Voronoi-Based Spatial Quadrature

Updated 11 November 2025

Voronoi-based spatial quadrature is a numerical integration technique that uses adaptive Voronoi tessellations to partition irregular high-dimensional spaces.
It employs methods such as simplex decomposition, Monte Carlo raycasting, and hybrid schemes to accurately evaluate integrals over complex domains.
These approaches enhance computational efficiency and robustness in applications like electronic-structure calculations, spatial statistics, and meshless PDE solvers.

Voronoi-based spatial quadrature denotes the family of numerical integration schemes built upon Voronoi (or weight-modified "power diagram") decompositions of ℝᵈ. By leveraging the adaptive, locally data-driven partitioning generated by Voronoi cells, these schemes offer significant advantages for the integration of functions over irregular, high-dimensional domains, including enhanced accuracy, robustness in the presence of singularities, and adaptive mesh refinement. Voronoi-based quadratures are prominent in spatial point-process model diagnostics, ab initio electronic-structure computations, high-dimensional PDE solvers, and meshless particle methods.

1. Voronoi Tessellation and Cell Construction

The Voronoi tessellation associated with a point set $\{x_1,\dots,x_N\}\subset\mathbb{R}^d$ partitions space into $N$ cells such that each point $x\in\mathbb{R}^d$ is assigned to the cell of its nearest generator under a specified metric. For unweighted tessellations, the metric is Euclidean: $C_i = \left\{ x \in \mathbb{R}^d : \|x - x_i\| \leq \|x - x_j\| \ \forall j\neq i \right\}.$ Weighted (radical-plane or power diagram) variants replace $\|x - x_i\|^2$ by $D_i(x) = \|x - x_i\|^2 - R_i^2$ , with non-negative radii $R_i$ reflecting atomic or kernel size. Each cell is then

$\Omega_i = \{ x \in \mathbb{R}^d : D_i(x) \leq D_j(x),\ \forall j \neq i \}.$

In meshless quadrature contexts, such as the partition-of-unity method, each kernel center $x_i$ with support tensor $H_i$ and radius $r$ defines a local support

$S_i = \{ x \in \Omega : \|H_i(x - x_i)\| \leq r \},$

with the actual cell $V_i$ formed by recursively clipping $S_i$ with bisector halfspaces induced by all neighbors $j$ whose supports overlap $S_i$ . Algorithms for these constructions utilize sweep-line, divide-and-conquer, or (in higher dimensions) randomized and raycasting techniques, with boundary cells typically clipped to the domain and handled separately due to altered statistical properties (Bray et al., 2015, Alam et al., 2011, Sikorski et al., 16 May 2024, Bassett et al., 2020).

2. Quadrature Schemes on Voronoi Cells

Three principal quadrature paradigms on Voronoi cells are distinguished:

Subdivision and Local Quadrature: Each polyhedral cell $V_i$ is decomposed into simplexes or pyramidal elements over its faces, and standard quadrature rules (e.g., Gauss–Legendre, symmetric rules) are applied to each. This approach achieves exactness for polynomial integrands up to the quadrature order. When singularities (e.g., Coulomb $1/|x|$) are present, the interior (spherical) region is handled analytically or with radial quadrature, while the remaining interstitial polyhedron is mapped to a bi-unit cube using isoparametric transforms for efficient tensor-product quadrature (Alam et al., 2011, Bassett et al., 2020).
Monte Carlo Raycasting: For high-dimensional Voronoi cells, random directions $y \in S^{d-1}$ are sampled. For each $k$ , the intersection point $r_k = x_i + \ell_i(y_k) y_k$ with the cell boundary is found via a RayCast routine. The Jacobian change of variables converts the surface or volume integral over $C_i$ (or $\partial C_i$ ) into sums over these rays, leading to estimators: $I_V(f;C_i) \approx \frac{S_{d-1}}{n\, m} \sum_{k=1}^n\sum_{s=1}^m f\bigl(x_i (1-t_s) + r_k t_s \bigr) t_s^{d-1} \ell_k^d/d,$ where $t_s\in[0,1]$ are sub-random samples along the ray (Sikorski et al., 16 May 2024). The error scales as $O(n^{-1/2})$ .
Hybrid Quadrature: Combines MC area estimation for facets with exact (Leibnitz-type) interpolation over vertices and centroids of cell interfaces, yielding a balance between speed and accuracy in moderate dimensions ( $5 \lesssim d \lesssim 8$ ).

