Spatially Correlated Scalar Fields

• Spatially correlated scalar fields are real-valued mappings on spatial domains defined by prescribed covariance functions, such as the Matérn model.
• They are commonly modeled as Gaussian random fields and realized via SPDEs, enabling efficient sampling with finite element methods and multilevel strategies.
• Applications include environmental statistics, engineering surrogate modeling, and quantum gravity analyses, supported by robust statistical inference and scalable computational techniques.

A spatially correlated scalar field is a real-valued or vector-valued assignment to points in a spatial (and possibly spatio-temporal) domain, for which the values at different positions exhibit a prescribed spatial correlation structure. Such fields govern uncertainty quantification, statistical inference, physical modeling, and predictive surrogates in computational science, engineering, statistical physics, and applied mathematics.

1. Mathematical Formulation and Covariance Structure

Let $D\subset\mathbb{R}^d$ be a domain. A spatially correlated scalar field is a mapping $\phi:D\times\Omega\to\mathbb{R}$ (or $\mathbb{R}^d$), where $\Omega$ is a probability space. The spatial correlation between points $x, x'\in D$ is specified by the two-point covariance function $$c_\phi(x,x') = \operatorname{Cov}[\phi(x),\phi(x')]$$.

A central class is Gaussian random fields, for which all finite-dimensional marginals are jointly normal. The Matérn class is particularly prevalent: $$c_\phi(x,x') = \sigma^2 \frac{2^{1-\nu}}{\Gamma(\nu)}(\kappa\|x-x'\|)^\nu K_\nu (\kappa\|x-x'\|)$$ where $\sigma^2$ is the marginal variance, $\nu > 0$ the smoothness exponent, $\kappa = \sqrt{2\nu}/\ell$ encodes the correlation length $\ell$, and $K_\nu$ is the modified Bessel function of the second kind (Ben-Yelun et al., 2023, Osborn et al., 2017, Osborn et al., 2017).

Functional data models operate in a separable Hilbert space $H$ (e.g., $L^2([0,1])$ or $L^2(D)$), with

$X: D \rightarrow H, \;\; X(s) \in H$

and operator-valued covariance $$C(s_1,s_2)(f) = \mathbb{E}[\langle X(s_1)\!-\!\mu(s_1),f\rangle( X(s_2)\!-\!\mu(s_2)) ]$$ (Martínez-Hernández et al., 2020).

2. Stochastic PDE Representation and Numerical Sampling

Spatially correlated Gaussian fields with Matérn covariance can be realized as solutions to stochastic partial differential equations (SPDEs). Whittle's theorem and its generalizations state that

$$(\kappa^2 - \Delta)^{\beta} r(x) = \frac{1}{\tau} g(x)$$

where $g(x)$ is spatial Gaussian white noise, $\beta = \nu/2 + d/4$, and $\tau$ normalizes the variance (Ben-Yelun et al., 2023, Osborn et al., 2017, Osborn et al., 2017). Integer $\beta$ yield recursive applications: $$(\kappa^2-\Delta) r^{(1)}(x) = \frac{1}{\tau} g(x); \quad (\kappa^2-\Delta) r^{(k)}(x) = r^{(k-1)}(x)$$ and $r(x) = r^{(\beta)}(x)$.

Discretization is typically via finite element methods (FEM):

• Assembly of mass ($M$) and stiffness ($A$) matrices, using basis $\{\phi_i\}$.
• The linear system $(\kappa^2 M + A) r^{(1)} = (1/\tau) g$, with $g \sim N(0,M)$.
• Sparse Cholesky or multifrontal factorizations permit scalable sampling, with cost $O(n^{1.5}\text{--} n^2)$ in 2D/3D (Ben-Yelun et al., 2023), and weak or strong scaling to $\sim 10^9$ degrees of freedom in parallel environments (Osborn et al., 2017).

Hierarchical and multilevel strategies decompose the sampling problem across coarsened mesh hierarchies, vital for multilevel Monte Carlo (MLMC) (Osborn et al., 2017, Osborn et al., 2017). In domain embedding, a structured mesh solves the SPDE, with $L^2$ projection onto an unstructured target mesh for full geometric flexibility (Osborn et al., 2017).

3. Statistical Inference, Estimation, and Surrogate Modeling

Empirical estimation of mean and covariance for spatially correlated functional data (Martínez-Hernández et al., 2020):

• Pointwise mean $\mu(s) = \mathbb{E}[X(s)]$.
• Covariance operator $C(s_1,s_2)$. Scalarizations such as the trace-covariogram $\sigma_{tr}(s_1,s_2)$ and trace-variogram $\gamma_{tr}(s_1,s_2)$ reduce operator complexity.

Estimation procedures include:

• Basis expansion $\{X(s_i;v)\approx \sum_{k=1}^K z_k(s_i)\eta_k(v)\}$ and kriging for $\{z_k(s)\}$.
• Weighted functional kriging with optimal weights solving constraints in the trace-covariogram.
• Penalization and regularization for high-dimensional $\mathcal{H}$.

