Non-linear Partition of Unity method

TL;DR

This work introduces the Non-linear Partition of Unity Method (NL-PUM), a framework that fuses Radial Basis Function (RBF) interpolation with WENO-inspired non-linear weights inside the Partition of Unity (PUM) setting to handle discontinuities in data. By computing local smoothness indicators $I_j$ via least-squares residuals and forming nonlinear weights $\alpha_j=(\epsilon+I_j)^{-t}$, NL-PUM upweights contributions from smooth regions while suppressing those near discontinuities, preserving the optimal $k+\frac{\nu}{2}$ convergence in smooth areas and achieving robust oscillation control near jumps. The authors establish error bounds for NL-PUM and validate the method with numerical experiments showing comparable performance to PUM on continuous data and marked improvements near discontinuities across various curve geometries and sampling schemes, aided by a two-pass discontinuity-treatment strategy. The approach leverages regular data partitioning and efficient data structures, making NL-PUM scalable and broadly applicable, with potential extensions to higher dimensions and practical domains such as image processing and fluid dynamics.

Abstract

This paper introduces the Non-linear Partition of Unity Method, a novel technique integrating Radial Basis Function interpolation and Weighted Essentially Non-Oscillatory algorithms. It addresses challenges in high-accuracy approximations, particularly near discontinuities, by adapting weights dynamically. The method is rooted in the Partition of Unity framework, enabling efficient decomposition of large datasets into subproblems while maintaining accuracy. Smoothness indicators and compactly supported functions ensure precision in regions with discontinuities. Error bounds are calculated and validate its effectiveness, showing improved interpolation in discontinuous and smooth regions. Some numerical experiments are performed to check the theoretical results.

Figures (7)

• Figure 1: (a) Approximation to Franke's function using PUM, (b), (c), (d) Approximation to function $f_1$, Eq. \ref{['frankesdisc']} using PUM.
• Figure 2: Black square: frontier of $\Omega$, red circumferences: frontiers of the patches, black curve: discontinuity, blue circumferences: contaminated patches, blue points: data, red stars: centers of the patches, green point (right): discontinuity point.
• Figure 3: Approximation of the function $f_1$ using PUM, and NL-PUM at grid points where we use $\phi_{M_2}$ for the RBF problems and $\phi_{W_2}$ for the partition of unity. Second and third rows are rotations of the plots presented in the first row.
• Figure 4: Approximation to functions $f_2$ using PUM, and NL-PUM at Halton's points using for RBF problems $\phi_{M_2}$ and for partition of unity $\phi_{W_2}$. The second row is a rotation of the plots presented in the first row.
• Figure 5: Approximation to function $f_3$ using PUM, and NL-PUM at Halton's points using for RBF problems $\phi_{M_2}$ and for partition of unity $\phi_{W_2}$.

Theorems & Definitions (8)

• Definition 1: FASSHAUER
• Lemma 2.1
• Theorem 2.2
• Proposition 2.3
• Theorem 3.1
• Definition 2: Contaminated patches
• Definition 3: Discontinuity points
• Theorem 5.1