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## 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

Received  August 2020 Revised  January 2021 Published  January 2021

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

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.

Citation: Karzan Berdawood, Abdeljalil Nachaoui, Rostam Saeed, Mourad Nachaoui, Fatima Aboud. An efficient D-N alternating algorithm for solving an inverse problem for Helmholtz equation. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021013
##### References:
 [1] L. Afraites, A. Hadri, A. Laghrib and M. Nachaoui, A high order PDE-constrained optimization for the image denoising problem, Inverse Problems in Science and Engineering (GIPE)., (2020). doi: 10.1080/17415977.2020.1867547.  Google Scholar [2] S. Avdonin, V. Kozlov, D. Maxwell and M. Truffer, Iterative methods for solving a nonlinear boundary inverse problem in glaciology, J. Inverse Ill-Posed Probl., 17 (2009), 239-258.  doi: 10.1515/JIIP.2009.018.  Google Scholar [3] K. A. Berdawood, A. Nachaoui, R. Saeed, M. Nachaoui and F. Aboud, An alternating procedure with dynamic relaxation for Cauchy problems governed by the modified Helmholtz equation, Advanced Mathematical Models & Applications, 5 (2020), 131-139.   Google Scholar [4] A. Bergam, A. Chakib, A. Nachaoui and M. Nachaoui, Adaptive mesh techniques based on a posteriori error estimates for an inverse Cauchy problem, Appl. Math. Comput., 346 (2019), 865-878.  doi: 10.1016/j.amc.2018.09.069.  Google Scholar [5] F. Berntsson, V. A. Kozlov, L. Mpinganzima and B. O. Turesson, An alternating iterative procedure for the Cauchy problem for the Helmholtz equation, Inverse Probl. Sci. Eng., 22 (2014), 45-62.  doi: 10.1080/17415977.2013.827181.  Google Scholar [6] A. Chakib, A. Nachaoui, M. Nachaoui and H. Ouaissa, On a fixed point study of an inverse problem governed by Stokes equation, Inverse Problems, 35 (2019), 015008, 30 pp. doi: 10.1088/1361-6420/aaedce.  Google Scholar [7] R. Chapko and B. T. Johansson, An alternating potential-based approach to the Cauchy problem for the Laplace equation in a planar domain with a cut, Comput. Methods Appl. Math., 8 (2008), 315-335.  doi: 10.2478/cmam-2008-0023.  Google Scholar [8] J. T. Chen and F. C. Wong, Dual formulation of multiple reciprocity method for the acoustic mode of a cavity with a thin partition, Journal of Sound and Vibration, 217 (1998), 75-95.  doi: 10.1006/jsvi.1998.1743.  Google Scholar [9] M. Choulli, Une Introduction aux Problèmes Inverses Elliptiques et Paraboliques, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-02460-3.  Google Scholar [10] L. Eldén, F. Berntsson and T. Regińska, Wavelet and Fourier methods for solving the sideways heat equation, SIAM J. Sci. Comput., 21 (2000), 2187-2205.  doi: 10.1137/S1064827597331394.  Google Scholar [11] A. Ellabib and A. Nachaoui, An iterative approach to the solution of an inverse problem in linear elasticity, Math. Comput. Simulation, 77 (2008), 189-201.  doi: 10.1016/j.matcom.2007.08.014.  Google Scholar [12] M. Essaouini, A. Nachaoui and S. El Hajji, Numerical method for solving a class of nonlinear elliptic inverse problems, J. Comput. Appl. Math., 162 (2004), 165-181.  doi: 10.1016/j.cam.2003.08.011.  Google Scholar [13] M. Essaouini, A. Nachaoui and S. El Hajji, Reconstruction of boundary data for a class of nonlinear inverse problems, J. Inverse Ill-Posed Probl., 12 (2004), 369-385.  doi: 10.1515/1569394042248238.  Google Scholar [14] G. J. Fix and S. P. Marin, Variational methods for underwater acoustic problems, J. Comput. Phys., 28 (1978), 253-270.  doi: 10.1016/0021-9991(78)90037-2.  