A Unified Framework for Efficient Kernel and Polynomial Interpolation

TL;DR

The paper develops a unified interpolation framework that blends compactly-supported Wendland kernels with multivariate polynomials to create a sparse, scalable interpolant. By formulating the problem as a kernel-plus-polynomial expansion, and by deriving efficient linear-algebra procedures for both the polynomial limit and the hybrid regime, the authors achieve accurate interpolation on Euclidean domains and manifolds, with notable edge- and boundary-behavior advantages. Key findings show that in 2D and 3D Euclidean domains, the unified interpolant often surpasses polynomial least-squares for rough targets and remains computationally favorable relative to global PHS+poly; on manifolds without boundaries, the advantage is diminished, while with boundaries the method can outperform both alternatives. The work offers practical algorithms for fast assembly and solves, highlights the impact of shape-parameter control and polynomial degree scaling, and points to extensions in PDE solvers, uncertainty quantification, and operator learning where trainable kernels can yield problem-adaptive sparsity and accuracy.

Abstract

We present a unified interpolation scheme that combines compactly-supported positive-definite kernels and multivariate polynomials. This unified framework generalizes interpolation with compactly-supported kernels and also classical polynomial least squares approximation. To facilitate the efficient use of this unified interpolation scheme, we present specialized numerical linear algebra procedures that leverage standard matrix factorizations. These procedures allow for efficient computation and storage of the unified interpolant. We also present a modification to the numerical linear algebra that allows us to generalize the application of the unified framework to target functions on manifolds with and without boundary. Our numerical experiments on both Euclidean domains and manifolds indicate that the unified interpolant is superior to polynomial least squares for the interpolation of target functions in settings with boundaries.

Figures (12)

• Figure 1: Boundary-clustered Poisson disk samples on the unit disk (left) and the unit ball (right). Interior nodes are shown in black and boundary nodes are shown in red.
• Figure 2: Relative $\ell_2$ error vs. $N^{1/d}$ (here, $N$) for $f(x) = |x|$ using the $C^2 (\mathbb{R}^3)$ Wendland kernel. (Left) Fixed-shape (FS) strategy: Shape parameters $\epsilon$ are tuned on the finest grid and reused. (Right) Fixed condition number (FC) strategy: $\epsilon$ is adjusted per grid to maintain fixed condition numbers ($K_t = 10^{12}, 10^8, 10^4$).
• Figure 3: Computational costs vs. $N$ for $f(x) = |x|$ using the $C^2 (\mathbb{R}^3)$ Wendland kernel. (Top row) Fixed-shape (FS) strategy: Shape parameters $\epsilon$ are tuned on the finest grid and reused. (Bottom row) Fixed condition number (FC) strategy: $\epsilon$ is adjusted per grid to maintain fixed condition numbers ($K_t = 10^{12}, 10^8, 10^4$). Left: assembly and solve time; Right: evaluation time.
• Figure 4: Relative $\ell_2$ error vs. $N$ for $f(x) = \frac{1}{1 + 25x^2}$ using the $C^2 (\mathbb{R}^3)$ Wendland kernel. (Left) Fixed-shape (FS) strategy: Shape parameters $\epsilon$ are tuned on the finest grid and reused. (Right) Fixed condition number (FC) strategy: $\epsilon$ is adjusted per grid to maintain fixed condition numbers ($K_t = 10^{12}, 10^8, 10^4$).
• Figure 5: Computational costs vs. $N$ for $f(x) = \frac{1}{1 + 25x^2}$ using the $C^2 (\mathbb{R}^3)$ Wendland kernel. (Top row) Fixed-shape (FS) strategy: Shape parameters $\epsilon$ are tuned on the finest grid and reused. (Bottom row) Fixed condition number (FC) strategy: $\epsilon$ is adjusted per grid to maintain fixed condition numbers ($K_t = 10^{12}, 10^8, 10^4$). Left: assembly and solve time; Right: evaluation time.