标题： 加权本质非振荡 Shepard 方法 显示英文标题

标题： Weighted Essentially Non-Oscillatory Shepard method

摘要： Shepard方法是一种经典的快速算法，用于在多个维度中对散乱数据进行插值。 这是一个在数值分析中重要且众所周知的技术，其主要思想是远离近似点的数据应较少影响最终的近似结果。 在$\mathbb{R}^n$附近沿$\mathbb{R}^{n-1}$上的超曲面逼近分段光滑函数对于Shepard方法或任何其他线性技术来说，在稀疏数据情况下具有挑战性，因为准确捕捉尖锐变化并避免振荡存在固有的困难。 这封信致力于构建一种非线性的Shepard方法，该方法利用加权本质无震荡插值方法（WENO）的基本思想。 所提出的方法旨在通过结合WENO的自适应和非线性加权机制来提高传统Shepard方法的精度和稳定性。 为了解决这一挑战，我们将非线性地修改一般Shepard方法中的权重函数，考虑任何权重函数，而不是仅仅依赖于距离平方的倒数。 这种方法有效减少了不连续附近的振荡，并提高了整体插值质量。 数值实验表明新方法在处理复杂数据集方面表现出色，使其成为科学计算和数据分析中各种应用的宝贵工具。 显示英文摘要

摘要： Shepard method is a fast algorithm that has been classically used to interpolate scattered data in several dimensions. This is an important and well-known technique in numerical analysis founded in the main idea that data that is far away from the approximation point should contribute less to the resulting approximation. Approximating piecewise smooth functions in $\mathbb{R}^n$ near discontinuities along a hypersurface in $\mathbb{R}^{n-1}$ is challenging for the Shepard method or any other linear technique for sparse data due to the inherent difficulty in accurately capturing sharp transitions and avoiding oscillations. This letter is devoted to constructing a non-linear Shepard method using the basic ideas that arise from the weighted essentially non-oscillatory interpolation method (WENO). The proposed method aims to enhance the accuracy and stability of the traditional Shepard method by incorporating WENO's adaptive and nonlinear weighting mechanism. To address this challenge, we will nonlinearly modify the weight function in a general Shepard method, considering any weight function, rather than relying solely on the inverse of the distance squared. This approach effectively reduces oscillations near discontinuities and improves the overall interpolation quality. Numerical experiments demonstrate the superior performance of the new method in handling complex datasets, making it a valuable tool for various applications in scientific computing and data analysis.