Proving the stability estimates of variational least-squares Kernel-Based methods
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866909428987461632 |
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| author | Chen, Meng Ling, Leevan Yun, Dongfang |
| author_facet | Chen, Meng Ling, Leevan Yun, Dongfang |
| contents | Motivated by the need for the rigorous analysis of the numerical stability of variational least-squares kernel-based methods for solving second-order elliptic partial differential equations, we provide previously lacking stability inequalities. This fills a significant theoretical gap in the previous work [Comput. Math. Appl. 103 (2021) 1-11], which provided error estimates based on a conjecture on the stability. With the stability estimate now rigorously proven, we complete the theoretical foundations and compare the convergence behavior to the proven rates. Furthermore, we establish another stability inequality involving weighted-discrete norms, and provide a theoretical proof demonstrating that the exact quadrature weights are not necessary for the weighted least-squares kernel-based collocation method to converge. Our novel theoretical insights are validated by numerical examples, which showcase the relative efficiency and accuracy of these methods on data sets with large mesh ratios. The results confirm our theoretical predictions regarding the performance of variational least-squares kernel-based method, least-squares kernel-based collocation method, and our new weighted least-squares kernel-based collocation method. Most importantly, our results demonstrate that all methods converge at the same rate, validating the convergence theory of weighted least-squares in our proven theories. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_07080 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Proving the stability estimates of variational least-squares Kernel-Based methods Chen, Meng Ling, Leevan Yun, Dongfang Numerical Analysis 65N12, 65D15, 65N15 Motivated by the need for the rigorous analysis of the numerical stability of variational least-squares kernel-based methods for solving second-order elliptic partial differential equations, we provide previously lacking stability inequalities. This fills a significant theoretical gap in the previous work [Comput. Math. Appl. 103 (2021) 1-11], which provided error estimates based on a conjecture on the stability. With the stability estimate now rigorously proven, we complete the theoretical foundations and compare the convergence behavior to the proven rates. Furthermore, we establish another stability inequality involving weighted-discrete norms, and provide a theoretical proof demonstrating that the exact quadrature weights are not necessary for the weighted least-squares kernel-based collocation method to converge. Our novel theoretical insights are validated by numerical examples, which showcase the relative efficiency and accuracy of these methods on data sets with large mesh ratios. The results confirm our theoretical predictions regarding the performance of variational least-squares kernel-based method, least-squares kernel-based collocation method, and our new weighted least-squares kernel-based collocation method. Most importantly, our results demonstrate that all methods converge at the same rate, validating the convergence theory of weighted least-squares in our proven theories. |
| title | Proving the stability estimates of variational least-squares Kernel-Based methods |
| topic | Numerical Analysis 65N12, 65D15, 65N15 |
| url | https://arxiv.org/abs/2312.07080 |