Math 2.0

Numerical analysis Definition

Level-set method

Builds on Level setGradientHamilton–Jacobi equationDivergenceMean curvatureViscosity solution

Definition

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

  1. Representation: choose a continuous function ϕ ⁣:ℝd×[0,∞)→ℝ\phi \colon \R^d \times [0, \infty) \to \R with Ω(t)={x:ϕ(x,t)<0}\Omega(t) = \set{x : \phi(x, t) < 0} and Γ(t)={x:ϕ(x,t)=0}\Gamma(t) = \set{x : \phi(x, t) = 0}, the zero level set of ϕ(⋅,t)\phi(\cdot, t).
  2. Evolution: with VV extended near Γ(t)\Gamma(t), ϕ\phi solves the level-set equation ϕt+V∣∇ϕ∣=0.\phi_t + V \abs{\nabla \phi} = 0. This is a Hamilton–Jacobi equation, understood in the viscosity sense where ϕ\phi is not smooth.
  3. Discretization: the equation is solved on a fixed grid, usually with upwind finite differences. The interface Γ(t)\Gamma(t) is read off as the zero set of the discrete ϕ\phi.

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

Remarks

Where ∇ϕ≠0\nabla \phi \ne 0, the outward unit normal is n=∇ϕ/∣∇ϕ∣n = \nabla \phi / \abs{\nabla \phi}, with ∇\nabla the gradient. The divergence κ=∇⋅n\kappa = \nabla \cdot n is the sum of the principal curvatures of Γ\Gamma, a multiple of its mean curvature, and it is used for motion by curvature with V=−κV = -\kappa. One usually keeps ϕ\phi close to a signed distance function, with ∣∇ϕ∣=1\abs{\nabla \phi} = 1, by periodic reinitialization.

Example: in ℝ2\R^2 let V=1V = 1 and ϕ(x,t)=∣x∣−(r0+t)\phi(x, t) = \abs{x} - (r_0 + t) with r0>0r_0 > 0. For x≠0x \ne 0 one has ϕt=−1\phi_t = -1 and ∣∇ϕ∣=1\abs{\nabla \phi} = 1, so ϕt+V∣∇ϕ∣=0\phi_t + V \abs{\nabla \phi} = 0, and Γ(t)\Gamma(t) is the circle of radius r0+tr_0 + t.

Non-example: ϕ(x,t)=∣x∣−r0\phi(x, t) = \abs{x} - r_0 represents a circle but fails clause 2 for V=1V = 1, since ϕt+∣∇ϕ∣=1≠0\phi_t + \abs{\nabla \phi} = 1 \ne 0. Its zero set does not move.

Reading path

To understand this page from the ground up, read these 15 pages in order. The foundations come first.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptHamilton–Jacobi equation, concept, no evidence levelHamilton–Jacobi equat…conceptMean curvature, concept, no evidence levelMean curvatureconceptViscosity solution, concept, no evidence levelViscosity solutionconceptTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptLevel set, concept, no evidence levelLevel setconceptPartial derivative, concept, no evidence levelPartial derivativeconceptVector field, concept, no evidence levelVector fieldconceptGradient, concept, no evidence levelGradientconceptDivergence, concept, no evidence levelDivergenceconceptLevel-set method, definition, no evidence levelLevel-set methoddefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Euclidean space object
  2. Function concept
  3. Hamilton–Jacobi equation concept
  4. Mean curvature concept
  5. Viscosity solution concept
  6. Total derivative concept
  7. Function of several variables concept
  8. Vector-valued function concept
  9. Differentiable function of several variables concept
  10. Level set concept
  11. Partial derivative concept
  12. Vector field concept
  13. Gradient concept
  14. Divergence concept
  15. Level-set method definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Level-set_method
About this page
TagKT0380H39M
URIm2://KT0380H39M
Kinddefinition
WritingW0 stub
MSC65M06, 35F21, 65D18
DefinesLevel-set method
UsesLevel set, Gradient, Hamilton–Jacobi equation, Divergence, Mean curvature, Viscosity solution

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