For surrogate prediction of spatial fields, the Linear Model of Coregionalization (LMC) posits

$y(x) = W f(x) + \varepsilon(x)$

with $W$ a mixing matrix, $f(x)$ independent latent GPs, and $$\operatorname{Cov}[y(x), y(x')] = W K_f(x,x')W^\top + \Sigma_\varepsilon \delta_{x x'}$$. When $y(x)$ is highly nonlinear, the Extended LMC (E-LMC) introduces an invertible neural network $h(\cdot)$ to linearize outputs before LMC is applied in latent space, retaining interpretability and scaling to high $d$ (Wang et al., 2022).

Empirical studies confirm substantial improvements (up to 40–80% reduction in predictive MSE) for E-LMC over state-of-the-art alternatives in complex PDE-driven spatial fields (Wang et al., 2022).

4. Correlation Functions, Non-Ergodicity, and Variance Decomposition

In non-ergodic systems, the field $\phi(r,t)$ admits a decomposition of the variance of observables and their spatial correlation functions (Wittmer et al., 2022):

• Total correlation $C_{total}(R)$;
• Internal correlation $C_{int}(R)$ (fluctuations within meta-basins);
• External correlation $C_{ext}(R)$ (fluctuations between meta-basins).

These are related by $C_{total} = C_{int} + C_{ext}$, with observed variances expressible as spatial averages over the respective functions. In the limit of long sampling times, the external correlation dominates and reflects the quenched spatial structure (e.g., power-law or exponential decay). For a lattice spring model with quenched disordered $k_i$, $C_{ext}(r)$ reduces to $C_k(r)$, the two-point correlator of the spring constants (Wittmer et al., 2022).

Global variances then scale with system size according to the range and form of spatial correlation:

• Short-range: $\sigma_{ext}^2\sim 1/N$.
• Power-law: $\sigma_{ext}^2\sim N^{-\alpha/d}$ for $C_k(r)\sim r^{-\alpha}$.

5. Physical and Geometric Contexts: Turbulence, Quantum Gravity, and Lattice Models

In passive scalar turbulence, spatiotemporal correlations exhibit characteristic crossovers (Gorbunova et al., 2021):

• For $t\ll\tau_0$, correlations are Gaussian in $pt$, $C(p,t)/C(p,0)\approx \exp(-\alpha_s (pt)^2)$, reflecting sweeping decorrelation.
• For $t\gg\tau_0$, exponential in $p^2 t$, $C(p,t)\sim\exp(-\alpha_\ell p^2 |t|)$, consistent with eddy-diffusivity.

Direct numerical simulation confirms these analytical predictions, including the connection to the Obukhov–Corrsin spectrum and crossover behavior as a function of the velocity field’s time-scale (Gorbunova et al., 2021).

In discrete quantum gravity, scalar fields on fluctuating 2D triangulations with an $R^2$-curvature term (coupling $\beta$) demonstrate that algebraic boundary correlation, $C(\theta)\sim (2\sin(\theta/2))^{-2\Delta}$, is robust to geometric fluctuations. The conformal scaling exponent $\Delta$ is determined by the bulk mass via the AdS/CFT relation. As $\beta$ decreases and curvature fluctuations increase, the algebraic decay persists, and even under strong quantum gravity corrections or fermionic backreaction (via integrating out Kähler–Dirac fermions), boundary conformal behavior is preserved (Asaduzzaman et al., 2021).

6. Computational Strategies and Algorithmic Scaling

Efficient sampling and manipulation of spatially correlated scalar fields in large-scale settings rely on exploiting the sparsity of discretized PDE operators:

• Sparse precision matrices derived from SPDE discretizations (Ben-Yelun et al., 2023).
• Multilevel and hierarchical strategies (algebraic or geometric coarsening) for MLMC estimators, with mesh-independent convergence via algebraic multigrid (AMG) (Osborn et al., 2017, Osborn et al., 2017).
• On non-matching meshes, $L^2$-projection pipelines enable efficient parallel data transfer, crucial for embedding and uncertainty propagation on unstructured application geometries (Osborn et al., 2017).

For design under uncertainty (e.g., robust topology optimization of lattice structures), first-order Taylor expansions enable moment estimation and sensitivity analysis without explicit formation of dense covariance matrices. All computations are implemented as sparse matrix solves, allowing scaling to tens of thousands of design and stochastic variables (Ben-Yelun et al., 2023).

7. Practical Application Domains and Software Ecosystem

Spatially correlated scalar fields underpin a range of applied and methodological domains:

• Environmental statistics (e.g., wind or rainfall spatial fields)—implemented in R via packages such as fda, mgcv, fdaPDE, geofd (functional kriging), Manifoldgstat, and INLA (Martínez-Hernández et al., 2020).
• Engineering surrogate modeling for topology optimization and field predictions (Ben-Yelun et al., 2023, Wang et al., 2022).
• Large-scale uncertainty quantification in porous media flow using MLMC and scalable SPDE sampling (Osborn et al., 2017, Osborn et al., 2017).
• Analysis of non-ergodic, disordered systems in physics (Wittmer et al., 2022).
• Quantum gravity and holographic boundary correlations (Asaduzzaman et al., 2021).

These advances enable statistical and computational tractability of high-dimensional, strongly correlated spatial data across scientific and engineering applications.