the reconstructed one, obtained by the algorithms 1 and 2 in (a) and by algorithms 3 and 4 in (b), for k=15
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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, conver...
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Citations
... Building on these findings, explicit regularized solutions for the Cauchy problem have been determined for various factorizations of the Helmholtz operator [18][19][20][21][22][23][24][25][26]. Additionally, readers can explore several boundary value problems in greater detail in references [3][4][5][6][7][8][9][10][11][12][13], [15], [17], [27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43] and [46][47]. The complexity of the Cauchy problem for elliptic equations underpins the challenge faced when attempting to derive general results applicable across varied scenarios. ...
... At this stage, we combine the assessments from (23) -(24) and (26) -(27), taking into account (21), which enables us to establish the estimation given in (12). □. ...
The research explores the ongoing assessment and reliability of the solution to the Cauchy problem related to the Helmholtz equation within a specific domain, using known values from a smooth segment of the domain's boundary as a reference point. This situation falls within the realm of mathematical physics where the solutions discovered do not consistently rely on the initial conditions. Highlighting practical applications, it is crucial not only to find an approximate solution but also to ascertain its derivative. Assuming that a solution exists and is continuously differentiable in a nearby region, accurate Cauchy data is scrutinized. A concrete formula has been developed to express both the solution and its derivative, along with a regularization approach for scenarios where ongoing approximations of the initial Cauchy data are provided under certain conditions, featuring a designated error threshold in the uniform metric rather than using the original data. Evaluations confirming the stability of the solution to the classical Cauchy problem have been presented.
... Subsequently, the associated algorithm was implemented using finite element approximation [29], and further enhanced through relaxation schemes [30]. Since then, various studies have been conducted, drawing inspiration from these algorithms to address numerous ill-posed problems [6,19,42,43,58]. ...
... Now we present the regularity results for the solution of problem (6) in H 2 ( ) which will be required in the subsequent sections ( for the theorem proof, see [23]). ...
... and using the H 2 regularity for the solution of the direct problem (6). While it is not the case for our U ad . ...
This paper delves into addressing an inverse Cauchy problem using the primal-dual method. This approach is motivated by the inherent challenges posed by the irregular nature of the unknown parameter, characterized as L1 type, which more accurately reflects real-world phenomena than traditional regularity assumptions. Our approach involves reformulating the inverse problem into a PDE-constrained optimization framework, incorporating fidelity and regularization terms in the L1 norm. Subsequently, we apply a primal-dual method to effectively navigate the non-differentiability of the cost function. In tandem with theoretical advancements, we rigorously establish the stability of the solution, particularly in the presence of noisy data, ensuring convergence. To substantiate the efficacy and stability of our algorithm, we undertake a series of numerical experiments spanning both two and three dimensions. These experiments not only demonstrate the performance of our methodology but also underscore its resilience in handling real-world noisy data scenarios. Finally, we present experimental results demonstrating the efficiency of our proposed approach. These results include instances with both noisy and noise-free data, represented in two and three dimensions.
... The advantage of adaptive methods is their ability to adapt to the specifics of the problem without the need for preliminary parameter adjustment, which makes them convenient and efficient. In Ref. [20], the authors also present an efficient algorithm for solving the Cauchy problem based on the relaxation of alternating iterations, which shows high accuracy and stability at various noise levels. This approach is especially effective at large wave numbers where traditional methods can give unstable solutions. ...
... In comparison with the results of Ref. [20], the Nesterov method requires a larger number of iterations when calculating without noise, but the achieved accuracy is almost the same. In the presence of noise, the Nesterov method demonstrates superiority in accuracy. ...
In this paper, the Cauchy problem for the Helmholtz equation, also known as the continuation problem, is considered. The continuation problem is reduced to a boundary inverse problem for a well-posed direct problem. A generalized solution to the direct problem is obtained and an estimate of its stability is given. The inverse problem is reduced to an optimization problem solved using the gradient method. The convergence of the Landweber method with respect to the functionals is compared with the convergence of the Nesterov method. The calculation of the gradient in discrete form, which is often used in the numerical solutions of the inverse problem, is described. The formulation of the conjugate problem in discrete form is presented. After calculating the gradient, an algorithm for solving the inverse problem using the Nesterov method is constructed. A computational experiment for the boundary inverse problem is carried out, and the results of the comparative analysis of the Landweber and Nesterov methods in a graphical form are presented.
... For the Cauchy problem associated with Helmholtz equation, several papers have indicated that the KMF algorithm loses its efficiency especially for large enough wavenumbers. Either it does not converge or the convergence becomes so slow that it can be assimilated to non-convergence [4,7,8]. ...
... In order to be able to solve these problems which cannot be treated by this algorithm, we have suggested in previous works [4] an iterative relaxation method that can circumvent these obstacles. As for the Poisson equation [48], convergence and acceleration intervals, depending on the data of the problem, have been found. ...
... (9) to (12). 4 ...
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.
... Using this splitting, iterative methods based on fixed point theory were developed [38]. These methods were extended to Cauchy problems for more general elliptic equations [2,9,8,24,25]. Other methods of solving the Cauchy problem are based on its reformulation as a control problem [17,40]: ...
Abstract. In this paper, we develop a Haar wavelet-based reconstruction method to recover missing data on an inaccessible part of the boundary from measured data on another accessible part. The technique is developed to solve inverse Cauchy problems governed by the Poisson equation which is severely
ill-posed. The new method is mathematically simple to implement and can be
easily applied to Cauchy problems governed by other partial differential equations appearing in various fields of natural science, engineering, and economics.
