Numerical computation of the cut locus <i>via</i> a variational approximation of the distance function

Générau, François; Oudet, Èdouard; Velichkov, Bozhidar

doi:10.1051/m2an/2021088

Cited by 5 publications

(4 citation statements)

References 17 publications

(13 reference statements)

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“…The difference in performance has to be attributed to the fact that the algorithm in [30] has a computational complexity that grows exponentially with the size of the triangulation, while our algorithm is affected by the classical polynomial computational complexity of FEM methods. Finally, we would like to note that the computational cost of our method is comparable if not better to the approach described in [27]. The use of implicit time-stepping in combination with Newton method as proposed in [17], which allows a drastic improvement in computational efficiency, is the next step in our future studies.…”

Section: Numerical Experimentsmentioning

confidence: 90%

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Computing the cut locus of a Riemannian manifoldviaoptimal transport

Facca

Berti

Fassò³

et al. 2022

ESAIM: M2AN

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In this paper, we give a new characterization of the cut locus of a point on a compact Riemannian manifold as the zero set of the optimal transport density solution of the Monge-Kantorovich equations, a PDE formulation of the optimal transport problem with cost equal to the geodesic distance. Combining this result with an optimal transport numerical solver, based on the so-called dynamical Monge-Kantorovich approach, we propose a novel framework for the numerical approximation of the cut locus of a point in a manifold. We show the applicability of the proposed method on a few examples settled on 2d-surfaces embedded in $R^{3}$, and discuss advantages and limitations.

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Section: Numerical Experimentsmentioning

confidence: 90%

“…To overcome these limitations, in [26] the authors propose a characterization of the cut locus as the limit in the Hausdorff sense of a variationally-defined thawed region around the cut locus. This allows the construction of a convergent finite-element-based numerical approximation of the cut locus which is described and analyzed in [27].…”

Section: Introductionmentioning

confidence: 99%

Computing the cut locus of a Riemannian manifoldviaoptimal transport

Facca

Berti

Fassò³

et al. 2022

ESAIM: M2AN

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“…Indeed, the method has been applied in [27] for the identification of the cut locus of a triangulated surface with respect of a point, i.e., the set of points where the distance function becomes non differentiable, or, equivalently, the minimizing geodesics are not unique. Moreover, our approach can be extended to the approximation of medial axes and Voronoi diagrams of general surfaces embedded in R 3 , quantities that are of great interest in computational geometry [34]. Finally, we would like to mention that this approach represents a new variationally based formulation for the solution of the Eikonal equation on triangulated surfaces and as such it is applicable for example to biological modeling [51,22] or travel time tomography of the Earth's surface [53].…”

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confidence: 99%

Numerical solution of the $ L^1 $-optimal transport problem on surfaces

Berti,

Facca,

Putti

2023

ACSE

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In this article we study the numerical solution of the L 1 -optimal transport problem on 2D surfaces embedded in R 3 by means of the dynamic Monge-Kantorovich(DMK) formulation. Our numerical approach uses the surface finite element method (SFEM) to approximate the elliptic partial differential equation (PDE) and related integrals involved in the surface DMK. The accuracy and efficiency of the proposed numerical scheme is tested against an exact solution on a 2D sphere. When using a mesh with edges aligned with the boundary of the support of the transported measures, the experimental results show errors that scale coherently with our choice of discretization spaces, achieving first order convergence for the calculation of the optimal transport solution. On the other hand, the errors in the computation of the Wasserstein-1 distance (an optimal transport-based distance for nonnegative measures) enjoy faster convergence, scaling almost cubically with h. However, if the mesh used is not adapted to the problem, this super-linear convergence is lost and the convergence toward the optimal transport solution reduces to O(h 0.6 ).We also experimentally compared our approach against a state of the art method based on the alternating direction method of multipliers (ADMM). The results obtained in the calculation of the Wasserstein-1 distance show that our proposed approach is more accurate and computationally more efficient than the ADMM-based method, independently on the mesh used.

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“…What is more, it can be used to show that, with the regularity theory for obstacle-type problems, u m is locally C 1,1 regular. For a broader study of problem (1.1) and its relation with the cut locus, we refer to [3].…”

mentioning

confidence: 99%