Polynomial Interpolation of a Vector Field on a Convex Polyhedral Domain

TL;DR

This paper introduces a computational method to reconstruct polynomial vector fields on convex polytopes from discrete data, ensuring the fields are tangent to the boundary and satisfy boundary conditions.

Contribution

It provides an explicit algebraic characterization of tangent polynomial vector fields on convex polytopes, enabling accurate reconstruction from samples.

Findings

Efficient algorithm for polynomial vector field reconstruction.

Exact boundary tangency conditions incorporated into the model.

Applicable to arbitrary dimensions and polyhedral 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.