An Aitken-like acceleration method applied to missing boundary data reconstruction for the Cauchy–Helmholtz problem

Amel Ben Abda; Riadh Ben Fatma; Damien Tromeur-Dervout

doi:10.1016/j.crma.2009.11.010

Numerical Analysis

An Aitken-like acceleration method applied to missing boundary data reconstruction for the Cauchy–Helmholtz problem
[Une méthode d'accélération de type Aitken appliquée à la reconstruction de données frontières manquantes sur le problème de Cauchy–Helmholtz]

Présenté par : Olivier Pironneau

Amel Ben Abda ¹ ; Riadh Ben Fatma ¹ ; Damien Tromeur-Dervout ²

¹ LAMSIN, École nationale d'ingénieurs de Tunis, université Tunis-El Manar, B.P. 37, 1002 Le Belvédère, Tunisia
² Université de Lyon, université Lyon 1, CNRS, institut Camille-Jordan, 43, boulevard du 11 Novembre 1918, 69622 Villeurbanne, France

Comptes Rendus. Mathématique, Volume 348 (2010) no. 1-2, pp. 93-97.

Résumé
Abstract

Cette Note concerne le problème mal-posé de Cauchy–Helmholtz. Ce problème est interprété en terme d'équation d'interface qu'on résout via une méthode d'Aitken–Schwarz. Des essais numériques illustrent l'efficacité de cette méthode.

This Note is concerned with the severely ill-posed Cauchy–Helmholtz problem. This Cauchy problem being rephrased through an “interfacial” equation, we resort to an Aitken–Schwarz method for solving this equation. Numerical trials highlight the efficiency of the present method.

Reçu le : 2008-12-05
Accepté le : 2009-11-19
Publié le : 2009-12-24

DOI : 10.1016/j.crma.2009.11.010

Affiliations des auteurs :

Amel Ben Abda ¹ ; Riadh Ben Fatma ¹ ; Damien Tromeur-Dervout ²

¹ LAMSIN, École nationale d'ingénieurs de Tunis, université Tunis-El Manar, B.P. 37, 1002 Le Belvédère, Tunisia
² Université de Lyon, université Lyon 1, CNRS, institut Camille-Jordan, 43, boulevard du 11 Novembre 1918, 69622 Villeurbanne, France

@article{CRMATH_2010__348_1-2_93_0,
     author = {Amel Ben Abda and Riadh Ben Fatma and Damien Tromeur-Dervout},
     title = {An {Aitken-like} acceleration method applied to missing boundary data reconstruction for the {Cauchy{\textendash}Helmholtz} problem},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {93--97},
     publisher = {Elsevier},
     volume = {348},
     number = {1-2},
     year = {2010},
     doi = {10.1016/j.crma.2009.11.010},
     language = {en},
}

TY  - JOUR
AU  - Amel Ben Abda
AU  - Riadh Ben Fatma
AU  - Damien Tromeur-Dervout
TI  - An Aitken-like acceleration method applied to missing boundary data reconstruction for the Cauchy–Helmholtz problem
JO  - Comptes Rendus. Mathématique
PY  - 2010
SP  - 93
EP  - 97
VL  - 348
IS  - 1-2
PB  - Elsevier
DO  - 10.1016/j.crma.2009.11.010
LA  - en
ID  - CRMATH_2010__348_1-2_93_0
ER  -

%0 Journal Article
%A Amel Ben Abda
%A Riadh Ben Fatma
%A Damien Tromeur-Dervout
%T An Aitken-like acceleration method applied to missing boundary data reconstruction for the Cauchy–Helmholtz problem
%J Comptes Rendus. Mathématique
%D 2010
%P 93-97
%V 348
%N 1-2
%I Elsevier
%R 10.1016/j.crma.2009.11.010
%G en
%F CRMATH_2010__348_1-2_93_0

Amel Ben Abda; Riadh Ben Fatma; Damien Tromeur-Dervout. An Aitken-like acceleration method applied to missing boundary data reconstruction for the Cauchy–Helmholtz problem. Comptes Rendus. Mathématique, Volume 348 (2010) no. 1-2, pp. 93-97. doi : 10.1016/j.crma.2009.11.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.11.010/

Bibliographie
Cité par

[1] R. Ben Fatma; M. Azaïez; A. Ben Abda; N. Gmati Missing boundary data recovering for the Helmholtz problem, C. R. Mecanique, Volume 335 (2007), pp. 787-792

[2] M.O. Bristeau; J. Erhel; P. Feat; R. Glowinski; J. Périaux Solving the Helmholtz equation at high-wave numbers on a parallel computer with a shared virtual memory, Internat. J. Supercomput. Appl., Volume 9 (1995) no. 1, pp. 18-28

[3] H.W. Engl; M. Hanke; A. Neubauer Regularisation of Inverse Problems, Kluwer Academic, Dordrecht, The Netherlands, 1996

[4] A. Frullone, D. Tromeur-Dervout, Adaptive acceleration of the Aitken–Schwarz Domain Decomposition on nonuniform nonmatching grids, preprint CDCSP07-00, submitted for publication

[5] M. Garbey Acceleration of the Schwarz method for elliptic problems, SIAM J. Sci. Comput., Volume 26 (2005), pp. 1871-1893

[6] M. Garbey; D. Tromeur-Dervout Two level domain decomposition for multi-cluster (H.K.T. Chan; T. Kako; O. Pironneau, eds.), Proc. Int. Conf. on Domain Decomposition Methods DD12, DDM org, 2001, pp. 325-340

[7] M. Garbey; D. Tromeur-Dervout On some Aitken-like acceleration of the Schwarz method, Internat. J. Numer. Methods Fluids, Volume 40 (2002) no. 12, pp. 1493-1513

[8] B.T. Johansson; V.A. Kozlov An alternating method for Cauchy problems for Helmholtz-type operators in non-homogeneous medium, IMA J. Appl. Math., Volume 74 (2009) no. 1, pp. 62-73

[9] V.A. Koslov; V.G. Maz'ya; A.V. Fomin An iterative method for solving the Cauchy problem for elliptic equations, Comput. Meth. Math. Phys., Volume 31 (1991), pp. 45-52

[10] D. Martin Documentation MELINA, ENSTA, May 2000 http://perso.univ-rennes1.fr/daniel.martin/melina/

[11] F. Nataf, Quasi Optimal Interface Conditions in Domain Decomposition Methods. Application to Problems with Extreme Contrasts in the Coefficients, R.I. 514, CMAP, Ecole Polytechnique, 2003

[12] Kim-Dang Phung Remarques sur l'observabilité pour de l'équation de Laplace, ESAIM Control Optim. Calc. Var., Volume 9 (2003), pp. 621-635 (Remarks on observability for the Laplace equation, in French; English, French summaries)

[13] T. Reginska; K. Reginski Approximate solution of a Cauchy problem for the Helmholtz equation, Inverse Problems, Volume 22 (2006), pp. 975-989

[14] D. Tromeur-Dervout Aitken–Schwarz method: acceleration of the convergence of the Schwarz method, Domain Decomposition Methods: Theory and Applications, GAKUTO Internat. Ser. Math. Sci. Appl., vol. 25, Gakkotosho, Tokyo, 2006, pp. 37-64

[15] S.R. Wu; J. Yu Application of BEM-based acoustic holography to radiation analysis of sound sources with arbitrarily shaped geometries, J. Acoust. Soc. Am., Volume 104 (1998), pp. 2054-2060

Cité par Sources :

Commentaires - Politique