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R-Adaptive Mesh Optimization to Enhance Finite Element Basis Compression

Graham Harper, Denis Ridzal, Tim Wildey

TL;DR

Two novel contributions to reduce the memory burden while maintaining performance in simulations based on finite element discretizations are presented, including dictionary-based data compression schemes that detect and exploit the structure of the discretization, due to redundancies across the finite element mesh.

Abstract

Modern computing systems are capable of exascale calculations, which are revolutionizing the development and application of high-fidelity numerical models in computational science and engineering. While these systems continue to grow in processing power, the available system memory has not increased commensurately, and electrical power consumption continues to grow. A predominant approach to limit the memory usage in large-scale applications is to exploit the abundant processing power and continually recompute many low-level simulation quantities, rather than storing them. However, this approach can adversely impact the throughput of the simulation and diminish the benefits of modern computing architectures. We present two novel contributions to reduce the memory burden while maintaining performance in simulations based on finite element discretizations. The first contribution develops dictionary-based data compression schemes that detect and exploit the structure of the discretization, due to redundancies across the finite element mesh. These schemes are shown to reduce memory requirements by more than 99 percent on meshes with large numbers of nearly identical mesh cells. For applications where this structure does not exist, our second contribution leverages a recently developed augmented Lagrangian sequential quadratic programming algorithm to enable r-adaptive mesh optimization, with the goal of enhancing redundancies in the mesh. Numerical results demonstrate the effectiveness of the proposed methods to detect, exploit and enhance mesh structure on examples inspired by large-scale applications.

R-Adaptive Mesh Optimization to Enhance Finite Element Basis Compression

TL;DR

Two novel contributions to reduce the memory burden while maintaining performance in simulations based on finite element discretizations are presented, including dictionary-based data compression schemes that detect and exploit the structure of the discretization, due to redundancies across the finite element mesh.

Abstract

Modern computing systems are capable of exascale calculations, which are revolutionizing the development and application of high-fidelity numerical models in computational science and engineering. While these systems continue to grow in processing power, the available system memory has not increased commensurately, and electrical power consumption continues to grow. A predominant approach to limit the memory usage in large-scale applications is to exploit the abundant processing power and continually recompute many low-level simulation quantities, rather than storing them. However, this approach can adversely impact the throughput of the simulation and diminish the benefits of modern computing architectures. We present two novel contributions to reduce the memory burden while maintaining performance in simulations based on finite element discretizations. The first contribution develops dictionary-based data compression schemes that detect and exploit the structure of the discretization, due to redundancies across the finite element mesh. These schemes are shown to reduce memory requirements by more than 99 percent on meshes with large numbers of nearly identical mesh cells. For applications where this structure does not exist, our second contribution leverages a recently developed augmented Lagrangian sequential quadratic programming algorithm to enable r-adaptive mesh optimization, with the goal of enhancing redundancies in the mesh. Numerical results demonstrate the effectiveness of the proposed methods to detect, exploit and enhance mesh structure on examples inspired by large-scale applications.

Paper Structure

This paper contains 13 sections, 2 theorems, 37 equations, 12 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background on finite element methods
  3. Trade-offs between computational complexity and storage cost
  4. Exploiting mesh redundancy to compress finite element data
  5. Enhancing mesh redundancy: Problem formulation
  6. Formulation of the objective function
  7. Formulation of the constraints
  8. Enhancing mesh redundancy: Algorithms
  9. Mesh optimization using ALESQP
  10. Computing cell targets
  11. Computing cell weights
  12. Numerical results
  13. Conclusion

Key Result

Lemma 1

\newlabellem:shapes0 Let $\hat{\phi} \in \mathcal{S}$, with $\mathcal{S} \in \{ H^1(\hat{T}), H(\mathrm{curl};\hat{T}), H(\mathrm{div};\hat{T}),L^2(\hat{T})\}$, be a finite element basis function on the reference domain. Let $D\in\{1,\mathrm{grad},\mathrm{curl},\mathrm{div}\}$ be an appropriate op

Figures (12)

  • Figure 1: Illustration of the reference to physical mapping $\Phi_T$ for a mesh cell $T$ and reference cell $\hat{T}$.
  • Figure 1: Three examples of mesh classes with redundancy: (left) structured/graded, (middle) geometrically patterned, and (right) extruded. The latter two meshes comprise large clusters of identical cells. The cells in the graded mesh have shapes that are identical up to a dimensional scaling.
  • Figure 1: Reference square and triangle geometries, both subsets of $[0,1]^2$.
  • Figure 1: r-adaptive mesh optimization of a randomly perturbed structured $20\times20$ grid. A structured grid is recovered with $10^{-21}$ misfit, losslessly compressing to 99.75%.
  • Figure 2: Tradeoffs between storing and recomputing basis data for a heat transfer simulation, where we show the percentage of basis data retained across the mesh. For the given mesh, containing significant redundancy, the cost of our proposed basis compression scheme is denoted by the $\textcolor{red}{\mathsf{x}}$ mark.
  • ...and 7 more figures

Theorems & Definitions (5)

  • Lemma 1
  • Proof 1
  • Remark 1
  • Lemma 1
  • Proof 2