A pseudocode segment for raycasting-based MC quadrature is:

for k in 1..n_rays:
    y = uniform_sample_on_S_{d-1}()
    (σ, r) = RayCast(i, y)
    ℓ = norm(r - x_i)
    for ℓs in 1..m_sub:
        t = random(0,1)
        x = x_i*(1-t) + r*t
        Fvol += f(x) * t**(d-1) * ℓ**d / d

and for cell construction with simplex decomposition:

for i in 1..N:
    determine support neighbors N_i
    P_i = EllipsoidBound(x_i, H_i, r)
    for j in N_i:
        P_i = ClipByHalfspace(P_i, plane{x : |x-x_i| = |x-x_j|})
    {T_{i,α}} = DecomposeToSimplexes(P_i)
    for T in {T_{i,α}}:
        {ξ_k, w_k} = ReferenceQuad(T)
        for k:
            x_q = MapToPhysical(T, ξ_k)
            w_q = w_k * |detJacobian(T, ξ_k)|
            accumulate w_{ij}, φ_j(x_q), ∇φ_j(x_q)

(Alam et al., 2011, Bassett et al., 2020).

3. Theoretical Properties and Statistical Foundations

Under a correct spatial point-process model, the vector of cell integrals $\{ \Lambda_i \}$ over Voronoi cells— $\Lambda_i = \int_{V_i} \lambda(x)\,dx$ —exhibits explicit probabilistic structure. For a stationary unit-rate Poisson process, the area $A$ of (interior) Voronoi cells is approximately distributed as $\Gamma(\alpha=3.569,\,\beta=3.569)$ , implying

$r_i = 1 - X_i,\qquad X_i \sim \Gamma(3.569, 3.569)$

with $\mathbb{E}[r_i] = 0$ and $\mathrm{Var}(r_i) = 3.569 / (3.569)^2$ (Bray et al., 2015). The approximate independence and mixing properties of Voronoi cell statistics underpin the applicability of aggregate normal approximations for sums and averages, without invoking simple CLTs for individual cells.

Partition-of-unity and polynomial-reproduction constraints in meshless quadrature ensure that all constant and linear fields are integrated exactly on each cell $V_i$ , enforcing local conservation and insuring spatial convergence order $n+1$ for RK-based bases. For frequent update/adaptation (e.g., moving particles), the cell structure seamlessly adjusts to evolving point distributions (Bassett et al., 2020).

4. Comparison with Grid-Based and Alternative Quadratures

Fixed rectangular (pixel) quadrature partitions suffer from two limitations: when pixel sizes are too small, expected counts are highly skewed, yielding many degenerate or near-Bernoulli integrals; when pixels are large, they average over inhomogeneities, sharply reducing the ability to detect local departures from model assumptions. Voronoi-adaptive quadrature, by adjusting cell size to local data density, avoids both extremes.

Quantitative studies with simulated Poisson and inhomogeneous fields indicate that Voronoi-based residual diagnostics retain power above 90% for moderate model misspecification, while pixel-based tests deteriorate due to the trade-off between cell count and power. In residuals for seismological models, Voronoi-based Kolmogorov–Smirnov–PIT tests capture local deviations in fault and periphery regions missed entirely by fixed-grid schemes. Hybrid MC/polygonal and pure MC raytracing quadrature methods are essential in high-dimensional integration, where grid-based cubature is infeasible (Bray et al., 2015, Sikorski et al., 16 May 2024).

The table below summarizes the essential trade-offs between the principal schemes:

Scheme	Dimensional Range	Accuracy	Computational Cost
Polygonal/simplex	$d \leq 5$	Exact (poly.)	Combinatorial in $d$
Monte Carlo Raycast	Any $d$	$O(n^{-1/2})$	Linear in $n_{\mathrm{rays}}$
Hybrid (MC + Linear)	$5 \leq d \leq 8$	1–2% (linear)	Intermediate

5. Applications and Performance Benchmarks

Spatial Residual Analysis: Voronoi-based quadrature has proven especially powerful in assessing lack of fit in point-process models for seismological hazard, detecting spatially structured under/overprediction patterns that are masked in grid-based residuals. MC studies indicate consistently higher sensitivity in model validation for homogeneous and inhomogeneous fields, with formal PIT histograms under Voronoi quadrature displaying sharper deviations from uniformity (Bray et al., 2015).

