Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Limits of non-local approximations to the Eikonal equation on manifolds

Jalal M. Fadili, Nicolas Forcadel, Rita Zantout

TL;DR

This work studies a time-dependent non-local Eikonal equation on a compact Riemannian manifold and its convergence to the local form as the non-local kernel is scaled. Using viscosity solution theory, Perron’s method, and geometric tools on manifolds, the authors establish well-posedness and regularity for both the local and non-local problems, along with explicit error bounds between the two formulations in both continuous time and Forward Euler discretization. They then transfer these results to sequences of random geometric graphs, proving almost-sure uniform convergence of graph-based solutions to the local viscosity solution as the graph size grows and the kernel scale shrinks at a controlled rate. The results provide a rigorous continuum limit for non-local graph approximations of Eikonal-type dynamics on manifolds, with quantified rates depending on kernel regularity, graph sampling density, and discretization parameters. Practically, this enables robust, scalable PDE-based front propagation and distance computations on manifold-structured data and graphs, with guaranteed convergence guarantees as data resolution increases.

Abstract

In this paper, we consider a non-local approximation of the time-dependent Eikonal equation defined on a Riemannian manifold. We show that the local and the non-local problems are well-posed in the sense of viscosity solutions and we prove regularity properties of these solutions in time and space. If the kernel is properly scaled, we then derive error bounds between the solution to the non-local problem and the one to the local problem, both in continuous-time and Forward Euler discretization. Finally, we apply these results to a sequence of random weighted graphs with n vertices. In particular, we establish that the solution to the problem on graphs converges almost surely uniformly to the viscosity solution of the local problem as the kernel scale parameter decreases at an appropriate rate when the number of vertices grows and the time step vanishes.

Limits of non-local approximations to the Eikonal equation on manifolds

TL;DR

This work studies a time-dependent non-local Eikonal equation on a compact Riemannian manifold and its convergence to the local form as the non-local kernel is scaled. Using viscosity solution theory, Perron’s method, and geometric tools on manifolds, the authors establish well-posedness and regularity for both the local and non-local problems, along with explicit error bounds between the two formulations in both continuous time and Forward Euler discretization. They then transfer these results to sequences of random geometric graphs, proving almost-sure uniform convergence of graph-based solutions to the local viscosity solution as the graph size grows and the kernel scale shrinks at a controlled rate. The results provide a rigorous continuum limit for non-local graph approximations of Eikonal-type dynamics on manifolds, with quantified rates depending on kernel regularity, graph sampling density, and discretization parameters. Practically, this enables robust, scalable PDE-based front propagation and distance computations on manifold-structured data and graphs, with guaranteed convergence guarantees as data resolution increases.

Abstract

In this paper, we consider a non-local approximation of the time-dependent Eikonal equation defined on a Riemannian manifold. We show that the local and the non-local problems are well-posed in the sense of viscosity solutions and we prove regularity properties of these solutions in time and space. If the kernel is properly scaled, we then derive error bounds between the solution to the non-local problem and the one to the local problem, both in continuous-time and Forward Euler discretization. Finally, we apply these results to a sequence of random weighted graphs with n vertices. In particular, we establish that the solution to the problem on graphs converges almost surely uniformly to the viscosity solution of the local problem as the kernel scale parameter decreases at an appropriate rate when the number of vertices grows and the time step vanishes.
Paper Structure (19 sections, 18 theorems, 133 equations)

This paper contains 19 sections, 18 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Contributions
  4. Related work
  5. Outline
  6. Notations and prerequisites on Riemannian manifolds
  7. Preliminaries on Riemannian manifolds
  8. Properties of the Riemannian distance
  9. Other notations
  10. Well-posedness and regularity results
  11. Problem \ref{['chap4_eikonal-eq']}
  12. Problem \ref{['chap4_eikonal-eq-discrete']}
  13. Consistency and error bounds
  14. Continuous time non-local to local error bound
  15. Forward Euler discrete time non-local to local error bound
  16. ...and 4 more sections

Key Result

Proposition 2.12

Let $\mathcal{M}$ be a compact Riemannian manifold. Then there exists a constant $r>0$ such that for every $x \in \mathcal{M}$, the exponential map $\mathrm{Exp}_x$ is defined on $B(0_x,r) \subset T_x\mathcal{M}$ and provides a $C^\infty$ diffeomorphism Moreover, the distance function is given by and for every $x \in \mathcal{M}$, the distance map $y \in \mathcal{M} \mapsto d_\mathcal{M}(x,y)$ i

Theorems & Definitions (53)

  • Definition 2.1: $C^3$-smooth manifold
  • Definition 2.2: Riemannian metric and norm
  • Definition 2.3: Differentiable functions on $\mathcal{M}$
  • Definition 2.4: Differential and gradient of a differentiable function
  • Remark 2.5
  • Definition 2.6: Length of a curve
  • Remark 2.7
  • Definition 2.8: Riemannian distance
  • Definition 2.9: Exponential map
  • Definition 2.10: Cut locus of a point
  • ...and 43 more