To take into account the ill-conditioning of the obtained linear system due to the ill-posedness of the Cauchy problem, a preconditioning strategy combined with a regularization has been developed. Comparing the numerical results produced by a meshless method based on the polynomial expansion with those
produced by the proposed technique illustrates the superiority of the latter.
Other numerical results show its effectiveness.
... Also, it is not clear how large the wave number can be taken without the convergence being impaired. Recently, (Berdawood et al., 2021(Berdawood et al., , 2022, propose new efficient alternating algorithms based on idea initially proposed in Jourhmane & Nachaoui (1999); to solve the Cauchy problem for the Poisson equation. They prove the convergence of the proposed procedures, for all values of wave number in the case of the Helmholtz equation and they show that their method can accelerate convergence in the case of the modified Helmholtz equation (Berdawood et al., 2020). ...
... The goal is to show that, in the case of the Cauchy problem governed by the Helmholtz equation, the procedure proposed here works well without restriction of the value of the wave number k. Contrary to some previous works, we will show, by a numerical analysis, that this method is stable, when the given data is noisy. In order to show the efficiency of the proposed procedure compared to other existing schemes concerned with the same problem, we treat examples of Berdawood et al. (2021Berdawood et al. ( , 2022. ...
The objective of this paper is to solve numerically a Cauchy problem defined on a two-dimensional domain occupied by a material satisfying the Helmholtz type equations and verifying additional Cauchy-type boundary conditions on the accessible part of the boundary. A meshless numerical method using an approximation of the solution based on the polynomial expansion is applied. To confirm the efficiency of the proposed method, different examples were considered and the obtained linear system was solved using the well-known CG and CGLS algorithms.
... The relaxation JN method introduced in the last papers drastically reduced the number of iterations required to achieve convergence. It was used in elasticity (Ellabib & Nachaoui, 2008;Marin & Johansson, 2010), and recently for Cauchy problem governed by Stocks equation (Chakib et al., 2018) and for the Helmholtz equation (Berdawood et al., 2020(Berdawood et al., , 2021Berdawood-Nachaoui et al., 2021). This relaxation JN method has been applied the to simulate the two-dimensional non linear elliptic problem (Essaouini et al., 2004a,b) and recently combined the domain decomposition method and the relaxation method to solve a Cauchy problems on inhomogenious material. ...
... The relaxation JN method introduced in the last papers drastically reduced the number of iterations required to achieve convergence. It was used in elasticity (Ellabib & Nachaoui, 2008;Marin & Johansson, 2010), and recently for Cauchy problem governed by Stocks equation (Chakib et al., 2018) and for the Helmholtz equation (Berdawood et al., 2020(Berdawood et al., , 2021Berdawood-Nachaoui et al., 2021). This relaxation JN method has been applied the to simulate the two-dimensional non linear elliptic problem (Essaouini et al., 2004a,b) and recently combined the domain decomposition method and the relaxation method to solve a Cauchy problems on inhomogenious material. ...
This paper discusses the recovering of both Dirichlet and Neumann data on some part of the domain
boundary, starting from the knowledge of these data on another part of the boundary for a family of quasi-linear
inverse problems. The nonlinear problem is reduced to a linear Cauchy problem for the Laplace equation coupled
with a sequence of nonlinear scalar equations. We solve the linear problem using a closed-form regularization
analytical solution. Various numerical examples and effects of added small perturbations into the input data are
investigated. The numerical results show that the method produces a stable reasonably approximate solution.
In this study, a heat conduction in fin-related inverse problem for the modified Helmholtz equation is taken into consideration. The purpose of this study is to estimate the temperature on an under-specified boundary (a part of the outer border of a given domain) by using the Cauchy data, on a portion of the boundary that is accessible (boundary temperature and heat flux). The suggested meshless approach is used to numerically solve this problem. By injecting a noise to the Cauchy data, the stability is verified.
In this paper, we present a numerical based on Haar wavelets to solve an inverse Cauchy problem governed by the Helmholtz equation. The problem involves reconstructing the boundary condition on an inaccessible boundary from the given Cauchy data on another part of the boundary. We discuss the formulation of the problem and the use of Haar wavelets. The proposed method involves approximating the solution using a finite sum of Haar wavelets and solving the resulting linear system of equations using a least-squares method. The effectiveness of the proposed method is demonstrated through numerical experiments. From these results, we demonstrate that the Haar wavelet method can be used to obtain an accurate solution to the problem.KeywordsInverse problemsIll-posed problemsCauchy problemHelmholtz equationMeshless methodHaar wavelet methodMulti scale preconditioning
In this paper, we present a novel method for solving an inverse problem that involves determining an unknown defect D compactly contained in a simply-connected bounded domain , given the Dirichlet temperature data u, the Neumann heat flux data on the boundary , and a Dirichlet boundary condition on the boundary . We assume that the temperature u satisfies the modified Helmholtz equation governing the conduction of heat in a fin. The proposed method involves dividing the problem into two subproblems: first, solving a Cauchy problem governed by the modified Helmholtz equation to determine the temperature u, followed by solving a series of nonlinear scalar equations to determine the coordinates of the points defining the boundary . Our numerical experiments demonstrate the effectiveness and accuracy of the proposed method in solving this challenging inverse problem.KeywordsModified Helmholtz’s equationInverse problemCauchy problemPolynomial expansionRegularisztion
