Um framework para solução do problema inverso do espalhamento baseado na Formulação Variacional Ultra Fraca

Fernanda Lúcia Sá Ferreira; Julius Monteiro de Barros Filho; Amaury Alvarez Cruz; Daniel Gregorio Alfaro Vigo

doi:10.35819/remat2024v10iespecialid7053

Authors

Fernanda Lúcia Sá Ferreira Centro Federal de Educação Tecnológica Celso Suckow da Fonseca (CEFET/RJ), Nova Iguaçu, RJ; Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, RJ, Brasil https://orcid.org/0000-0002-0921-2494
Julius Monteiro de Barros Filho Centro Federal de Educação Tecnológica Celso Suckow da Fonseca (CEFET/RJ), Nova Iguaçu, RJ; Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, RJ, Brasil https://orcid.org/0000-0003-2688-920X
Amaury Alvarez Cruz Universidade Federal do Rio de Janeiro (UFRJ), Programa de Pós-Graduação em Informática (PPGI), Rio de Janeiro, RJ, Brasil https://orcid.org/0000-0002-5513-7974
Daniel Gregorio Alfaro Vigo Universidade Federal do Rio de Janeiro (UFRJ), Programa de Pós-Graduação em Informática (PPGI), Rio de Janeiro, RJ, Brasil https://orcid.org/0000-0002-3280-8720

DOI:

https://doi.org/10.35819/remat2024v10iespecialid7053

Keywords:

inverse problems, mild-slope equation, ultra weak variational formulation

Abstract

This study focuses on two-dimensional inverse scattering problems, where information about an inaccessible region is inferred from measurements taken in accessible areas. The main focus is on the iterative resolution of the refraction-diffraction coefficient in the Helmholtz equation, using a regularized minimum square problem to cope with the ill-posed nature of the problem. To solve the corresponding direct problem, the Ultra Weak Variational Formulation (UWVF) method was applied, a method known for its computational efficiency, requiring fewer resources and allowing for analytical calculations. The methodology developed here was successfully applied to determine the bathymetry in coastal regions from the knowledge of waves in deep waters and along the coastline, yielding reliable and precise results.

Downloads

Download data is not yet available.

Author Biographies

Fernanda Lúcia Sá Ferreira, Centro Federal de Educação Tecnológica Celso Suckow da Fonseca (CEFET/RJ), Nova Iguaçu, RJ; Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, RJ, Brasil

http://lattes.cnpq.br/5576155484322896
Julius Monteiro de Barros Filho, Centro Federal de Educação Tecnológica Celso Suckow da Fonseca (CEFET/RJ), Nova Iguaçu, RJ; Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, RJ, Brasil

http://lattes.cnpq.br/9004019590106442
Amaury Alvarez Cruz, Universidade Federal do Rio de Janeiro (UFRJ), Programa de Pós-Graduação em Informática (PPGI), Rio de Janeiro, RJ, Brasil

http://lattes.cnpq.br/0370476774197561
Daniel Gregorio Alfaro Vigo, Universidade Federal do Rio de Janeiro (UFRJ), Programa de Pós-Graduação em Informática (PPGI), Rio de Janeiro, RJ, Brasil

http://lattes.cnpq.br/8694890749217765

References

ALVAREZ, A. C.; GARCÍA, G. C.; SARKIS, M. The ultra weak variational formulation for the modified mild-slope equation. Applied Mathematical Modelling. v. 52, p. 28-41, 2017. DOI: https://doi.org/10.1016/j.apm.2017.07.018.

BAO, G.; LI, P. Shape reconstruction of inverse medium scattering for the Helmholtz equation. In: WANG, Y.; YAGOLA, A.; YANG C. (ed.). Computational Methods for Applied Inverse Problems. Berlin, Boston: De Gruyter, 2012. cap. 12. DOI: https://doi.org/10.1515/9783110259056.283.

BERKHOFF, J. C. W. Computation of combined refraction - diffraction. Coastal Engineering Proceedings. v. 1, n. 13, p. 23, Jan. 1972. DOI: https://doi.org/10.9753/ICCE.V13.23.

