An efficient D-N alternating algorithm for solving an inverse problem for Helmholtz equation

Karzan Berdawood; Abdeljalil Nachaoui; Rostam Saeed; Mourad Nachaoui; Fatima Aboud

doi:10.3934/dcdss.2021013

Article Contents

2022, Volume 15, Issue 1: 57-78. Doi: 10.3934/dcdss.2021013

This issue Previous Article Extended Krylov subspace methods for solving Sylvester and Stein tensor equations Next Article Galerkin method of weakly damped cubic nonlinear Schrödinger with Dirac impurity, and artificial boundary condition in a half-line

An efficient D-N alternating algorithm for solving an inverse problem for Helmholtz equation

1.
Department of Mathematics, College of Science, Salahaddin University-Erbil, Iraq
2.
Laboratoire de Mathématiques Jean Leray, UMR6629 CNRS/Université de Nantes, France
3.
Laboratoire de Mathématiques et Applications, Université Sultan Moulay Slimane, Béni-Mellal, Maroc
4.
Department of Mathematics, College of Science, University of Diyala, Iraq

^* Corresponding author: Karzan Berdawood, Department of Mathematics, College of Science, Salahaddin University-Erbil, Iraq
^* Corresponding author: Karzan Berdawood, Department of Mathematics, College of Science, Salahaddin University-Erbil, Iraq

Received: July 31, 2020

Revised: December 31, 2020

Early access: January 2021

Published: January 2022

The first author is supported by Split-Site program between Salahaddin University-Erbil and Nantes University

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Abstract

Data completion known as Cauchy problem is one most investigated inverse problems. In this work we consider a Cauchy problem associated with Helmholtz equation. Our concerned is the convergence of the well-known alternating iterative method [25]. Our main result is to restore the convergence for the classical iterative algorithm (KMF) when the wave numbers are considerable. This is achieved by, some simple modification for the Neumann condition on the under-specified boundary and replacement by relaxed Neumann ones. Moreover, for the small wave number $ k $, when the convergence of KMF algorithm's [25] is ensured, our algorithm can be used as an acceleration of convergence.

In this case, we present theoretical results of the convergence of this relaxed algorithm. Meanwhile it, we can deduce the convergence intervals related to the relaxation parameters in different situations. In contrast to the existing results, the proposed algorithm is simple to implement converges for all choice of wave number.

We approach our algorithm using finite element method to obtain an accurate numerical results, to affirm theoretical results and to prove it's effectiveness.

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Mathematics Subject Classification: Primary: 35J25, 35R30; Secondary: 65N21.

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Figure 1. Results obtained by Algorithm 1 and Algorithm 2, at $ y = b $ for $ k = \sqrt{ 15} $

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Figure 2. Algorithm 2: Variation of iterations number at the convergence for $ k = \sqrt{15} $

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Figure 3. Comparison of relative errors in Algorithm 2, for $ \theta = 1, $ $ \theta = 1.6 $ in the case $ k = \sqrt{ 15} $

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Figure 4. Stopping criteria and relative error (6.3) in Algorithm 1 for $ k = \sqrt{ 25.5} $

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Figure 5. Exact and reconstructed solutions obtained by Algorithm 2, at $ y = b $ for $ k = \sqrt{ 25.5} $

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Figure 6. Comparison of relative errors in Algorithm 2, for $ \theta = 0.1 $ and $ \theta = 0.98 $ in the case $ k = \sqrt{ 25.5} $

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Figure 7. Exact and reconstructed solutions from Algorithm 2, at $ y = b $ (a) $ k = \sqrt{35}, $ $ \theta = 0.5 $ (b) $ k = \sqrt{ 52}, $ $ \theta = 0.14 $

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Figure 8. diverges of the Algorithm 2 for $ \theta = 1.62 $ in the case $ k = \sqrt{ 15} $

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Figure 9. Exact and reconstructed noisy solutions obtained by Algorithm 2, at $ y = b $ for $ k = \sqrt{15} $ and $ \theta = 1.6 $

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Figure 10. Exact and reconstructed noisy solutions obtained by Algorithm 2, at $ y = b $ for $ k = \sqrt{52} $ and $ \theta = 0.14 $

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Table 1. Convergence intervals for different values of $ k $

Wave numbers	Algorithm 1	Algorithm 2	Relaxed intervals
$ k=\sqrt{15} $	converges	converges	$ (0, 1.6168) $
$ k=\sqrt{25.5} $	diverges	converges	$ (0, 0.9893) $
$ k=\sqrt{35} $	diverges	converges	$ (0, 0.5791) $
$ k=\sqrt{52} $	diverges	converges	$ (0, 0.1450) $

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