2026/07/23 by Adeeba Haider, Roberto Cavoretto, Juan Ruiz-Álvarez +1
Computer Science · Mathematics · #math.NA #cs.NA
Scattered data approximation with discontinuities is challenging due to the Gibbs phenomenon, which significantly reduces accuracy near interfaces. The recently introduced Non-Linear Partition of Unity Method (NL-PUM) addresses this by combining Radial Basis Function (RBF) interpolation with a non-linear Weighted Essentially Non-Oscillatory (WENO) strategy. While effective, NL-PUM's performance relies heavily on two fixed hyperparameters: the RBF shape parameter and the patch radius. This work extends NL-PUM by adapting both hyperparameters locally using Leave-One-Out Cross-Validation (LOOCV) minimized via Global Optimization with Optimistic Improvement (GOOI). Our main innovation is a smoothness indicator linking a discontinuity-aware shrinkage process to LOOCV-based radius selection: patches in smooth regions remain unchanged, while those near discontinuities automatically shrink to avoid crossing the interface. The resulting method, LOOCV-NL-PUM-GOOI, requires no prior knowledge of interface geometry and introduces no extra cost beyond standard adaptive shape parameter selection. Numerical experiments on synthetic test functions with jump discontinuities and a real-data application to Norwegian Fjords elevation data confirm that this approach substantially reduces approximation error near discontinuities while preserving full accuracy in smooth regions.