具有可变光滑度的Whittle-Matérn场
Whittle-Matérn Fields with Variable Smoothness
AI总结：
本文提出了一种具有空间变化光滑度的非局部Whittle-Matérn场模型，通过变分框架证明了解的存在性和正则性，并展示了有限元采样方法及数值实验。
摘要：
我们引入并分析了一种非局部的Whittle-Matérn高斯场的非局部推广，其中光滑度参数通过分数阶次s=s(x)∈[s̲,s̲]⊂(0,1)在空间中变化。该模型通过一个积分形式算子定义，其核由修正贝塞尔函数构造，局部奇异性由对称指数β(x,y)=(s(x)+s(y))/2控制。这种可变阶非局部公式不同于经典常阶伪微分设置，提出了新的分析和数值挑战。我们开发了一种新的变分框架，适应于该核，证明了在截断有界域上的弱解存在性和唯一性，并推导了高斯（谱）解的Sobolev正则性，由最小局部阶控制：实现位于H^r(G)中，对于每个r<2s̲-d/2（此处H^r(G)表示有界域G的Sobolev空间），因此当s̲>d/4时位于L_2(G)中。我们还提出了一种有限元采样方法用于积分模型，推导了误差估计，并在一维中提供了数值实验，展示了空间变化的光滑度对样本协方差的影响。计算方面和可扩展实现的方向进行了讨论。
英文摘要：
We introduce and analyze a nonlocal generalization of Whittle--Matérn Gaussian fields in which the smoothness parameter varies in space through the fractional order, $s=s(x)\in[\underline s,\overline s]\subset(0,1)$. The model is defined via an integral-form operator whose kernel is constructed from the modified Bessel function of the second kind and whose local singularity is governed by the symmetric exponent $β(x,y)=(s(x)+s(y))/2$. This variable-order nonlocal formulation departs from the classical constant-order pseudodifferential setting and raises new analytic and numerical challenges. We develop a novel variational framework adapted to the kernel, prove existence and uniqueness of weak solutions on truncated bounded domains, and derive Sobolev regularity of the Gaussian (spectral) solution controlled by the minimal local order, with realizations lying in $H^r(\mathcal G)$ for every $r<2\underline s-\tfrac{d}{2}$ with $d\ge2$ or $\underline s\le1/2$, with a slightly reduced range in the remaining case (i.e. $d=1$ and $\underline s > 1/2$), hence in $L_2(\mathcal G)$ when $\underline s>d/4$. Here $H^r(\mathcal G)$ denotes the Sobolev space on the bounded domain $\mathcal G$. We also present a finite-element sampling method for the integral model, derive error estimates, and provide numerical experiments in one dimension that illustrate the impact of spatially varying smoothness on the solution covariance. Computational aspects and directions for scalable implementations are discussed.
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