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$\mathcal{H}$-matrix approximability of inverses of discretizations of the fractional Laplacian

Michael Karkulik, Jens Markus Melenk

TL;DR

The paper addresses the computational challenge of nonlocal fractional Laplacian discretizations by proving that the inverse stiffness matrix of a Galerkin discretization using piecewise linear elements on quasiuniform meshes admits a blockwise low-rank $\mathcal{H}$-matrix representation with exponential convergence in the block rank. It leverages the Caffarelli-Silvestre extension and a Beppo-Levi space framework to connect the fractional problem to a weighted elliptic PDE and its trace, enabling rigorous approximation results for $\mathbf{A}^{-1}$ and providing a practical path to fast solvers and preconditioners. The main contributions include a precise blockwise low-rank approximation theorem, a detailed Beppo-Levi analytical foundation, and numerical experiments in 2D validating exponential decay of the inverse approximation error with rank, suggesting significant computational savings for nonlocal PDEs. The findings have important implications for scalable solvers in nonlocal models and point to extensions to higher-degree discretizations and $\mathcal{H}^2$-matrix formats for even greater efficiency.

Abstract

The integral version of the fractional Laplacian on a bounded domain is discretized by a Galerkin approximation based on piecewise linear functions on a quasi-uniform mesh. We show that the inverse of the associated stiffness matrix can be approximated by blockwise low-rank matrices at an exponential rate in the block rank.

$\mathcal{H}$-matrix approximability of inverses of discretizations of the fractional Laplacian

TL;DR

The paper addresses the computational challenge of nonlocal fractional Laplacian discretizations by proving that the inverse stiffness matrix of a Galerkin discretization using piecewise linear elements on quasiuniform meshes admits a blockwise low-rank -matrix representation with exponential convergence in the block rank. It leverages the Caffarelli-Silvestre extension and a Beppo-Levi space framework to connect the fractional problem to a weighted elliptic PDE and its trace, enabling rigorous approximation results for and providing a practical path to fast solvers and preconditioners. The main contributions include a precise blockwise low-rank approximation theorem, a detailed Beppo-Levi analytical foundation, and numerical experiments in 2D validating exponential decay of the inverse approximation error with rank, suggesting significant computational savings for nonlocal PDEs. The findings have important implications for scalable solvers in nonlocal models and point to extensions to higher-degree discretizations and -matrix formats for even greater efficiency.

Abstract

The integral version of the fractional Laplacian on a bounded domain is discretized by a Galerkin approximation based on piecewise linear functions on a quasi-uniform mesh. We show that the inverse of the associated stiffness matrix can be approximated by blockwise low-rank matrices at an exponential rate in the block rank.

Paper Structure

This paper contains 11 sections, 15 theorems, 102 equations, 9 figures.

Table of Contents

  1. Introduction
  2. The fractional Laplacian and the Caffarelli-Silvestre extension
  3. Notation
  4. Main results
  5. Model problem and discretization
  6. Blockwise low-rank approximation
  7. The Beppo-Levi space $\mathcal{B}^1_\alpha(\mathbb{R}^{d+1}_+$)
  8. Technical details and proofs
  9. $\mathcal{H}$-matrix approximability
  10. Numerical experiments
  11. Conclusions and extensions

Key Result

Theorem 2.2

Let $\eta>0$ be a fixed admissibility parameter and $q\in(0,1)$. Let $(\tau,\sigma)$ be a cluster pair with $\eta$-admissible bounding boxes. Then, for each $k\in\mathbb{N}$, there exist matrices $\mathbf{X}_{\tau\sigma}\in\mathbb{R}^{\vert \tau \vert\times r}$ and $\mathbf{Y}_{\tau\sigma}\in\mathbb The constants $C_{\rm dim}$ and $C_{\rm apx}$ depend only on $d$, $\Omega$, the shape-regularity of

Figures (9)

  • Figure 1: Square domain, $s=0.25$, 2674 elements.
  • Figure 2: Square domain, $s=0.25$, 17130 elements.
  • Figure 3: Square domain, $s=0.5$, 2674 elements.
  • Figure 4: Square domain, $s=0.5$, 17130 elements.
  • Figure 5: Square domain, $s=0.75$, 2674 elements.
  • ...and 4 more figures

Theorems & Definitions (34)

  • Remark 2.1
  • Theorem 2.2
  • Definition 2.3: cluster tree
  • Definition 2.4: far field, near field, and sparsity constant
  • Theorem 2.5
  • proof
  • Remark 2.6
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • ...and 24 more