Google Scholar [15] J. Hadamard, Lectures on Cauchy's Problem in Linear Partial Differential Equations, Dover Publications, New York, 1953.  Google Scholar [16] Q. Hua, Y. Gu, W. Qu, W. Chen and C. Zhang, A meshless generalized finite difference method for inverse Cauchy problems associated with three-dimensional inhomogeneous Helmholtz-type equations, Eng. Anal. Bound. Elem., 82 (2017), 162-171.  doi: 10.1016/j.enganabound.2017.06.005.  Google Scholar [17] F. Ihlenburg and I. Babuška, Finite element solution of the Helmholtz equation with high wave number Part I: The $h$-version of the FEM, Comput. Math. Appl., 30 (1995), 9-37.  doi: 10.1016/0898-1221(95)00144-N.  Google Scholar [18] F. Ihlenburg and I. Babuška, Finite element solution of the Helmholtz equation with high wave number part II: The $h$-$p$ version of the FEM, SIAM J. Numer. Anal., 34 (1997), 315-358.  doi: 10.1137/S0036142994272337.  Google Scholar [19] B. T. Johansson and V. A. Kozlov, An alternating method for Cauchy problems for Helmholtz-type operators in non-homogeneous medium, IMA J. Appl. Math., 74 (2009), 62-73.  doi: 10.1093/imamat/hxn013.  Google Scholar [20] B. T. Johansson and L. Marin, Relaxation of alternating iterative algorithms for the Cauchy problem associated with the modified Helmholtz equation, CMC Comput. Mater. Continua, 13 (2009), 153-189.   Google Scholar [21] M. Jourhmane and A. Nachaoui, A relaxation algorithm for solving a Cauchy problem, Inverse Problems in Engineering, Engineering, 1 (1996), 151-158.   Google Scholar [22] M. Jourhmane and A. Nachaoui, An alternating method for an inverse Cauchy problem, Numer. Algorithms, 21 (1999), 247-260.  doi: 10.1023/A:1019134102565.  Google Scholar [23] M. Jourhmane and A. Nachaoui, Convergence of an alternating method to solve the Cauchy problem for Poisson's equation, Appl. Anal., 81 (2002), 1065-1083.  doi: 10.1080/0003681021000029819.  Google Scholar [24] D. A. Juraev, On a regularized solution of the Cauchy problem for matrix factorizations of the Helmholtz equation, Advanced Mathematical Models & Applications, 4 (2019), 86-96.   Google Scholar [25] V. A. Kozlov, V. G. Maz'ya and A. V. Fomin, An iterative method for solving the Cauchy problem for elliptic equations, Zh. Vychisl. Mat. i Mat. Fiz., 31 (1991), 64-74.   Google Scholar [26] Z. P. Li, C. Xu, M. Lan and Z. Qian, A mollification method for a Cauchy problem for the Helmholtz equation, Int. J. Comput. Math., 95 (2018), 2256-2268.  doi: 10.1080/00207160.2017.1380193.  Google Scholar [27] S. Lyaqini, M. Quafafou and M. Nachaoui et al., Supervised learning as an inverse problem based on non-smooth loss function, Knowl. Inf. Syst., 62 (2020), 3039-3058.  doi: 10.1007/s10115-020-01439-2.  Google Scholar [28] L. Marin, A relaxation method of an alternating iterative (MFS) algorithm for the Cauchy problem associated with the two-dimensional modified Helmholtz equation, Numer. Methods Partial Differential Equations, 28 (2012), 899-925.  doi: 10.1002/num.20664.  Google Scholar [29] L. Marin, L. Elliott, P. J. Heggs, D. B. Ingham, D. Lesnic and X. Wen, An alternating iterative algorithm for the Cauchy problem associated to the Helmholtz equation, Comput. Methods Appl. Mech. Engrg., 192 (2003), 709-722.  doi: 10.1016/S0045-7825(02)00592-3.  Google Scholar [30] L. Marin and B. T. Johansson, A relaxation method of an alternating iterative algorithm for the Cauchy problem in linear isotropic elasticity, Comput. Methods Appl. Mech. Engrg., 199 (2010), 3179-3196.  doi: 10.1016/j.cma.2010.06.024.  Google Scholar [31] M. Nachaoui, Parameter learning for combined first and second order total variation for image reconstruction, Advanced Mathematical Models & Applications, 5 (2020), 53-69.   