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Efficiently parallelizable kernel-based multi-scale algorithm

Federico Lot, Christian Rieger

TL;DR

This work reformulates the kernel-based multiscale method for scattered data as a single large block-triangular linear system, enabling efficient parallel solution by decoupling a block-diagonal (solved with Conjugate Gradient) from a lower-triangular part (solved with Jacobi). By proving decay properties of the interlevel interactions, it introduces a thresholding strategy that compresses storage with provable error control, and provides detailed complexity bounds that reveal practical gains on large hierarchies of point clouds. Theoretical results are complemented by numerical experiments using Wendland RBFs and GPU acceleration, demonstrating accurate reconstructions and scalable performance up to hundreds of millions of points. The approach offers a scalable route to high-resolution multiscale kernel approximations in applications across approximation theory, PDEs, and machine learning contexts where scattered data arise.

Abstract

The kernel-based multi-scale method has been proven to be a powerful approximation method for scattered data approximation problems which is computationally superior to conventional kernel-based interpolation techniques. The multi-scale method is based of an hierarchy of point clouds and compactly supported radial basis functions, typically Wendland functions. There is a rich body of literature concerning the analysis of this method including error estimates. This article addresses the efficient parallelizable implementation of those methods. To this end, we present and analyse a monolithic approach to compute the kernel-based multi-scale approximation.

Efficiently parallelizable kernel-based multi-scale algorithm

TL;DR

This work reformulates the kernel-based multiscale method for scattered data as a single large block-triangular linear system, enabling efficient parallel solution by decoupling a block-diagonal (solved with Conjugate Gradient) from a lower-triangular part (solved with Jacobi). By proving decay properties of the interlevel interactions, it introduces a thresholding strategy that compresses storage with provable error control, and provides detailed complexity bounds that reveal practical gains on large hierarchies of point clouds. Theoretical results are complemented by numerical experiments using Wendland RBFs and GPU acceleration, demonstrating accurate reconstructions and scalable performance up to hundreds of millions of points. The approach offers a scalable route to high-resolution multiscale kernel approximations in applications across approximation theory, PDEs, and machine learning contexts where scattered data arise.

Abstract

The kernel-based multi-scale method has been proven to be a powerful approximation method for scattered data approximation problems which is computationally superior to conventional kernel-based interpolation techniques. The multi-scale method is based of an hierarchy of point clouds and compactly supported radial basis functions, typically Wendland functions. There is a rich body of literature concerning the analysis of this method including error estimates. This article addresses the efficient parallelizable implementation of those methods. To this end, we present and analyse a monolithic approach to compute the kernel-based multi-scale approximation.

Paper Structure

This paper contains 12 sections, 20 theorems, 177 equations, 5 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notation and basic setting
  3. Matrix formulation
  4. The block diagonal system
  5. The lower block triangular system
  6. Thresholding
  7. Error coupling
  8. Computational cost
  9. Numerical Experiments
  10. Bounds on M
  11. Truncation
  12. Parallelization on a GPU

Key Result

Theorem 4

Let $\Omega \subset \mathbf{R}^{d}$ and let $X_{N(\ell)}$ for $1\le \ell \le L$ denote a sequence of discrete point sets in $\Omega$. We obtain for all $f \in C(\Omega)$ and

Figures (5)

  • Figure 1: Comparison of the theoretical bound of $\|M_{L}\|_{2 \to 2}$ from Lemma \ref{['lem:Mbound']} with its numerical value.
  • Figure 2: normalized norm difference from $M_{L}$ and $\tilde{M}_{L}$.
  • Figure 3: non-zero elements ratio from $M_{L}$ and $\tilde{M}_{L}$.
  • Figure 4: Time required for solve the system \ref{['eq:split']} with different data sizes
  • Figure 5: Efficiency of the system solution with different data sizes

Theorems & Definitions (41)

  • Definition 3
  • Theorem 4
  • proof
  • Theorem 5
  • Corollary 6
  • proof
  • Lemma 7
  • proof
  • Lemma 8
  • proof
  • ...and 31 more