---
tag: KT0380H39M
kind: definition
name: Level-set method
msc: [65M06, 35F21, 65D18]
defines: 7KWR08FJET
uses: [0B2RZ0778N, 0ABK9V9Y64, 5TRXQKP7DY, 0MWV6ZTK6S, PA78NFEPCF, H1ERJ3T75T]
sources:
  - url: https://en.wikipedia.org/wiki/Level-set_method
---

## Definition

Let $\Gamma(t) \subset \R^d$, $t \ge 0$, be a family of closed hypersurfaces that bound open sets $\Omega(t)$.
Let $V(x, t)$ be a given speed of $\Gamma(t)$ in its outward normal direction.
The *level-set method* represents and evolves $\Gamma(t)$ as follows.

1. Representation: choose a continuous function $\phi \colon \R^d \times [0, \infty) \to \R$ with $\Omega(t) = \set{x : \phi(x, t) < 0}$ and $\Gamma(t) = \set{x : \phi(x, t) = 0}$, the zero [[level set|0B2RZ0778N]] of $\phi(\cdot, t)$.
2. Evolution: with $V$ extended near $\Gamma(t)$, $\phi$ solves the level-set equation
$$\phi_t + V \abs{\nabla \phi} = 0.$$
This is a [[Hamilton–Jacobi equation|5TRXQKP7DY]], understood in the [[viscosity sense|H1ERJ3T75T]] where $\phi$ is not smooth.
3. Discretization: the equation is solved on a fixed grid, usually with upwind finite differences. The interface $\Gamma(t)$ is read off as the zero set of the discrete $\phi$.

If the interface is carried by a velocity field $v(x, t)$, the evolution equation is $\phi_t + v \cdot \nabla \phi = 0$, which is the case $V = v \cdot n$.

## Remarks

Where $\nabla \phi \ne 0$, the outward unit normal is $n = \nabla \phi / \abs{\nabla \phi}$, with $\nabla$ the [[gradient|0ABK9V9Y64]].
The [[divergence|0MWV6ZTK6S]] $\kappa = \nabla \cdot n$ is the sum of the principal curvatures of $\Gamma$, a multiple of its [[mean curvature|PA78NFEPCF]], and it is used for motion by curvature with $V = -\kappa$.
One usually keeps $\phi$ close to a signed distance function, with $\abs{\nabla \phi} = 1$, by periodic reinitialization.

Example: in $\R^2$ let $V = 1$ and $\phi(x, t) = \abs{x} - (r_0 + t)$ with $r_0 > 0$.
For $x \ne 0$ one has $\phi_t = -1$ and $\abs{\nabla \phi} = 1$, so $\phi_t + V \abs{\nabla \phi} = 0$, and $\Gamma(t)$ is the circle of radius $r_0 + t$.

Non-example: $\phi(x, t) = \abs{x} - r_0$ represents a circle but fails clause 2 for $V = 1$, since $\phi_t + \abs{\nabla \phi} = 1 \ne 0$.
Its zero set does not move.