Google Scholar [32] M. Nachaoui, Étude Théorique et Approximation Numérique d'un Problème Inverse de Transfert de la Chaleur, Doctoral Dissertation. (tel-00678032) Nantes unversity, 2011. Google Scholar [33] M. Nachaoui, A. Chakib and A. Nachaoui, An efficient evolutionary algorithm for a shape optimization problem, Applied and Computational Mathematics, 19 (2020), 220-244.   Google Scholar [34] A. Nachaoui and M. Nachaoui, Iterative methods for Forward and Inverse Bioelelectric Field Problem, International Conference on Applied Mathematics, Modeling and Life Sciences, Icamls'18, Marmara University, Istanbul, Turkey. (hal-02599556) Oct 2018. Google Scholar [35] A. Nachaoui, M. Nachaoui, A. Chakib and M. A. Hilal, Some novel numerical techniques for an inverse Cauchy problem, J. Comput. Appl. Math., 381, (2021), 113030, 21 pp. doi: 10.1016/j.cam.2020.113030.  Google Scholar [36] A. Nachaoui, M. Nachaoui and T. Tadumadze, Electrical Potentials Measured on the Surface of the Knee for Detecting Osteoarthritis-Induced Cartilage Degeneration, Second International Conference of Mathematics in Erbil (SICME2019), 2019. Google Scholar [37] C. R. Vogel, Computational Methods for Inverse Problems, Society for Industrial and Applied Mathematics, 2002. doi: 10.1137/1.9780898717570.  Google Scholar [38] C. Yu, Z. Zhou and M. Zhuang, An acoustic intensity-based method for reconstruction of radiated fields, The Journal of the Acoustical Society of America, 123 (2008), 1892-1901.  doi: 10.1121/1.2875046.  Google Scholar

show all references

##### References:
 [1] L. Afraites, A. Hadri, A. Laghrib and M. Nachaoui, A high order PDE-constrained optimization for the image denoising problem, Inverse Problems in Science and Engineering (GIPE)., (2020). doi: 10.1080/17415977.2020.1867547.  Google Scholar [2] S. Avdonin, V. Kozlov, D. Maxwell and M. Truffer, Iterative methods for solving a nonlinear boundary inverse problem in glaciology, J. Inverse Ill-Posed Probl., 17 (2009), 239-258.  doi: 10.1515/JIIP.2009.018.  Google Scholar [3] K. A. Berdawood, A. Nachaoui, R. Saeed, M. Nachaoui and F. Aboud, An alternating procedure with dynamic relaxation for Cauchy problems governed by the modified Helmholtz equation, Advanced Mathematical Models & Applications, 5 (2020), 131-139.   Google Scholar [4] A. Bergam, A. Chakib, A. Nachaoui and M. Nachaoui, Adaptive mesh techniques based on a posteriori error estimates for an inverse Cauchy problem, Appl. Math. Comput., 346 (2019), 865-878.  doi: 10.1016/j.amc.2018.09.069.  Google Scholar [5] F. Berntsson, V. A. Kozlov, L. Mpinganzima and B. O. Turesson, An alternating iterative procedure for the Cauchy problem for the Helmholtz equation, Inverse Probl. Sci. Eng., 22 (2014), 45-62.  doi: 10.1080/17415977.2013.827181.  Google Scholar [6] A. Chakib, A. Nachaoui, M. Nachaoui and H. Ouaissa, On a fixed point study of an inverse problem governed by Stokes equation, Inverse Problems, 35 (2019), 015008, 30 pp. doi: 10.1088/1361-6420/aaedce.  Google Scholar [7] R. Chapko and B. T. Johansson, An alternating potential-based approach to the Cauchy problem for the Laplace equation in a planar domain with a cut, Comput. Methods Appl. Math., 8 (2008), 315-335.  doi: 10.2478/cmam-2008-0023.  Google Scholar [8] J. T. Chen and F. C. Wong, Dual formulation of multiple reciprocity method for the acoustic mode of a cavity with a thin partition, Journal of Sound and Vibration, 217 (1998), 75-95.  doi: 10.1006/jsvi.1998.1743.  Google Scholar [9] M. Choulli, Une Introduction aux Problèmes Inverses Elliptiques et Paraboliques, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-02460-3.  