Hamiltonian Fast Marching: A Numerical Solver for Anisotropic and Non-Holonomic Eikonal PDEs

Mirebeau, Jean-Marie; Portegies, Jorg M.

doi:10.5201/ipol.2019.227

Cited by 38 publications

(48 citation statements)

References 58 publications

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“…This can be done for the isotropic case (where ξ ( x , ν ) is independent of ν ) using for example the supergradient marching algorithm of Benmansour et al, 20 which applies automatic differentiation to a fast‐marching algorithm 21 . One approach to the anisotropic case could be to generalise Benmansour et al 20 to anisotropic metrics, using an anisotropic fast‐marching algorithm such as those presented in other studies, 8,22,23 but we do not pursue this here.…”

Section: Derivation Of the Inverse Problemsmentioning

confidence: 99%

An inverse problem for Voronoi diagrams: A simplified model of non‐destructive testing with ultrasonic arrays

Bourne

Mulholland

Sahu

et al. 2020

Math Methods in App Sciences

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In this paper, we study the inverse problem of recovering the spatially varying material properties of a solid polycrystalline object from ultrasonic travel time measurements taken between pairs of points lying on the domain boundary. We consider a medium of constant density in which the orientation of the material's lattice structure varies in a piecewise constant manner, generating locally anisotropic regions in which the wave speed varies according to the incident wave direction and the material's known slowness curve. This particular problem is inspired by current challenges faced by the ultrasonic non‐destructive testing of polycrystalline solids. We model the geometry of the material using Voronoi tessellations and study two simplified inverse problems where we ignore wave refraction. In the first problem, the Voronoi geometry itself and the orientations associated to each region are unknowns. We solve this nonsmooth, nonconvex optimisation problem using a multistart non‐linear least squares method. Good reconstructions are achieved, but the method is shown to be sensitive to the addition of noise. The second problem considers the reconstruction of the orientations on a fixed square mesh. This is a smooth optimisation problem but with a much larger number of degrees of freedom. We prove that the orientations can be determined uniquely given enough boundary measurements and provide a numerical method that is more stable with respect to the addition of noise.

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Section: Derivation Of the Inverse Problemsmentioning

confidence: 99%

An inverse problem for Voronoi diagrams: A simplified model of non‐destructive testing with ultrasonic arrays

Bourne

Mulholland

Sahu

et al. 2020

Math Methods in App Sciences

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“…Computing geodesic distances in two and higher dimensional models in a robust manner is challenging due to complex forms that the geodesic equations may take. Recent advances in [ 21 , 42 ] have overcame some of these issues by focusing on solving for the distance function first from which minimal geodesic paths are extracted. Specifically, the author develops numerical schemes to solve Eikonal equations for the distance function of a Riemannian manifold numerically on uniform grids.…”

Section: Higher Dimensional Examplesmentioning

confidence: 99%

“…However, coordinate transformations to and from normal coordinates make this technique quite complicated. We will utilize simple robust methods for computing geodesic distances in model spaces of dimension two to four developed in [ 21 ]. Here, the authors construct a Hamiltonian Fast Marching technique that solves an Eikonal equation for the distance function of a Riemannian manifold.…”

Section: Introductionmentioning

confidence: 99%

Clustering Financial Return Distributions Using the Fisher Information Metric

Taylor

2019

Entropy

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Information geometry provides a correspondence between differential geometry and statistics through the Fisher information matrix. In particular, given two models from the same parametric family of distributions, one can define the distance between these models as the length of the geodesic connecting them in a Riemannian manifold whose metric is given by the model's Fisher information matrix. One limitation that has hindered the adoption of this similarity measure in practical applications is that the Fisher distance is typically difficult to compute in a robust manner. We review such complications and provide a general form for the distance function for one parameter model. We next focus on higher dimensional extreme value models including the generalized Pareto and generalized extreme value distributions that will be used in financial risk applications. Specifically, we first develop a technique to identify the nearest neighbors of a target security in the sense that their best fit model distributions have minimal Fisher distance to the target. Second, we develop a hierarchical clustering technique that utilizes the Fisher distance. Specifically, we compare generalized extreme value distributions fit to block maxima of a set of equity loss distributions and group together securities whose worst single day yearly loss distributions exhibit similarities.

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“…The directional derivatives are then approximated using upwind finite differences as in (2.4) below, and the coupled system of equations resulting from the discretized PDE is solved in a single pass over the domain [65]. It would be too long to describe here the approximation procedure [63,65] yielding (2.3), which involves a relaxation parameter ε > 0 and techniques from algorithmic geometry. Nevertheless let us mention the meta parameters (I, J, K) used: Reeds-Shepp (1, 3, 1), Euler-Mumford (0, 27, 1), Dubins (0, 6, 2).…”

Section: Globally Optimal Paths With a Curvature Penaltymentioning

confidence: 99%

Partial differential equations and variational methods for geometric processing of images

Buet¹,

Mirebeau²,

Gennip³

et al. 2019

The SMAI Journal of Computational Mathematics

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Hamiltonian Fast Marching: A Numerical Solver for Anisotropic and Non-Holonomic Eikonal PDEs

Cited by 38 publications

References 58 publications

An inverse problem for Voronoi diagrams: A simplified model of non‐destructive testing with ultrasonic arrays

An inverse problem for Voronoi diagrams: A simplified model of non‐destructive testing with ultrasonic arrays

Clustering Financial Return Distributions Using the Fisher Information Metric

Partial differential equations and variational methods for geometric processing of images

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