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Proving the stability estimates of variational least-squares Kernel-Based methods

Meng Chen, Leevan Ling, Dongfang Yun

TL;DR

This work delivers a rigorous stability analysis for variational least-squares kernel-based methods solving second-order elliptic PDEs by proving the stability inequality $C^{-1} h^{2q}\|u^h\|^2_{H^{q+2}(\Omega)} \leq \|\mathcal{L}u^h\|^2_{L^2(\Omega)} + h^{-3} \|u^h\|^2_{L^2(\partial\Omega)}$ for $0\le q \le \tau-2$, thereby filling a gap left by prior conjectures. It develops discrete stability via carefully chosen collocation sets $Y$ and $Z$ and then propagates these results to the continuous setting, enabling error and condition-number estimates for VLS; it further extends the theory to Weighted LS (WLS) by establishing norm equivalences between continuous and weighted-discrete norms and showing convergence without exact quadrature weights. The paper also proves a uniform $L^2$-norm bound in the trial space through discrete sampling and demonstrates the convergent behavior of WLS kernels, supported by comprehensive numerical experiments. Collectively, these results validate that variational and kernel-based collocation methods converge at the same rates, even with large mesh ratios, and illustrate practical implementations that avoid the burden of exact quadrature weights while maintaining accuracy and stability.

Abstract

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.

Proving the stability estimates of variational least-squares Kernel-Based methods

TL;DR

This work delivers a rigorous stability analysis for variational least-squares kernel-based methods solving second-order elliptic PDEs by proving the stability inequality for , thereby filling a gap left by prior conjectures. It develops discrete stability via carefully chosen collocation sets and and then propagates these results to the continuous setting, enabling error and condition-number estimates for VLS; it further extends the theory to Weighted LS (WLS) by establishing norm equivalences between continuous and weighted-discrete norms and showing convergence without exact quadrature weights. The paper also proves a uniform -norm bound in the trial space through discrete sampling and demonstrates the convergent behavior of WLS kernels, supported by comprehensive numerical experiments. Collectively, these results validate that variational and kernel-based collocation methods converge at the same rates, even with large mesh ratios, and illustrate practical implementations that avoid the burden of exact quadrature weights while maintaining accuracy and stability.

Abstract

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.
Paper Structure (8 sections, 6 theorems, 55 equations, 6 figures)

This paper contains 8 sections, 6 theorems, 55 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Assumptions and preliminaries
  3. Proof of stability estimates
  4. Implications to least-squares collocation methods
  5. A uniform bound for $L^2(\Omega)$-norm in the trial space
  6. Weighted-LS kernel-based collocation method is convergent
  7. Numerical Examples
  8. Conclusion

Key Result

Theorem 1

Let $d\leq 3$ and $\tau\geq k\geq 4$ (hence, $\tau>d/2$). Suppose $\Omega\subset \mathbb{R}^d$ is a bounded domain with $C^k$-boundary $\partial\Omega$ satisfying an interior cone condition. Let $\mathcal{L}$ in the PDE eq:bvp be a second-order strongly elliptic operator satisfying boundary regulari where $C$ is a generic constant independent of $u$ and $u^h_{{VLS}}$.

Figures (6)

  • Figure 1: An illustration of the distribution of data points contributing to the lower Riemann sum based on a partition with equally sized subdoamins with edge length $\delta$. The red points showcase a distribution that maximizes the fill distance when some points are located at opposite corners of one quadrant/cuboid region. The blue points show a distribution that minimizes the fill distance when all the points are in the centers of all quadrant/cuboid regions.
  • Figure 2: Distribution of $N_Y=40^2$ sine-transformed (PDE 1), signed-squared (PDE 2), and NodeLab Mishra-Node:19 generated (PDE4) collocation points, overlaying the color contour of the corresponding source function $f$ (PDEs 1 and 2).
  • Figure 3: Relative $L^2$ convergence profiles of VLS-Tp, WLS-Id, and WLS-Rd for PDE 1 using the Matérn kernel with smoothness orders $\tau=4,5,6$. Errors associated with oversampling ratio $\gamma \in \{1.0,1.5,2.0\}$ are shown individually, while those with $\gamma\in\{ 2.5,3.0,3.5,4.0\}$ are collectively represented in a shaded area. The predicted convergence rate from Theorem \ref{['main thm']} is represented by the reference slope.
  • Figure 4: Relative $L^2$ convergence profiles for PDE 2, presented in the same format as in Figure \ref{['fig err1']}.
  • Figure 5: Relative $L^2$ convergence profiles for Example 2, solving the modified Helmholtz equation $\Delta u - u = f$ with a Gaussian function centered at $[0,0]$ as the exact solution, presented in the same format as in Figure \ref{['fig err1']}.
  • ...and 1 more figures

Theorems & Definitions (10)

  • Theorem 1
  • Proposition 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof