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Proving the stability estimates of variational least-squares kernel-based methods
Computers & Mathematics with Applications ( IF 2.9 ) Pub Date : 2024-12-18 , DOI: 10.1016/j.camwa.2024.12.008
Meng Chen, Leevan Ling, Dongfang Yun

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.

中文翻译:


证明基于变分最小二乘核的方法的稳定性估计



由于需要对基于变分最小二乘核的方法的数值稳定性进行严格分析,以求解二阶椭圆偏微分方程,我们提供了以前缺乏的稳定性不等式。这填补了之前工作 [Comput. Math. Appl. 103 (2021) 1-11] 中的一个重要理论空白,该工作根据稳定性的猜想提供了误差估计。稳定性估计现在得到了严格的验证,我们完成了理论基础,并将收敛行为与经过验证的速率进行了比较。此外,我们建立了另一个涉及加权离散范数的稳定性不等式,并提供了一个理论证明,证明精确的正交权重对于基于加权最小二乘核的搭配方法收敛不是必需的。我们新颖的理论见解通过数值示例进行了验证,这些示例展示了这些方法在具有大网格比的数据集上的相对效率和准确性。结果证实了我们对基于变分最小二乘核的方法、基于最小二乘核的搭配方法以及我们新的基于加权最小二乘核的搭配方法的性能的理论预测。最重要的是,我们的结果表明所有方法都以相同的速率收敛,验证了我们经过验证的理论中加权最小二乘法的收敛理论。
更新日期:2024-12-18
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