Polynomial Interpolation of a Vector Field on a Convex Polyhedral Domain

TL;DR

The paper tackles exact polynomial-vector-field interpolation on convex polyhedral domains under a no-penetration boundary condition. By homogenizing the domain to a cone and linking tangency to the logarithmic derivation module, it identifies the tangent-field space with a syzygy module of the cone's Jacobian ideal and computes a basis via Schreyer’s Gröbner-basis algorithm. This basis enables a straightforward linear-least-squares fit to data that exactly enforces boundary tangency for any degree bound $k$, and supports additional linear constraints such as divergence-free or curl-free conditions. The framework applies in arbitrary dimension and is demonstrated with a pentagon example, illustrating exact constraint satisfaction and controllable model complexity. The approach uniquely integrates algebro-geometric tools with data-driven vector-field interpolation, yielding physically consistent reconstructions in confined domains.

Abstract

We present a computational method for reconstructing a vector field on a convex polytope $\mathcal{P} \subset \mathbb{R}^d$ of arbitrary dimension from discrete samples. We specifically address the scenario where the vector field is subject to a no-penetration (slip) boundary condition, requiring it to be tangent to the boundary $\partial \mathcal{P}$. Given a degree bound $k$, our algorithm computes a polynomial vector field of degree at most $k$ that fits the observed data in the least-squares sense while exactly satisfying the tangency constraints. Central to our approach is an explicit characterization of the module of polynomial vector fields tangent to $\partial \mathcal{P}$, derived using algebraic concepts from the theory of hyperplane arrangements.

Figures (7)

• Figure 1: Examples of convex polyhedral spaces in $\mathbb{R}^2$ and a tangential vector field defined on them.
• Figure 2: The cone $\widehat{\mathcal{P}}$ in $\mathbb{R}^{3}$ corresponding to a trapezoid $\mathcal{P}$ in $\mathbb{R}^2$. First, $\mathcal{P}$ is embedded into $\mathbb{R}^{3}$ on the plane $x_0=1$. By joining the origin with each facet of $\mathcal{P}$, we obtain hyperplanes in $\mathbb{R}^{3}$ that share the origin as the apex. An element of ${\mathrm{Poly_{\partial}}({\mathcal{P}})}$ can be extended to a field tangent to $\widehat{\mathcal{P}}$ simply by scaling by the "height" $x_0$. The field thus obtained is parallel to the plane $x_0=0$. Conversely, any homogeneous tangential field to $\widehat{\mathcal{P}}$ can be made parallel to $x_0=0$ by subtracting an $\mathbb{R}[x_1,\ldots,x_d]$-multiple of $\xi_E=(x_1,\ldots,x_d)$, which is a "radial" vector field that is tangent to any plane going through the origin. By restricting the parallel field to $x_0=1$, we obtain an element of ${\mathrm{Poly_{\partial}}({\mathcal{P}})}$.
• Figure 3: The best fitting vector field for the first four observations at $x_{s1},\ldots,x_{s4}$ (indicated by red arrows), using degree $4$ (left) and degree $5$ (right). For degree $4$, the fit exhibits a noticeable deviation from the observations, as reflected in the higher error value in \ref{['eq:error']} of about $2.7$. In contrast, with degree $5$, the fitted vector field exactly interpolates the observations, resulting in zero error.
• Figure 4: We gradually increase the number of observation points and compute the best fitting fields of degree $5$. The error is zero for every case.
• Figure 5: Polynomial approximation of a vortex-based flow field at lower degrees $k \in \{4,5,6,8\}$. Left column: Ground truth field. Subsequent columns: Approximations at increasing degrees. Row 1: Velocity vectors at 50 sample points with RMSE from velocity-based fitting. Row 2: Vorticity distribution from vorticity-based fitting. The diverging colormap (blue-white-red) emphasizes regions of positive and negative rotation. Row 3: Streamlines showing global flow structure. Lower-degree approximations fail to capture fine-scale circulation features.

Theorems & Definitions (30)

• Definition 3.1
• Definition 3.2
• Example 3.3
• Lemma 4.1
• proof
• Corollary 4.2
• proof
• Lemma 5.1
• proof
• Definition 5.2