Limits and consistency of non-local and graph approximations to the Eikonal equation

Fadili, Jalal M.; Forcadel, Nicolas; Nguyen, Thi Tuyen; Zantout, Rita

doi:10.48550/arxiv.2105.01977

Cited by 3 publications

(6 citation statements)

References 37 publications

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“…Developments in analysis are also possible, as our formal PDE limits in the regular grid setting lead to questions of convergence together with existence and uniqueness theory. Another direction is to extend the theory to random graph constructions (as demonstrated in the numerical results), possibly by applying constructions found [18,13,17]. Investigation of properties of the path sets that lead to optimal travel times could also allow new interpretation of results.…”

Section: Discussionmentioning

confidence: 99%

“…Current analytical results include an investigation of viscosity solutions for Hamilton-Jacobi equations on networks [6] and an approximation scheme for an eikonal equation on a network [5], producing an approximation of shortest paths to the boundary. In addition, limits and consistency of non-local and graph approximations to the time-dependent (local) eikonal equation have been studied in [17].…”

Section: Introductionmentioning

confidence: 99%

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Models for information propagation on graphs

Dunbar¹,

Elliott²,

Kreußer³

2022

Preprint

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In this work we propose and unify classes of different models for information propagation over graphs. In a first class, propagation is modeled as a wave which emanates from a set of known nodes at an initial time, to all other unknown nodes at later times with an ordering determined by the time at which the information wave front reaches nodes. A second class of models is based on the notion of a travel time along paths between nodes. The time of information propagation from an initial known set of nodes to a node is defined as the minimum of a generalized travel time over subsets of all admissible paths. A final class is given by imposing a local equation of an eikonal form at each unknown node, with boundary conditions at the known nodes. The solution value of the local equation at a node is coupled the neighbouring nodes with smaller solution values. We provide precise formulations of the model classes in this graph setting, and prove equivalences between them. Motivated by the connection between first arrival time model and the eikonal equation in the continuum setting, we demonstrate that for graphs in the particular form of grids in Euclidean space mean field limits under grid refinement of certain graph models lead to Hamilton-Jacobi PDEs. For a specific parameter setting, we demonstrate that the solution on the grid approximates the Euclidean distance.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Models for information propagation on graphs

Dunbar¹,

Elliott²,

Kreußer³

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Finally, there has been a lot of recent activity proving a rigorous connection between graphbased variational methods on graphs and their continuum limits. While many of these works have focused on graph Laplacians and total variation energies and their associated statistical problems (see, e.g., [20,21,23,24]), a few recent works have also made steps in this direction in the context of eikonal equations [8,18]. Discussion of empirical approximations of the boundary of domains, an issue we encounter in Section 3 can also be found in [8], [53].…”

Section: Related Literaturementioning

confidence: 99%

“…All of these steps have been rigorously justified within the context of graph Laplacians, and we expect that similar rigorous justification would hold here. Analyses of this type have been carried out in [20], for the Laplace-Beltrami operator, and in [18], for a time-dependent eikonal equation.…”

Section: Difference Scheme On Point Cloudsmentioning

confidence: 99%

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Eikonal depth: an optimal control approach to statistical depths

Molina-Fructuoso¹,

Murray²

2022

Preprint

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Statistical depths provide a fundamental generalization of quantiles and medians to data in higher dimensions. This paper proposes a new type of globally defined statistical depth, based upon control theory and eikonal equations, which measures the smallest amount of probability density that has to be passed through in a path to points outside the support of the distribution: for example spatial infinity. This depth is easy to interpret and compute, expressively captures multi-modal behavior, and extends naturally to data that is non-Euclidean. We prove various properties of this depth, and provide discussion of computational considerations. In particular, we demonstrate that this notion of depth is robust under an aproximate isometrically constrained adversarial model, a property which is not enjoyed by the Tukey depth. Finally we give some illustrative examples in the context of two-dimensional mixture models and MNIST.

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Ratio convergence rates for Euclidean first-passage percolation: Applications to the graph infinity Laplacian

Bungert¹,

Calder²,

Tim³

2022

Preprint

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In this paper we prove the first quantitative convergence rates for the graph infinity Laplace equation for length scales at the connectivity threshold. In the graph-based semi-supervised learning community this equation is also known as Lipschitz learning. The graph infinity Laplace equation is characterized by the metric on the underlying space, and convergence rates follow from convergence rates for graph distances. At the connectivity threshold, this problem is related to Euclidean first passage percolation, which is concerned with the Euclidean distance function d h (x, y) on a homogeneous Poisson point process on R d , where admissible paths have step size at most h > 0. Using a suitable regularization of the distance function and subadditivity we prove that d hs (0, se1)/s → σ as s → ∞ almost surely where σ ≥ 1 is a dimensional constant and hs log(s) 1 d . A convergence rate is not available due to a lack of approximate superadditivity when hs → ∞. Instead, we prove convergence rates for the ratio d h (0,se 1 ) d h (0,2se 1 ) → 1 2 when h is frozen and does not depend on s. Combining this with the techniques that we developed in (Bungert, Calder, Roith, IMA Journal of Numerical Analysis, 2022), we show that this notion of ratio convergence is sufficient to establish uniform convergence rates for solutions of the graph infinity Laplace equation at percolation length scales.

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Limits and consistency of non-local and graph approximations to the Eikonal equation

Cited by 3 publications

References 37 publications

Models for information propagation on graphs

Models for information propagation on graphs

Eikonal depth: an optimal control approach to statistical depths

Ratio convergence rates for Euclidean first-passage percolation: Applications to the graph infinity Laplacian

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