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Kernel Multi-Grid on Manifolds

Thomas Hangelbroek, Christian Rieger

TL;DR

This article applies the framework of the geometric multigrid method with a $\tau\ge 2$-cycle to scattered, quasi-uniform point clouds on the surface and shows that the resulting linear algebra can be accelerated by using the Lagrange function decay, with convergence rates which are obtained by a rigorous analysis.

Abstract

Kernel methods for solving partial differential equations on surfaces have the advantage that those methods work intrinsically on the surface and yield high approximation rates if the solution to the partial differential equation is smooth enough. Localized Lagrange bases have proven to alleviate the computational complexity of usual kernel methods to some extent, although the efficient numerical solution of the ill-conditioned linear systems of equations arising from kernel-based Galerkin solutions to PDEs has not been addressed in the literature so far. In this article we apply the framework of the geometric multigrid method with a $τ\ge 2$-cycle to scattered, quasi-uniform point clouds on the surface. We show that the resulting linear algebra can be accelerated by using the Lagrange function decay, with convergence rates which are obtained by a rigorous analysis. In particular, we can show that the computational cost to solve the linear system scales log-linear in the degrees of freedom.

Kernel Multi-Grid on Manifolds

TL;DR

This article applies the framework of the geometric multigrid method with a -cycle to scattered, quasi-uniform point clouds on the surface and shows that the resulting linear algebra can be accelerated by using the Lagrange function decay, with convergence rates which are obtained by a rigorous analysis.

Abstract

Kernel methods for solving partial differential equations on surfaces have the advantage that those methods work intrinsically on the surface and yield high approximation rates if the solution to the partial differential equation is smooth enough. Localized Lagrange bases have proven to alleviate the computational complexity of usual kernel methods to some extent, although the efficient numerical solution of the ill-conditioned linear systems of equations arising from kernel-based Galerkin solutions to PDEs has not been addressed in the literature so far. In this article we apply the framework of the geometric multigrid method with a -cycle to scattered, quasi-uniform point clouds on the surface. We show that the resulting linear algebra can be accelerated by using the Lagrange function decay, with convergence rates which are obtained by a rigorous analysis. In particular, we can show that the computational cost to solve the linear system scales log-linear in the degrees of freedom.
Paper Structure (27 sections, 21 theorems, 147 equations, 2 algorithms)

This paper contains 27 sections, 21 theorems, 147 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Outline of the paper
  3. Problem Set Up
  4. Manifold
  5. Sobolev spaces
  6. Elliptic operator
  7. Galerkin methods
  8. Kernel approximation
  9. The Lagrange basis and stiffness matrix
  10. The stiffness matrix
  11. The diagonal of the stiffness matrix
  12. The smoothing property
  13. $L_2$ stability of the smoothing operator
  14. Smoothing properties
  15. The direct kernel multigrid method
  16. ...and 12 more sections

Key Result

Lemma 1

Suppose the family $(V_h)$ has the property that for all $\tilde{u}\in W_2^2(\mathbb{M})$, the distance in $W_2^1(\mathbb{M})$ from $V_h$ satisfies $\mathrm{dist}_{\|\cdot\|_{W^{1}_{2}(\mathbb{M})}} \left(\tilde{u},V_h \right) \le C h \left\| \tilde{u} \right\|_{W^{2}_{2}(\mathbb{M})}$. Then for

Theorems & Definitions (45)

  • Remark 1
  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • ...and 35 more