TY - JOUR

T1 - Approximating geodesics via random points

AU - Davis, Erik

AU - Sethuraman, Sunder

N1 - Publisher Copyright:
Copyright © 2017, The Authors. All rights reserved.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2017/11/18

Y1 - 2017/11/18

N2 - Given a 'cost' functional F on paths γ in a domain D ⊂ Rd, in the form F(γ) = R01 f(γ(t), γ' (t))dt, it is of interest to approximate its minimum cost and geodesic paths. Let X1,..., Xn be points drawn independently from D according to a distribution with a density. Form a random geometric graph on the points where Xi and Xj are connected when 0 < |Xi − Xj| < ε, and the length scale ε = εn vanishes at a suitable rate. For a general class of functionals F, associated to Finsler and other distances on D, using a probabilistic form of Gamma convergence, we show that the minimum costs and geodesic paths, with respect to types of approximating discrete 'cost' functionals, built from the random geometric graph, converge almost surely in various senses to those corresponding to the continuum cost F, as the number of sample points diverges. In particular, the geodesic path convergence shown appears to be among the first results of its kind.60D05, 58E10, 62-07, 49J55, 49J45, 53C22, 05C82

AB - Given a 'cost' functional F on paths γ in a domain D ⊂ Rd, in the form F(γ) = R01 f(γ(t), γ' (t))dt, it is of interest to approximate its minimum cost and geodesic paths. Let X1,..., Xn be points drawn independently from D according to a distribution with a density. Form a random geometric graph on the points where Xi and Xj are connected when 0 < |Xi − Xj| < ε, and the length scale ε = εn vanishes at a suitable rate. For a general class of functionals F, associated to Finsler and other distances on D, using a probabilistic form of Gamma convergence, we show that the minimum costs and geodesic paths, with respect to types of approximating discrete 'cost' functionals, built from the random geometric graph, converge almost surely in various senses to those corresponding to the continuum cost F, as the number of sample points diverges. In particular, the geodesic path convergence shown appears to be among the first results of its kind.60D05, 58E10, 62-07, 49J55, 49J45, 53C22, 05C82

KW - Consistency

KW - Distance

KW - Finsler

KW - Gamma convergence

KW - Geodesic

KW - Random geometric graph

KW - Scaling limit

KW - Shortest path

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M3 - Article

AN - SCOPUS:85092876095

JO - Nuclear Physics A

JF - Nuclear Physics A

SN - 0375-9474

ER -