MGARD (MultiGrid Adaptive Reduction of Data) is an algorithm for compressing and refactoring scientific data, based on the theory of multigrid methods. The core algorithm is built around stable multilevel decompositions of conforming piecewise linear $C^0$ finite element spaces, enabling accurate error control in various norms and derived quantities of interest. In this work, we extend this construction to arbitrary order Lagrange finite elements $\mathbb{Q}_p$, $p \geq 0$, and propose a reformulation of the algorithm as a lifting scheme with polynomial predictors of arbitrary order. Additionally, a new formulation using a compactly supported wavelet basis is discussed, and an explicit construction of the proposed wavelet transform for uniform dyadic grids is described.

翻译：MGARD（多网格自适应数据缩减）是一种基于多网格方法理论的科学数据压缩与重构算法。其核心算法围绕协调分片线性$C^0$有限元空间的稳定多级分解构建，能够在不同范数及感兴趣的导出量中实现精确误差控制。本文将该构造推广至任意阶拉格朗日有限元$\mathbb{Q}_p$（$p \geq 0$），并提出将算法重新表述为采用任意阶多项式预测器的提升格式。此外，讨论了使用紧支撑小波基的新表述，并描述了在均匀二进网格上实现所提出小波变换的显式构造方法。