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$\mathcal{H}^2$-matrices for translation-invariant kernel functions

Steffen Börm, Janne Henningsen

TL;DR

The paper addresses the storage bottleneck in boundary element method discretizations with translation-invariant kernels by proposing a translation-invariant modification of $\mathcal{H}^2$-matrices. It constructs a hierarchy of axis-aligned boxes with uniform levels and uses tensor Chebyshev interpolation; translation symmetry allows reuse of far-field coupling blocks and reduces memory movement. The main contributions are proving that far-field storage per level scales as $O(k^2 \log(n))$, deriving explicit storage bounds for leaf, transfer, nearfield, and coupling components, and validating the approach with numerical experiments showing significant memory savings without sacrificing accuracy. This work enables efficient, scalable BEM-level computations for large-scale translation-invariant problems.

Abstract

Boundary element methods for elliptic partial differential equations typically lead to boundary integral operators with translation-invariant kernel functions. Taking advantage of this property is fairly simple for particle methods, e.g., Nystrom-type discretizations, but more challenging if the supports of basis functions have to be taken into account. In this article, we present a modified construction for $\mathcal{H}^2$-matrices that uses translation-invariance to significantly reduce the storage requirements. Due to the uniformity of the boxes used for the construction, we need only a few uncomplicated assumptions to prove estimates for the resulting storage complexity.

$\mathcal{H}^2$-matrices for translation-invariant kernel functions

TL;DR

The paper addresses the storage bottleneck in boundary element method discretizations with translation-invariant kernels by proposing a translation-invariant modification of -matrices. It constructs a hierarchy of axis-aligned boxes with uniform levels and uses tensor Chebyshev interpolation; translation symmetry allows reuse of far-field coupling blocks and reduces memory movement. The main contributions are proving that far-field storage per level scales as , deriving explicit storage bounds for leaf, transfer, nearfield, and coupling components, and validating the approach with numerical experiments showing significant memory savings without sacrificing accuracy. This work enables efficient, scalable BEM-level computations for large-scale translation-invariant problems.

Abstract

Boundary element methods for elliptic partial differential equations typically lead to boundary integral operators with translation-invariant kernel functions. Taking advantage of this property is fairly simple for particle methods, e.g., Nystrom-type discretizations, but more challenging if the supports of basis functions have to be taken into account. In this article, we present a modified construction for -matrices that uses translation-invariance to significantly reduce the storage requirements. Due to the uniformity of the boxes used for the construction, we need only a few uncomplicated assumptions to prove estimates for the resulting storage complexity.
Paper Structure (4 sections, 11 theorems, 96 equations, 7 figures, 6 tables)

This paper contains 4 sections, 11 theorems, 96 equations, 7 figures, 6 tables.

Table of Contents

  1. Introduction
  2. H2-matrices for translation-invariant kernel functions
  3. Complexity estimates
  4. Numerical experiments

Key Result

Lemma 1

Storing the leaf matrices $(V_{t})_{t \in \mathscr{L}_{T_{I}}}$ and $(W_{s})_{s \in \mathscr{L}_{T_{J}}}$ requires not more than $2kn$ units of storage.

Figures (7)

  • Figure 1: Relation between the boxes of $T_{I}$ and their support bounding boxes.
  • Figure 2: The translations of the boxes of $T_{I}$ and the translations of their sons.
  • Figure 3: For $t \in \widehat{T}^{(6)}$ the ball $\mathsf{B}_{2}[\beta_{t}, \rho^{(6)}]$ contains all inadmissible $s \in \widehat{T}^{(6)}$.
  • Figure 4: Storage requirements of $\widetilde{G}$ on $\Gamma_{S}$ in MB.
  • Figure 5: Storage requirements of $\widetilde{K}$ on $\Gamma_{S}$ in MB.
  • ...and 2 more figures

Theorems & Definitions (13)

  • Definition 1: Tree notations
  • Lemma 1: Storage requirements of the leaf matrices
  • Lemma 2
  • Theorem 1: Storage requirements of the transfer matrices
  • Lemma 3
  • Lemma 4: Sparsity
  • Lemma 5: Difference bound for $T_{I \times J}$
  • Theorem 2: Storage requirements of the coupling matrices
  • Lemma 6: Sparsity of $T_{I \times J}$
  • Lemma 7: Storage requirements of the nearfield matrices
  • ...and 3 more