Monotone discretizations of levelset convex geometric PDEs

We introduce a novel algorithm that converges to level set convex viscosity solutions of high-dimensional Hamilton–Jacobi equations. The algorithm is applicable to a broad class of curvature motion PDEs, as well as a recently developed Hamilton–Jacobi equation for the Tukey depth, which is a statist...

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Bibliographic Details
Published in:Numerische Mathematik Vol. 156; no. 6; pp. 1987 - 2029
Main Authors: Calder, Jeff, Lee, Wonjun
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2024
Springer Nature B.V
Subjects:
Algorithms
Approximation
Cloud point curves
Data points
Discretization
Hamilton-Jacobi equation
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Numerical Analysis
Numerical and Computational Physics
Partial differential equations
Simulation
Theoretical
Two dimensional analysis
Two dimensional flow
65N25: Numerical methods for eigenvalue problems for boundary value problems involving PDEs
35F21: Hamilton-Jacobi equations
35D40: Viscosity solutions to PDEs
ISSN:0029-599X, 0945-3245
Online Access:Get full text
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Summary:We introduce a novel algorithm that converges to level set convex viscosity solutions of high-dimensional Hamilton–Jacobi equations. The algorithm is applicable to a broad class of curvature motion PDEs, as well as a recently developed Hamilton–Jacobi equation for the Tukey depth, which is a statistical depth measure of data points. A main contribution of our work is a new monotone scheme for approximating the direction of the gradient, which allows for monotone discretizations of pure partial derivatives in the direction of, and orthogonal to, the gradient. We provide a convergence analysis of the algorithm on both regular Cartesian grids and unstructured point clouds in any dimension, and present numerical experiments that demonstrate the effectiveness of the algorithm in approximating solutions of the affine flow in two dimensions and the Tukey depth measure of high-dimensional datasets such as MNIST and FashionMNIST.
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ISSN:0029-599X
0945-3245
DOI:10.1007/s00211-024-01444-5