Lifting MGARD: construction of (pre)wavelets on the interval using polynomial predictors of arbitrary order

TL;DR

This paper extends MGARD, a data compression algorithm, to higher-order finite elements by developing a lifting scheme with polynomial predictors, and introduces a wavelet basis formulation for improved data analysis.

Contribution

It introduces a novel lifting scheme for (pre)wavelets on the interval using arbitrary order polynomial predictors, expanding MGARD's applicability to higher-order finite element spaces.

Findings

Constructed explicit wavelet transform for uniform dyadic grids.

Extended MGARD to arbitrary order Lagrange finite elements.

Discussed a new compactly supported wavelet basis formulation.

Abstract

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.