Whittle-Matérn Fields with Variable Smoothness

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.

Figures (5)

• Figure 1: We consider three different functions of $s(x)$: Left, middle and right panels display a step function, a Gaussian bump and an oscillatory ramp, respectively.
• Figure 2: Comparison between the Matérn covariance and the samples covariance $C_s = A^{-1} M A^{-T}$ with $\kappa = 2.5$. For the constant case (top left), the Matérn covariance is computed with $\nu=2s-1/2$ while for the variable cases Cases 1--3 it is computed with $\nu = 2\,\left\langle s(x)\right\rangle-d/2$.
• Figure 3: The images show 100 generated samples (out of the 1000 samples) in all three cases when $\kappa=2.50$ (left column) and $\kappa=0.25$ (right column).
• Figure 4: Comparison of the samples covariance profiles $C_s(x,\cdot)$ for the variable--smoothness case $s = s(x)$ in Case 1 and the two constant--smoothness cases $s = \underline{s} = 0.35$ and $s = \overline{s} = 0.85$. The panels shows the covariances at $x=-1.5$, $x=0$ and $x=1.5$.
• Figure 5: The samples covariance $C_s$ for $\kappa=2.5$ (top row) and $\kappa=0.25$ (bottom row) for cases 1--3.

Theorems & Definitions (34)

• Definition 2.1: Gaussian white noise on $\mathcal{G}$
• Remark 2.2
• Theorem 3.1: Completeness
• Proposition 3.2: Density of smooth compactly supported functions
• Theorem 3.3: Equivalence of energy spaces for different $\kappa$
• Remark 3.4
• Theorem 3.5: Form representation
• Proposition 3.6
• Remark 3.7
• Proposition 3.8: Stochastic RHS