A Numerical Approximation Method for the Inverse Problem of the Three-Dimensional Laplace Equation
In this article, an inverse problem with regards to the Laplace equation with non-homogeneous Neumann boundary conditions in a three-dimensional case is investigated. To deal with this problem, a regularization method (mollification method) with the bivariate de la Vallée Poussin kernel is...
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| Format: | Article |
| Language: | English |
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MDPI AG
2019-05-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/7/6/487 |
| _version_ | 1818203535955722240 |
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| author | Shangqin He Xiufang Feng |
| author_facet | Shangqin He Xiufang Feng |
| author_sort | Shangqin He |
| collection | DOAJ |
| description | In this article, an inverse problem with regards to the Laplace equation with non-homogeneous Neumann boundary conditions in a three-dimensional case is investigated. To deal with this problem, a regularization method (mollification method) with the bivariate de la Vallée Poussin kernel is proposed. Stable estimates are obtained under a priori bound assumptions and an appropriate choice of the regularization parameter. The error estimates indicate that the solution of the approximation continuously depends on the noisy data. Two experiments are presented, in order to validate the proposed method in terms of accuracy, convergence, stability, and efficiency. |
| first_indexed | 2024-12-12T03:26:54Z |
| format | Article |
| id | doaj.art-cae9ad946de445cda7432fb364bc2c95 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-12-12T03:26:54Z |
| publishDate | 2019-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-cae9ad946de445cda7432fb364bc2c952022-12-22T00:40:01ZengMDPI AGMathematics2227-73902019-05-017648710.3390/math7060487math7060487A Numerical Approximation Method for the Inverse Problem of the Three-Dimensional Laplace EquationShangqin He0Xiufang Feng1School of Mathematics and Statistics, NingXia University, Yinchuan 750021, ChinaSchool of Mathematics and Statistics, NingXia University, Yinchuan 750021, ChinaIn this article, an inverse problem with regards to the Laplace equation with non-homogeneous Neumann boundary conditions in a three-dimensional case is investigated. To deal with this problem, a regularization method (mollification method) with the bivariate de la Vallée Poussin kernel is proposed. Stable estimates are obtained under a priori bound assumptions and an appropriate choice of the regularization parameter. The error estimates indicate that the solution of the approximation continuously depends on the noisy data. Two experiments are presented, in order to validate the proposed method in terms of accuracy, convergence, stability, and efficiency.https://www.mdpi.com/2227-7390/7/6/487three-dimensional Laplace equationill-posedde la Vallée Poussin kernelmollification methodregular parametererror estimate |
| spellingShingle | Shangqin He Xiufang Feng A Numerical Approximation Method for the Inverse Problem of the Three-Dimensional Laplace Equation Mathematics three-dimensional Laplace equation ill-posed de la Vallée Poussin kernel mollification method regular parameter error estimate |
| title | A Numerical Approximation Method for the Inverse Problem of the Three-Dimensional Laplace Equation |
| title_full | A Numerical Approximation Method for the Inverse Problem of the Three-Dimensional Laplace Equation |
| title_fullStr | A Numerical Approximation Method for the Inverse Problem of the Three-Dimensional Laplace Equation |
| title_full_unstemmed | A Numerical Approximation Method for the Inverse Problem of the Three-Dimensional Laplace Equation |
| title_short | A Numerical Approximation Method for the Inverse Problem of the Three-Dimensional Laplace Equation |
| title_sort | numerical approximation method for the inverse problem of the three dimensional laplace equation |
| topic | three-dimensional Laplace equation ill-posed de la Vallée Poussin kernel mollification method regular parameter error estimate |
| url | https://www.mdpi.com/2227-7390/7/6/487 |
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