Electronic-Structure Integration: In ab initio KKR, KKR-CPA, and EMTO codes, Voronoi-based isoparametric mapping plus tensor-product Gauss quadrature achieves "machine-precision" ( $\sim 10^{-13}$ ) for charge and potential integrals in milliseconds per Voronoi polyhedron, offering five or more orders of magnitude speedup and seven or more orders of accuracy over shape-function methods. Treatment of Coulomb singularities is robust: the spherical region is separated and handled analytically, while the interstitial region is mapped and integrated without loss of convergence (Alam et al., 2011).

Meshless PDEs and Transport: The meshless-local Petrov–Galerkin method with Voronoi-based integration achieves second-order spatial convergence for linear reproducing kernels, with local mass conservation and automatic adaptation of quadrature to evolving point distributions. The method is directly applicable to meshless discretization of the discrete-ordinates transport equation and other PDEs (Bassett et al., 2020).

High-Dimensional Domains: Raycasting-based MC integration is feasible for arbitrary $d$ and outperforms qhull-based exact tessellation for $d \gtrsim 6$ in computational speed. Hybrid schemes efficiently bridge the gap in moderate $d$ , with errors ≲2% for interface integrals (Sikorski et al., 16 May 2024).

6. Implementation Considerations and Practical Guidance

Selection of Quadrature Method: For $d \leq 5$ and when high accuracy for linear or polynomial integrands is essential, exact polygonal/simplex-based quadrature is preferred. For $d > 10$ , MC raycasting is the only practical method, and error can be arbitrarily reduced at $O(n^{-1/2})$ cost by increasing $n_{\mathrm{rays}}$ (Sikorski et al., 16 May 2024). In intermediate $d$ , hybrid methods provide an optimal compromise.
Handling Singularities: For functions with point singularities (e.g., in quantum chemistry), split the Voronoi cell at the singular region, treat it analytically or by 1D/2D quadrature, and use isoparametric mapping elsewhere (Alam et al., 2011).
Adaptive Integration Mesh: In particle or meshless methods, re-building the Voronoi tessellation on each timestep offers automatic refinement/coarsening, requiring only local kernels (e.g., PolyClipper for efficient clipping) and local neighbor search (e.g., via hash-grid or SPH-tree) (Bassett et al., 2020).
Performance Scaling: Voronoi cell and boundary construction can be completed in $\sim 10^{-3}$ s per cell (isoparametric), MC integration adds $\sim 10^{-4}$ s per cell for reasonable $n_{\mathrm{rays}}$ , and overall scaling is linear in the number of cells for fixed $d$ . For electronic-structure, overall speedup over shape-function methods exceeds $10^5\times$ at $10^7$ -fold higher accuracy (Alam et al., 2011).
Practical Recommendations: Precompute Voronoi cells exactly for $d \leq 6$ or use partial RayCast/raycasting for $d > 6$ . Meshless and finite-volume codes benefit by treating each cell as a control volume with locally adapted quadrature (Sikorski et al., 16 May 2024).

7. Significance and Research Impact

Voronoi-based spatial quadratures have transformed diagnostics in spatial-statistical modeling, leading to higher-powered goodness-of-fit tests and sharper localization of model deficiencies. In physics, the ability to perform numerically robust, highly accurate, and non-grid-aligned integration underpins state-of-the-art advances in electronic-structure computations and meshless discretization for high-dimensional PDEs. The flexibility of MC raycasting extends the reach of numerical quadrature well beyond the capabilities of grid-based and combinatorial simplex decompositions, opening avenues in high-dimensional stochastic simulation, machine learning kernels, and adaptive meshless methods.

A plausible implication is that the continued development of scalable, adaptive Voronoi-based quadrature methods will further extend the range of feasible computations in high-dimensional and geometrically complex domains. The integration of these schemes with emerging randomized and hybrid approaches suggests their applicability across a broad spectrum of computational mathematics and applied sciences.