Whittle-Matérn Fields with Variable Smoothness
🤖 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.
Application Category
📝 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
Research questions and friction points this paper is trying to address.
Whittle-Matérn fields
variable smoothness
nonlocal operators
Gaussian random fields
spatially varying parameters
Innovation
Methods, ideas, or system contributions that make the work stand out.
variable smoothness
nonlocal operators
Whittle-Matérn fields
fractional Sobolev regularity
finite element sampling
🔎 Similar Papers
No similar papers found.
