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

Viktor Reshniak, Evan Ferguson, Qian Gong, Nicolas Vidal, Rick Archibald, Scott Klasky

TL;DR

This work reframes MGARD as a wavelet lifting transform on the interval, enabling stable multilevel decompositions for arbitrary-order Lagrange elements $Q_p$ via polynomial predictors. It introduces a matrix-based formulation and refinement equations that connect primal and dual bases through interpolation and update operators, and derives compactly supported wavelet bases suitable for uniform dyadic grids and tensor grids. Key contributions include explicit refinement and duality relations, stable piecewise-polynomial predictors with computable Gramm matrices, and a detailed treatment of normalization, projection operators, and tensor-grid extensions. The results demonstrate how predictor order and data sampling affect coefficient decay and compression performance, with implications for adaptive, high-order data reduction in scientific computing. The framework offers a foundation for improved error control and future work on data-adaptive order selection and rigorous analysis.

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.

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

TL;DR

This work reframes MGARD as a wavelet lifting transform on the interval, enabling stable multilevel decompositions for arbitrary-order Lagrange elements via polynomial predictors. It introduces a matrix-based formulation and refinement equations that connect primal and dual bases through interpolation and update operators, and derives compactly supported wavelet bases suitable for uniform dyadic grids and tensor grids. Key contributions include explicit refinement and duality relations, stable piecewise-polynomial predictors with computable Gramm matrices, and a detailed treatment of normalization, projection operators, and tensor-grid extensions. The results demonstrate how predictor order and data sampling affect coefficient decay and compression performance, with implications for adaptive, high-order data reduction in scientific computing. The framework offers a foundation for improved error control and future work on data-adaptive order selection and rigorous analysis.

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 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 , , 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.
Paper Structure (18 sections, 3 theorems, 73 equations, 10 figures, 1 table)

This paper contains 18 sections, 3 theorems, 73 equations, 10 figures, 1 table.

Table of Contents

  1. MGARD wavelet transform
  2. Matrix formulation
  3. Refinement equations
  4. Primal basis
  5. Dual basis
  6. Uniform dyadic grids on $\Omega:=[0,1]$
  7. Normalization
  8. Interpolating projectors
  9. Piecewise constant predictor
  10. Piecewise polynomial predictors with stable projectors.
  11. Tensor grids
  12. Results
  13. Piecewise polynomials.
  14. Order comparison.
  15. Conclusion and future work
  16. ...and 3 more sections

Key Result

Theorem 1

Assume that $\varphi,\tilde{\varphi}$ are compactly supported biorthogonal scaling functions with $\varphi\in L^r(\Omega)$, $\tilde{\varphi}\in L^{r'}(\Omega)$ for $r\in[1,\infty]$, $1/r+1/r'=1$, or that $\varphi\in C^0(\Omega)$ and $\tilde{\varphi}$ is a Radon measure in which case $r=\infty$. Then for all $s>0$ such that where $n$ is the order of polynomial reproduction in $V_j$ and $t$ is such

Figures (10)

  • Figure 1: A subdivision of two domains into elements $\mathcal{T}$ (gray) and subdomains $\mathcal{S}$ (black).
  • Figure 2: Projections into the space of piecewise polynomials on the grid with $2^3$ elements.
  • Figure 3: Schematic illustration of MGARD lifting steps.
  • Figure 4: One-dimensional coarse (solid) and surplus (dashed) Lagrange basis functions at two adjacent grid levels $\mathcal{G}_{c/f}$. The corresponding one-dimensional elements $\tau_{c/f}^q$ and grid spacings $h_{c/f}$ are also shown.
  • Figure 5: The primal and dual basis functions induced by $\mathcal{P}_j=\mathcal{I}_j$ at level $1$. For clarity, only one basis vector per element is shown for the dual wavelet basis $\tilde{\psi}$.
  • ...and 5 more figures

Theorems & Definitions (9)

  • Theorem 1: cohen2003numerical
  • Remark 1.1
  • Remark 1.2
  • Definition 1.3
  • Definition 1.4
  • Lemma A.1
  • proof
  • Lemma A.2
  • proof