Paper · Oct 10, 2024

Median filter method for mean curvature flow using a random Jacobi algorithm

Anton Ullrich, Tim Laux

math.APWider flowsarXiv:2410.07776
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Abstract

We present an efficient scheme for level set mean curvature flow using a domain discretization and median filters. For this scheme, we show convergence in -norm under mild assumptions on the number of points in the discretization. In addition, we strengthen the weak convergence result for the MBO thresholding scheme applied to data clustering of Lelmi and one of the authors. This is done through a strong convergence of the discretized heat flow in the optimal regime. Different boundary conditions are also discussed.

Comments: 56 pages, 7 figures

Mean curvature flowBoundaries & non-compactPhase fields & multiphase flows

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The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper A new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness (Sebastian Hensel and Tim Laux, 2021)2Hensel 2021 BV solutions for mean curvature flow with constant contact angle: Allen-Cahn approximation and weak-strong uniqueness (Sebastian Hensel and Tim Laux, 2021)6Hensel 2021 Strong convergence of the thresholding scheme for the mean curvature flow of mean convex sets (Jakob Fuchs and Tim Laux, 2022)1Fuchs 2022 Generic level sets in mean curvature flow are BV solutions (Anton Ullrich and Tim Laux, 2023)4Ullrich 2023 Quantitative convergence of the nonlocal Allen–Cahn equation to volume-preserving mean curvature flow (Milan Kroemer and Tim Laux, 2023)5Kroemer 2023 Weighted, Multiphase, Volume-Preserving Mean Curvature Flow as Limit of the MBO Scheme on Manifolds (Fabius Krämer, 2025)8Krämer 2025 Equivalence of weak solution concepts for mean curvature flow (Tim Laux and Anton Ullrich, 2025)7Laux 2025 Convergence of a vector-valued Allen-Cahn system to Brakke's multiphase mean curvature flow (Tim Laux and Keisuke Takasao, 2026)3Laux 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Strong convergence of the thresholding scheme for the mean curvature flow of mean convex sets

    Jakob Fuchs and Tim LauxWider flows PDF

    Topics Phase fields & multiphase flows · Same author Tim Laux · Shared terms “thresholding scheme”, “strong convergence”, “discretization”

  2. A new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness

    Sebastian Hensel and Tim LauxWider flows PDF

    Topics Phase fields & multiphase flows · Same author Tim Laux

  3. Convergence of a vector-valued Allen-Cahn system to Brakke's multiphase mean curvature flow

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    Topics Phase fields & multiphase flows · Same author Tim Laux

  4. Generic level sets in mean curvature flow are BV solutions

    Anton Ullrich and Tim LauxWider flows PDF

    Topics Mean curvature flow · Same authors Anton Ullrich, Tim Laux · Shared terms “level set”, “addition”

  5. Quantitative convergence of the nonlocal Allen–Cahn equation to volume-preserving mean curvature flow

    Milan Kroemer and Tim LauxWider flows PDF

    Topics Phase fields & multiphase flows · Same author Tim Laux

  6. BV solutions for mean curvature flow with constant contact angle: Allen-Cahn approximation and weak-strong uniqueness

    Sebastian Hensel and Tim LauxWider flows PDF

    Topics Phase fields & multiphase flows · Same author Tim Laux

  7. Equivalence of weak solution concepts for mean curvature flow

    Tim Laux and Anton UllrichWider flows PDF

    Topics Mean curvature flow · Same authors Anton Ullrich, Tim Laux · Shared terms “level set”

  8. Weighted, Multiphase, Volume-Preserving Mean Curvature Flow as Limit of the MBO Scheme on Manifolds

    Fabius KrämerWider flows PDF

    Topics Phase fields & multiphase flows · Shared terms “thresholding scheme”, “MBO”, “discretization”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989) Phase fields & multiphase flows Interfaces moving by curvature can be approximated by smooth phase fields: the Allen–Cahn equation's transition layers converge to mean curvature… Start with: Two-Dimensional Motion of Idealized Grain Boundaries (Mullins, 1956)

In the guidebook

  1. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.