用于逼近由高斯白噪声驱动的分数阶椭圆微分方程的质量集中与数值求积
Mass Lumping and Numerical Quadrature for Approximation of Fractional Elliptic Differential Equations Driven by Gaussian White Noise
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中文总结 AI 辅助
本文针对分数阶椭圆SPDE，推导结合有限元、数值求积与质量集中的逼近收敛速率，分析空间变化量正则性对收敛的影响，通过数值实验验证结果。
中文摘要 AI 辅助
分数阶椭圆型随机偏微分方程（SPDE）在统计学和机器学习中被广泛用于对高斯随机场进行计算高效且灵活的建模。SPDE方法的计算效率依赖于有限元逼近结合数值求积与质量集中，这两种技术可在推理过程中实现稀疏矩阵方法。尽管已有诸多研究探讨了分数阶SPDE的有限元逼近，但实际应用中所用的质量集中与求积逼近的效果尚未得到充分分析。为填补这一空白，本文推导了结合有限元离散、数值求积与质量集中的分数阶SPDE数值逼近的收敛速率。具体而言，本文在涵盖SPDE方法中使用质量集中的主要场景的通用框架下，得到了协方差函数均方误差的显式收敛速率；还分析了形如$L^\beta(\tau u)=\mathcal{W}$的非平稳方差控制因子，其中$\tau$为空间变化量，推导了协方差误差估计以说明$\tau$的正则性如何影响收敛速率。作为具体示例，本文给出了有界欧氏域、黎曼流形和度量图上随机场的相关结果，并通过数值实验验证了理论结果。
英文摘要
Fractional elliptic stochastic partial differential equations (SPDEs) are widely used in statistics and machine learning for computationally efficient and flexible modeling of Gaussian random fields. The computational efficiency of the SPDE approach relies on finite element approximations combined with numerical quadrature and mass lumping, which enable sparse matrix methods during inference. Although many works have studied finite element approximations of fractional SPDEs, the effect of the mass lumping and quadrature approximations used in practice has not been fully analyzed. To fill this gap, we derive convergence rates for numerical approximations of fractional SPDEs based on finite element discretizations combined with numerical quadrature and mass lumping. Specifically, we obtain explicit convergence rates for the mean-squared error of the covariance function in a general framework that covers the main settings where mass lumping is used in the SPDE approach. We also analyze non-stationary variance-control factors of the form $L^β(τu)=\mathcal{W}$, where $τ$ is spatially varying, and derive covariance error estimates showing how the regularity of $τ$ affects the convergence rate. As specific examples, we provide results for random fields on bounded Euclidean domains, Riemannian manifolds, and metric graphs. Numerical experiments are presented that confirm the theoretical results.
