用于具有不连续性的散乱数据插值的自适应非线性单位分解方法
Adaptive Non-Linear Partition of Unity Methods for Scattered Data Interpolation with Discontinuities
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中文总结 AI 辅助
研究具有不连续性的散乱数据插值问题，核心方法是通过留一法交叉验证结合全局优化自适应调整非线性单位分解方法的超参数，主要贡献是降低不连续性附近逼近误差，在平滑区域保持精度且无需界面几何先验知识。
中文摘要 AI 辅助
具有不连续性的散乱数据逼近具有挑战性，因为吉布斯现象会显著降低界面附近的精度。最近引入的非线性单位分解方法（NL-PUM）通过将径向基函数（RBF）插值与非线性加权基本无振荡（WENO）策略相结合来解决此问题。虽然有效，但NL-PUM的性能严重依赖于两个固定超参数：RBF形状参数和补丁半径。本文通过使用乐观改进全局优化（GOOI）最小化的留一法交叉验证（LOOCV）局部调整这两个超参数来扩展NL-PUM。主要创新是一个平滑度指标，将不连续性感知收缩过程与基于LOOCV的半径选择联系起来：平滑区域的补丁保持不变，而靠近不连续性的补丁自动收缩以避免穿过界面。由此产生的方法LOOCV-NL-PUM-GOOI不需要界面几何的先验知识，并且除了标准自适应形状参数选择之外不会引入额外成本。对具有跳跃不连续性的合成测试函数的数值实验以及对挪威峡湾高程数据的实际应用证实，该方法在不连续性附近显著降低了逼近误差，同时在平滑区域保持了完全精度。
英文摘要
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.