Google Scholar [10] L. Eldén, F. Berntsson and T. Regińska, Wavelet and Fourier methods for solving the sideways heat equation, SIAM J. Sci. Comput., 21 (2000), 2187-2205.  doi: 10.1137/S1064827597331394.  Google Scholar [11] A. Ellabib and A. Nachaoui, An iterative approach to the solution of an inverse problem in linear elasticity, Math. Comput. Simulation, 77 (2008), 189-201.  doi: 10.1016/j.matcom.2007.08.014.  Google Scholar [12] M. Essaouini, A. Nachaoui and S. El Hajji, Numerical method for solving a class of nonlinear elliptic inverse problems, J. Comput. Appl. Math., 162 (2004), 165-181.  doi: 10.1016/j.cam.2003.08.011.  Google Scholar [13] M. Essaouini, A. Nachaoui and S. El Hajji, Reconstruction of boundary data for a class of nonlinear inverse problems, J. Inverse Ill-Posed Probl., 12 (2004), 369-385.  doi: 10.1515/1569394042248238.  Google Scholar [14] G. J. Fix and S. P. Marin, Variational methods for underwater acoustic problems, J. Comput. Phys., 28 (1978), 253-270.  doi: 10.1016/0021-9991(78)90037-2.  Google Scholar [15] J. Hadamard, Lectures on Cauchy's Problem in Linear Partial Differential Equations, Dover Publications, New York, 1953.  Google Scholar [16] Q. Hua, Y. Gu, W. Qu, W. Chen and C. Zhang, A meshless generalized finite difference method for inverse Cauchy problems associated with three-dimensional inhomogeneous Helmholtz-type equations, Eng. Anal. Bound. Elem., 82 (2017), 162-171.  doi: 10.1016/j.enganabound.2017.06.005.  Google Scholar [17] F. Ihlenburg and I. Babuška, Finite element solution of the Helmholtz equation with high wave number Part I: The $h$-version of the FEM, Comput. Math. Appl., 30 (1995), 9-37.  doi: 10.1016/0898-1221(95)00144-N.  Google Scholar [18] F. Ihlenburg and I. Babuška, Finite element solution of the Helmholtz equation with high wave number part II: The $h$-$p$ version of the FEM, SIAM J. Numer. Anal., 34 (1997), 315-358.  doi: 10.1137/S0036142994272337.  Google Scholar [19] B. T. Johansson and V. A. Kozlov, An alternating method for Cauchy problems for Helmholtz-type operators in non-homogeneous medium, IMA J. Appl. Math., 74 (2009), 62-73.  doi: 10.1093/imamat/hxn013.  Google Scholar [20] B. T. Johansson and L. Marin, Relaxation of alternating iterative algorithms for the Cauchy problem associated with the modified Helmholtz equation, CMC Comput. Mater. Continua, 13 (2009), 153-189.   Google Scholar [21] M. Jourhmane and A. Nachaoui, A relaxation algorithm for solving a Cauchy problem, Inverse Problems in Engineering, Engineering, 1 (1996), 151-158.   Google Scholar [22] M. Jourhmane and A. Nachaoui, An alternating method for an inverse Cauchy problem, Numer. Algorithms, 21 (1999), 247-260.  doi: 10.1023/A:1019134102565.  Google Scholar [23] M. Jourhmane and A. Nachaoui, Convergence of an alternating method to solve the Cauchy problem for Poisson's equation, Appl. Anal., 81 (2002), 1065-1083.  doi: 10.1080/0003681021000029819.  Google Scholar [24] D. A. Juraev, On a regularized solution of the Cauchy problem for matrix factorizations of the Helmholtz equation, Advanced Mathematical Models & Applications, 4 (2019), 86-96.   Google Scholar [25] V. A. Kozlov, V. G. Maz'ya and A. V. Fomin, An iterative method for solving the Cauchy problem for elliptic equations, Zh. Vychisl. Mat. i Mat. Fiz., 31 (1991), 64-74.   Google Scholar [26] Z. P. Li, C. Xu, M. Lan and Z. Qian, A mollification method for a Cauchy problem for the Helmholtz equation, Int. J. Comput. Math., 95 (2018), 2256-2268.  doi: 10.1080/00207160.2017.1380193.  Google Scholar [27] S. Lyaqini, M. Quafafou and M. Nachaoui et al., Supervised learning as an inverse problem based on non-smooth loss function, Knowl. Inf. Syst., 62 (2020), 3039-3058.  doi: 10.1007/s10115-020-01439-2.  