Whittle-Mat\'{e}rn Fields with Variable Smoothness

TL;DR

This paper introduces a variable-smoothness nonlocal Gaussian field model, analyzes its mathematical properties, and develops a finite-element sampling method, demonstrating how spatially varying smoothness affects sample covariances.

Contribution

It presents a novel variable-order nonlocal Gaussian field framework, with theoretical analysis, existence proofs, regularity results, and a finite-element sampling method.

Findings

Realizations are in Sobolev spaces depending on minimal local smoothness.

The model captures spatially varying smoothness effects.

Numerical experiments illustrate covariance impacts.

Abstract

We introduce and analyze a nonlocal generalization of Whittle--Mat\'ern 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 $\beta(x,y)=(s(x)+s(y))/2$. 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.