Optimal triangulations for piecewise linear approximations of non-convex variable products

Abstract

We show optimal triangulations for piecewise linear (PWL) approximations of indefinite quadratic functions over the plane. Optimal triangulations have minimum triangle density while allowing a PWL approximation that fulfills a prescribed error bound measured in the L-infinity norm. In 2000, Pottmann et al. proved optimal triangulations for PWL interpolations and conjectured that these are also optimal for general PWL approximations. This conjecture was refuted in 2018 by Atariah et al., who allowed a constant deviation at the vertices of the triangles and decreased the triangle density by roughly 3%, though they left open whether their construction was optimal. In this paper, we resolve this open question: allowing varying deviations at vertices reduces the optimal triangle density by 25% compared to Atariah et al., and we prove this is globally optimal. We thus show that the potential of general PWL approximations is significantly lower for indefinite than for definite quadratic functions, where the triangle density can be halved when allowing general approximations compared to interpolations. Furthermore, we prove that among parallelogram tilings -- triangulations built from translated copies of a triangle and its point-reflection -- the constant-deviation construction of Atariah et al. is optimal when continuity of the PWL approximation is required. We conjecture that this optimality extends to all continuous triangulations, not just those based on parallelograms.

Figures (10)

• Figure 1: Illustration of pwl. approximations. (a) The saddle surface $F(x,y) = xy$. (b) A continuous pwl. approximation: the linear pieces meet along shared edges, forming a connected surface. (c) A general (discontinuous) pwl. approximation: linear pieces may have different values at shared vertices, creating gaps. In (b) and (c), the triangulation is shown projected onto the $xy$-plane. General approximations allow larger triangles for the same error bound.
• Figure 2: Illustration of triangle density. The triangulation $\mathcal{T}$ (blue) covers the plane. To compute the density, we count triangles intersecting the square $Q_\ell$ (highlighted in green) and divide by $\ell^2$. As $\ell \to \infty$, this ratio converges to the triangle density $\delta(\mathcal{T})$.
• Figure 3: The optimal triangle for $\varepsilon$-approximation of $F(x,y) = xy$. Each vertex $v_i$ has a deviation$d_i := f(v_i) - F(v_i)$, the signed difference between the approximating plane and the surface at that vertex. Each edge carries an edge product$k_{ij} := (x_j - x_i)(y_j - y_i)$, which measures the curvature of $F$ along that edge (positive for "ascending" edges, negative for the "descending" edge). Note the symmetry: $v_2$ and $v_3$ are reflections across the line $x=y$, and $d_2 = d_3$. We call this the "LHH" (Low-High-High) deviation pattern: one low deviation ($d_1 = -\varepsilon$) and two equal high deviations ($d_2 = d_3 = 7\varepsilon/9$). These quantities are defined formally in Section \ref{['sec:error-analysis']}.
• Figure 4: Left: We show how to translate and then possibly reflect a triangle to get a triangle touching the origin and in the first orthant. Right: We show that with that triangle, we can tile the plane.
• Figure 5: The surface $F(x,y) = xy$ (blue mesh) and linear approximations $L$ with different deviation patterns. The deviations $d_i$ measure the signed vertical distance from surface to approximation at each vertex: (a) all positive (green, overestimation), (b) all negative (red, underestimation), (c) mixed signs (orange, general approximation).

Theorems & Definitions (69)

• Theorem 1.1: General pwl. approximations
• Theorem 1.2: Pwl. underestimations and overestimations
• Theorem 1.3: Continuous pwl. approximations -- parallelogram tilings
• Theorem 1.4: Continuous pwl. under/overestimations -- parallelogram tilings
• Definition 2.1: Triangle
• Definition 2.2: Triangulation
• Definition 2.3: Locally Finite Triangulation and Piecewise Linear Functions
• Definition 2.4: Approximation Error
• Definition 2.5: Triangle Density Atariah:2018
• Remark 2.6