An Adaptive Selection Method for Shape Parameters in MQ-RBF Interpolation for Two-Dimensional Scattered Data and Its Application to Integral Equation Solving
The paper proposes an adaptive selection method for the shape parameter in the multi-quadratic radial basis function (MQ-RBF) interpolation of two-dimensional (2D) scattered data and achieves good performance in solving integral equations (O-MQRBF). The effectiveness of MQ-RBF interpolation for 2D s...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-05-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/7/6/448 |
| _version_ | 1797594729220669440 |
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| author | Jian Sun Ling Wang Dianxuan Gong |
| author_facet | Jian Sun Ling Wang Dianxuan Gong |
| author_sort | Jian Sun |
| collection | DOAJ |
| description | The paper proposes an adaptive selection method for the shape parameter in the multi-quadratic radial basis function (MQ-RBF) interpolation of two-dimensional (2D) scattered data and achieves good performance in solving integral equations (O-MQRBF). The effectiveness of MQ-RBF interpolation for 2D scattered data largely depends on the choice of the shape parameter. However, currently, the most appropriate parameter is chosen by empirical techniques or trial and error, and there is no widely accepted method. Fourier transform can linearly represent 2D scattering data as a combination of sine and cosine functions. Therefore, the paper employs an improved stochastic walk optimization algorithm to determine the optimal shape parameters for sine functions and their linear combinations, generating a dataset. Based on this dataset, the paper trains a particle swarm optimization backpropagation neural network (PSO-BP) to construct an optimal shape parameter selection model. The adaptive model accurately predicts the ideal shape parameters of the Fourier expansion of 2D scattering data, significantly reducing computational cost and improving interpolation accuracy. The adaptive method forms the basis of the O-MQRBF algorithm for solving one-dimensional integral equations. Compared with traditional methods, this algorithm significantly improves the precision of the solution. Overall, this study greatly facilitates the development of MQ-RBF interpolation technology and its widespread use in solving integral equations. |
| first_indexed | 2024-03-11T02:27:33Z |
| format | Article |
| id | doaj.art-56ce4beef102407db474724598168927 |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-11T02:27:33Z |
| publishDate | 2023-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-56ce4beef102407db4747245981689272023-11-18T10:29:26ZengMDPI AGFractal and Fractional2504-31102023-05-017644810.3390/fractalfract7060448An Adaptive Selection Method for Shape Parameters in MQ-RBF Interpolation for Two-Dimensional Scattered Data and Its Application to Integral Equation SolvingJian Sun0Ling Wang1Dianxuan Gong2College of Science, North China University of Science and Technology, Tangshan 063210, ChinaCollege of Science, North China University of Science and Technology, Tangshan 063210, ChinaCollege of Science, North China University of Science and Technology, Tangshan 063210, ChinaThe paper proposes an adaptive selection method for the shape parameter in the multi-quadratic radial basis function (MQ-RBF) interpolation of two-dimensional (2D) scattered data and achieves good performance in solving integral equations (O-MQRBF). The effectiveness of MQ-RBF interpolation for 2D scattered data largely depends on the choice of the shape parameter. However, currently, the most appropriate parameter is chosen by empirical techniques or trial and error, and there is no widely accepted method. Fourier transform can linearly represent 2D scattering data as a combination of sine and cosine functions. Therefore, the paper employs an improved stochastic walk optimization algorithm to determine the optimal shape parameters for sine functions and their linear combinations, generating a dataset. Based on this dataset, the paper trains a particle swarm optimization backpropagation neural network (PSO-BP) to construct an optimal shape parameter selection model. The adaptive model accurately predicts the ideal shape parameters of the Fourier expansion of 2D scattering data, significantly reducing computational cost and improving interpolation accuracy. The adaptive method forms the basis of the O-MQRBF algorithm for solving one-dimensional integral equations. Compared with traditional methods, this algorithm significantly improves the precision of the solution. Overall, this study greatly facilitates the development of MQ-RBF interpolation technology and its widespread use in solving integral equations.https://www.mdpi.com/2504-3110/7/6/448MQ-RBFshape parametersFourier transformPSO-BPadaptive methodintegral equation |
| spellingShingle | Jian Sun Ling Wang Dianxuan Gong An Adaptive Selection Method for Shape Parameters in MQ-RBF Interpolation for Two-Dimensional Scattered Data and Its Application to Integral Equation Solving Fractal and Fractional MQ-RBF shape parameters Fourier transform PSO-BP adaptive method integral equation |
| title | An Adaptive Selection Method for Shape Parameters in MQ-RBF Interpolation for Two-Dimensional Scattered Data and Its Application to Integral Equation Solving |
| title_full | An Adaptive Selection Method for Shape Parameters in MQ-RBF Interpolation for Two-Dimensional Scattered Data and Its Application to Integral Equation Solving |
| title_fullStr | An Adaptive Selection Method for Shape Parameters in MQ-RBF Interpolation for Two-Dimensional Scattered Data and Its Application to Integral Equation Solving |
| title_full_unstemmed | An Adaptive Selection Method for Shape Parameters in MQ-RBF Interpolation for Two-Dimensional Scattered Data and Its Application to Integral Equation Solving |
| title_short | An Adaptive Selection Method for Shape Parameters in MQ-RBF Interpolation for Two-Dimensional Scattered Data and Its Application to Integral Equation Solving |
| title_sort | adaptive selection method for shape parameters in mq rbf interpolation for two dimensional scattered data and its application to integral equation solving |
| topic | MQ-RBF shape parameters Fourier transform PSO-BP adaptive method integral equation |
| url | https://www.mdpi.com/2504-3110/7/6/448 |
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