Google Scholar [28] L. Marin, A relaxation method of an alternating iterative (MFS) algorithm for the Cauchy problem associated with the two-dimensional modified Helmholtz equation, Numer. Methods Partial Differential Equations, 28 (2012), 899-925.  doi: 10.1002/num.20664.  Google Scholar [29] L. Marin, L. Elliott, P. J. Heggs, D. B. Ingham, D. Lesnic and X. Wen, An alternating iterative algorithm for the Cauchy problem associated to the Helmholtz equation, Comput. Methods Appl. Mech. Engrg., 192 (2003), 709-722.  doi: 10.1016/S0045-7825(02)00592-3.  Google Scholar [30] L. Marin and B. T. Johansson, A relaxation method of an alternating iterative algorithm for the Cauchy problem in linear isotropic elasticity, Comput. Methods Appl. Mech. Engrg., 199 (2010), 3179-3196.  doi: 10.1016/j.cma.2010.06.024.  Google Scholar [31] M. Nachaoui, Parameter learning for combined first and second order total variation for image reconstruction, Advanced Mathematical Models & Applications, 5 (2020), 53-69.   Google Scholar [32] M. Nachaoui, Étude Théorique et Approximation Numérique d'un Problème Inverse de Transfert de la Chaleur, Doctoral Dissertation. (tel-00678032) Nantes unversity, 2011. Google Scholar [33] M. Nachaoui, A. Chakib and A. Nachaoui, An efficient evolutionary algorithm for a shape optimization problem, Applied and Computational Mathematics, 19 (2020), 220-244.   Google Scholar [34] A. Nachaoui and M. Nachaoui, Iterative methods for Forward and Inverse Bioelelectric Field Problem, International Conference on Applied Mathematics, Modeling and Life Sciences, Icamls'18, Marmara University, Istanbul, Turkey. (hal-02599556) Oct 2018. Google Scholar [35] A. Nachaoui, M. Nachaoui, A. Chakib and M. A. Hilal, Some novel numerical techniques for an inverse Cauchy problem, J. Comput. Appl. Math., 381, (2021), 113030, 21 pp. doi: 10.1016/j.cam.2020.113030.  Google Scholar [36] A. Nachaoui, M. Nachaoui and T. Tadumadze, Electrical Potentials Measured on the Surface of the Knee for Detecting Osteoarthritis-Induced Cartilage Degeneration, Second International Conference of Mathematics in Erbil (SICME2019), 2019. Google Scholar [37] C. R. Vogel, Computational Methods for Inverse Problems, Society for Industrial and Applied Mathematics, 2002. doi: 10.1137/1.9780898717570.  Google Scholar [38] C. Yu, Z. Zhou and M. Zhuang, An acoustic intensity-based method for reconstruction of radiated fields, The Journal of the Acoustical Society of America, 123 (2008), 1892-1901.  doi: 10.1121/1.2875046.  Google Scholar
Results obtained by Algorithm 1 and Algorithm 2, at $y = b$ for $k = \sqrt{ 15}$
Algorithm 2: Variation of iterations number at the convergence for $k = \sqrt{15}$
Comparison of relative errors in Algorithm 2, for $\theta = 1,$ $\theta = 1.6$ in the case $k = \sqrt{ 15}$
Stopping criteria and relative error (6.3) in Algorithm 1 for $k = \sqrt{ 25.5}$
Exact and reconstructed solutions obtained by Algorithm 2, at $y = b$ for $k = \sqrt{ 25.5}$
Comparison of relative errors in Algorithm 2, for $\theta = 0.1$ and $\theta = 0.98$ in the case $k = \sqrt{ 25.5}$
Exact and reconstructed solutions from Algorithm 2, at $y = b$ (a) $k = \sqrt{35},$ $\theta = 0.5$ (b) $k = \sqrt{ 52},$ $\theta = 0.14$
diverges of the Algorithm 2 for $\theta = 1.62$ in the case $k = \sqrt{ 15}$
Exact and reconstructed noisy solutions obtained by Algorithm 2, at $y = b$ for $k = \sqrt{15}$ and $\theta = 1.6$
Exact and reconstructed noisy solutions obtained by Algorithm 2, at $y = b$ for $k = \sqrt{52}$ and $\theta = 0.14$
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 $(0, 0.1450)$
 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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