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

Authors:
Karzan Berdawood at Salahaddin University-Erbil
Karzan Berdawood
Abdeljalil Nachaoui at Nantes Université
Abdeljalil Nachaoui
Mourad Nachaoui at Université Sultan Moulay Slimane
Mourad Nachaoui
Fatima Aboud at University of Diyala
Fatima Aboud
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Abstract and Figures

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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Received: 21 June 2020 Revised: 16 September 2020 Accepted: 30 December 2020 Published on: 29 April 2021
DOI: 10.1002/num.22793
RESEARCH ARTICLE
An effective relaxed alternating procedure for
Cauchy problem connected with Helmholtz Equation
Karzan A. Berdawood1,2 Abdeljalil Nachaoui2Mourad Nachaoui2,3
Fatima Aboud4
1Department of Mathematics, Salahaddin
University, Erbil, Iraq
2Laboratoire de Mathématiques Jean Leray
UMR6629 CNRS, Université de Nantes 2 rue de
la Houssinière, Nantes, France
3Laboratoire de Mathématiques et Applications,
Université Sultan Moulay slimane, Faculté des
Sciences et Techniques, Béni-Mellal, Morocco
4University of Diyala, Diyala, Iraq
Correspondence
M. Nachaoui, Laboratoire de Mathématiques Jean
Leray UMR6629 CNRS / Université de Nantes 2
rue de la Houssinière, BP92208 44322 Nantes,
France.
Email: mourad.nachaoui@univ-nantes.fr
Abstract
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 imple-
ment, 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 sta-
bility, consistency and convergence of this algorithm. This
confirms the efficiency of the proposed method.
KEYWORDS
data completion, Helmholtz equation, inverse Cauchy
problem,numerical simulation, relaxedalternating iterative
method
1INTRODUCTION
The Helmholtz equation is an elliptic Partial Differential Equation (PDE), which represents
time-independent solutions of the wave equation. It is often encountered in many branches of science
and engineering. This equation is used to model a wide variety of physical phenomena. These include
among others, wave propagation, vibration phenomena, aeroacoustics, under-water acoustics, seis-
mic inversion, electromagnetic, as well as heat conduction problems. Efficient and accurate numerical
approximation to the Helmholtz equation is significant to scientific computation. This explains the
1888 ©2021 Wiley Periodicals LLC wileyonlinelibrary.com/journal/num Numer Methods Partial Differential Eq. 2023;39:1888–1914.
... 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 . ...
On the resolution of the non-smooth inverse Cauchy problem by the primal-dual method
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... 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 ...
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... 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). □. ...
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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.
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... 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. ...
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... 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. ...
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... 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]: ...
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