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Volume 30, Issue 2
A Projection Method with Regularization for the Cauchy Problem of the Helmholtz Equation

Yunyun Ma, Fuming Ma & Heping Dong

DOI: 10.4208/jcm.1109-m3491

J. Comp. Math., 30 (2012), pp. 157-176.

Published online: 2012-04

[An open-access article; the PDF is free to any online user.]

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  • Abstract

This paper is concerned with the reconstruction of the radiation wave field in the exterior of a bounded two- or three-dimensional domain from the knowledge of Cauchy data on a part of the boundary of the aforementioned domain. It is described by the Cauchy problem for the Helmholtz equation. By using the Dirichlet-to-Neumann map, this problem is transformed into an operator equation with compact operator. We rigorously justify the asymptotic behaviors of singular values of the compact operator. Then a projection method with regularization is applied to solve the operator equation, and the convergence of the regularization method is discussed. Finally, several numerical examples are presented to illustrate the approach. The results demonstrate that the algorithm is effective.

  • Keywords

Helmholtz equation, Cauchy Problem, Projection method, Regularization.

  • AMS Subject Headings

35R25, 35R30, 78A40.

  • Copyright

COPYRIGHT: © Global Science Press

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  • TXT
@Article{JCM-30-157, author = {Yunyun Ma, Fuming Ma and Heping Dong}, title = {A Projection Method with Regularization for the Cauchy Problem of the Helmholtz Equation}, journal = {Journal of Computational Mathematics}, year = {2012}, volume = {30}, number = {2}, pages = {157--176}, abstract = {

This paper is concerned with the reconstruction of the radiation wave field in the exterior of a bounded two- or three-dimensional domain from the knowledge of Cauchy data on a part of the boundary of the aforementioned domain. It is described by the Cauchy problem for the Helmholtz equation. By using the Dirichlet-to-Neumann map, this problem is transformed into an operator equation with compact operator. We rigorously justify the asymptotic behaviors of singular values of the compact operator. Then a projection method with regularization is applied to solve the operator equation, and the convergence of the regularization method is discussed. Finally, several numerical examples are presented to illustrate the approach. The results demonstrate that the algorithm is effective.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1109-m3491}, url = {http://global-sci.org/intro/article_detail/jcm/8423.html} }
TY - JOUR T1 - A Projection Method with Regularization for the Cauchy Problem of the Helmholtz Equation AU - Yunyun Ma, Fuming Ma & Heping Dong JO - Journal of Computational Mathematics VL - 2 SP - 157 EP - 176 PY - 2012 DA - 2012/04 SN - 30 DO - http://doi.org/10.4208/jcm.1109-m3491 UR - https://global-sci.org/intro/article_detail/jcm/8423.html KW - Helmholtz equation, Cauchy Problem, Projection method, Regularization. AB -

This paper is concerned with the reconstruction of the radiation wave field in the exterior of a bounded two- or three-dimensional domain from the knowledge of Cauchy data on a part of the boundary of the aforementioned domain. It is described by the Cauchy problem for the Helmholtz equation. By using the Dirichlet-to-Neumann map, this problem is transformed into an operator equation with compact operator. We rigorously justify the asymptotic behaviors of singular values of the compact operator. Then a projection method with regularization is applied to solve the operator equation, and the convergence of the regularization method is discussed. Finally, several numerical examples are presented to illustrate the approach. The results demonstrate that the algorithm is effective.

Yunyun Ma, Fuming Ma and Heping Dong. (2012). A Projection Method with Regularization for the Cauchy Problem of the Helmholtz Equation. Journal of Computational Mathematics. 30 (2). 157-176. doi:10.4208/jcm.1109-m3491
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