BORGES, C.; GILLMAN, A.; GREENGARD, L. High resolution inverse scattering in two dimensions using recursive linearization. SIAM Journal on Imaging Sciences. v. 10, n. 2, p. 641-664, 2017. DOI: https://doi.org/10.1137/16M1093562.

CESSENAT, O.; DESPRÉS, B. Application of an ultra weak variational formulation of elliptic pdes to the two-dimensional helmholtz problem. SIAM Journal on Numerical Analysis. v. 35, n. 1, p. 255-299, 1998. DOI: https://doi.org/10.1137/S0036142995285873.

CESSENAT, O.; DESPRÉS, B. Using plane waves as base functions for solving time harmonic equations with the ultra weak variational formulation. Journal of Computational Acoustics. v. 11, n. 2, p. 227-238, 2003. DOI: https://doi.org/10.1142/S0218396X03001912.

CHAMBERLAIN, P.; PORTER, D. The modified mild-slope equation. Journal of Fluid Mechanics. v. 291, p. 393-407, 1995. DOI: https://doi.org/10.1017/S0022112095002758.

CHEN, H. S.; HOUSTON, J. R. Calculation of Water Oscillation in Coastal Harbors. HARBS and HARBD User's Manual. Vicksburg, Mississippi: U. S. Army Engineer Research and Development Center (ERDC) (Instruction Report, CERC-87-2), 1987. Disponível em: https://hdl.handle.net/20.500.11970/111457. Acesso em: 25 jun. 2024.

COLLINS, R. Nondestructive testing of materials. Amsterdam: IOS Press, 1995. v. 8.

FASSIEH, K. M. A numerical technique to estimate water depths from remotely sensed water wave characteristics. ISRN Oceanography. v. 2013, 2013.

GILBERT, R. P.; XU, Y. An inverse problem for harmonic acoustics in stratified oceans. Journal of Mathematical Analysis and Applications, v. 176, n. 1, p. 121-137, 1993. DOI: https://doi.org/10.1006/jmaa.1993.1203.

GYLYS-COLWELL, F. An inverse problem for the Helmholtz equation. Inverse Problems. v. 12, n. 2, p. 139-156, 1996. DOI: https://doi.org/10.1088/0266-5611/12/2/003.

HUTTUNEN T.; MONK, P.; KAIPIO, J. p. Computational aspects of the ultra-weak variational formulation. Journal of Computational Physics. v. 182, n. 1, p. 27-46, 2002. DOI: https://doi.org/10.1006/jcph.2002.7148.

LE ROUX, J. P. An extension of the airy theory for linear waves into shallow water. Coastal Engineering. v. 55, n. 4, p. 295-301, 2008. DOI: https://doi.org/10.1016/j.coastaleng.2007.11.003.

PASTORINO, M. Medical and industrial applications of inverse scattering based microwave imaging techniques. 2008 IEEE International Workshop on Imaging Systems and Techniques. Chania, Greece: IEEE, 2008. p. 34-38. DOI: https://doi.org/10.1109/IST.2008.4659936.

RADDER, A. C. On the parabolic equation method for water wave transformation. Journal of Fluid Mechanics. v. 95, n. 1, p. 159-176, 1979. DOI: https://doi.org/10.1017/S0022112079001397.

TARANTOLA, A. Inverse problem theory and methods for model parameter estimation. USA: SIAM, 2005.

TIKHONOV, A. N.; ARSENIN, V. Y. Solutions of ill-posed problems. Washington: V. H. Winston & Sons, 1977.

TSAY, T.-K.; LIU, P. L.-F. A finite element model for wave refraction and diffraction. Applied Ocean Research. v. 5, n. 1, p. 30-37, 1983. DOI: https://doi.org/10.1016/0141-1187(83)90055-X.

VASAN, V.; DECONINCK, B. The inverse water wave problem of bathymetry detection. Journal of Fluid Mechanics. v. 714, p. 562-590, 2013. DOI: https://doi.org/10.1017/jfm.2012.497.

WANG, Y. Regularization for inverse models in remote sensing. Progress in Physical Geography: Earth and Environment. v. 36, n. 1, p. 38-59, 2012. DOI: https://doi.org/10.1177/0309133311420320.