Numerical approximation of positive power curvature flow via deterministic games

TL;DR

This paper introduces a numerical scheme based on deterministic two-person games to approximate the level set solution for the positive power curvature flow of closed curves, demonstrating convergence to the viscosity solution.

Contribution

It develops a novel game-based numerical method for positive power curvature flow and proves its convergence to the level set solution.

Findings

Convergence of the value functions to the viscosity solution.

A proposed numerical scheme for calculating the value function.

Validation of the method through theoretical analysis.

Abstract

We approximate the level set solution for the motion of an embedded closed curve in the plane with normal speed $\max(0, \kappa)^{\ga}$ where $\kappa$ is the curvature of the curve and $\frac{1}{3}<\ga<1$ by the value functions of a family of deterministic two person games. We show convergence of the value functions to the viscosity solution of the level set equation and propose a numerical scheme for the calculation of the value function.