Whittle-Matérn Fields with Variable Smoothness

2026-02-18

AI Summary

This work proposes a novel Gaussian random field model based on a variable-order nonlocal integral operator, overcoming the limitation of classical Whittle–Matérn fields that employ constant-order pseudo-differential operators and thus cannot capture spatially varying smoothness. By allowing the fractional order parameter to vary with spatial location, the model enables flexible local control of smoothness. A variational framework adapted to this operator is established using a nonlocal kernel constructed from modified Bessel functions, and the existence and uniqueness of weak solutions are proven on bounded domains with truncation. Furthermore, Sobolev regularity bounds for sample paths are derived. Efficient sampling is achieved via finite element discretization, and one-dimensional numerical experiments demonstrate that spatially varying smoothness significantly alters the covariance structure, thereby surpassing the representational constraints of traditional constant-order models.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Kernel Methods
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

We introduce and analyze a nonlocal generalization of Whittle--Matérn Gaussian fields in which the smoothness parameter varies in space through the fractional order, $s=s(x)\in[\underline{s}\,,\bar{s}]\subset(0,1)$. 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: realizations lie in $H^r(G)$ for every $r<2\underline{s}-\tfrac{d}{2}$ (here $H^r(G)$ denotes the Sobolev space on the bounded domain $G$), hence in $L_2(G)$ when $\underline s>d/4$. 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 samples covariances. Computational aspects and directions for scalable implementations are discussed.

Problem

• Whittle-Matérn fields
• variable smoothness
• nonlocal operators
• Gaussian random fields
• spatially varying parameters

Innovation

• variable smoothness
• nonlocal operators
• Whittle-Matérn fields
• fractional Sobolev regularity
• finite element sampling